Noncommutative Lattices Skew Lattices, Skew Boolean Algebras and Beyond 2 3 5 7 11 13 17 19 23 29 31 37 41 89 97 101 103 107 109 113 127 131 137 139 149 151 157 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 Jonathan E. Leech famnit lectures ■ famnitova predavanja ■ 4 Jonathan Leech graduated from the University of Hawaii and earned a PhD at the University of California, Los Angeles. He has taught mathematics at the University of Tennessee, later at Missouri Western State University and finally at Westmont College in Santa Barbara, California. He was a Visiting Professor at Case Western Reserve University, the Universidad de Granada in Spain and Universidade Mackenzie in Brazil. He has been a scholar in residence at both the University of Sidney and the University of Tasmania in Australia. Throughout his academic career Professor Leech has studied algebraic structures related to semigroups, with much of his emphasis being on the theory of noncommutative lattices and of skew lattices in particular. He laid the foundations of the modern theory of noncommutative lattices in a number of (co)authored seminal publications. His work has inspired many mathematicians around the world to pursue research in this area. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Famnit Lectures │ Famnitova predavanja │ 4 Noncommutative Lattices Skew Lattices, Skew Boolean Algebras and Beyond Jonathan E. Leech Jonathan E. Leech Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Famnit Lectures │ Famnitova predavanja │ 4 │ ISSN 2335-3708 Reviewers and Editors Karin Cvetko - Vah Michael Kinyon Tomaž Pisanski Technical Support Nino Bašić Published by Založba Univerze na Primorskem/University of Primorska Press For publisher: Prof. Klavdija Kutnar, PhD, rector Titov trg 4, SI-6000 Koper Editor-in-chief Jonatan Vinkler Managing editor Alen Ježovnik Koper, 2020 ISBN 978-961-293-027-1 (pdf) http://www.hippocampus.si/ISBN/978-961-293-027-1.pdf ISBN 978-961-293-028-8 (html) http://www.hippocampus.si/ISBN/978-961-293-028-8/index.html DOI: https://doi.org/10.26493/978-961-293-027-1 © 2020 Založba Univerze na Primorskem/University of Primorska Press Kataložni zapis o publikaciji (CIP) pripravili v Narodni in univerzitetni knjižnici v Ljubljani COBISS.SI-ID=35005187 ISBN 978-961-293-027-1 (pdf) ISBN 978-961-293-028-8 (html) Table of Contents I: PRELIMINARIES ................................................................................................................ 11 1.1 Lattice ........................................................................................................................................ 13 1.2 Bands ......................................................................................................................................... 20 1.3 Noncommutative lattices – initial observations ........................................................... 28 References ........................................................................................................................................ 39 II: SKEW LATTICES .............................................................................................................. 41 2.1 Fundamental results .............................................................................................................. 43 2.2 Instances of commutative behavior .................................................................................. 47 2.3 Normal skew lattices .............................................................................................................. 53 2.4 Primitive skew lattices and skew lattice structure ..................................................... 59 2.5 Partial skew lattices and coset projections .................................................................... 71 2.6 Decompositions of normal, symmetric skew lattices ................................................. 76 References ........................................................................................................................................ 86 III: QUASILATTICES, PARALATTICES & THEIR CONGRUENCES ........................... 87 3.1 Congruences on quasilattices .............................................................................................. 89 3.2 Antilattices that are simple as algebras .......................................................................... 94 3.3 Regular quasilattices ............................................................................................................. 96 3.4 Paralattices and refined quasilattices ............................................................................. 98 3.5 The effects of the distributive identities ....................................................................... 105 3.6 Deriving simple antilattices from magic squares ....................................................... 108 References ...................................................................................................................................... 117 IV: SKEW BOOLEAN ALGEBRAS .................................................................................... 119 4.1 Skew Boolean algebras ........................................................................................................ 121 4.2 Finiteness, orthosums and free algebras ...................................................................... 126 4.3 Connections with strongly distributive skew lattices .............................................. 134 4.4 Skew Boolean algebras with intersections ................................................................... 138 4.5 Omega algebras and skew Boolean covers ................................................................... 152 References ...................................................................................................................................... 161 5 V: FURTHER TOPICS IN SKEW LATTICES ................................................................. 163 5.1 Symmetric skew lattices ..................................................................................................... 166 5.2 Distributive identities in the symmetric case ............................................................. 169 5.3 Cancellation in skew lattices ............................................................................................. 171 5.4 Categorical skew lattices .................................................................................................... 178 5.5 Distributive skew lattices ................................................................................................... 187 5.6 Midpoints and distributive skew chains ....................................................................... 195 5.7 Counting theorems and cancellative skew lattices .................................................... 200 References ...................................................................................................................................... 206 VI: SKEW LATTICES IN RINGS ........................................................................................ 207 6.1 Quadratic skew lattices in rings ....................................................................................... 209 6.2 ∇-bands and cubic skew lattices ...................................................................................... 219 6.3 The question of s-closure ................................................................................................. 230 6.4 Idempotent-closed rings ..................................................................................................... 233 6.5 Decomposing E( R) and R ..................................................................................................... 243 6.6 Idempotent-closed rings of matrices .............................................................................. 251 References ...................................................................................................................................... 259 VII: FURTHER TOPICS IN SKEW BOOLEAN ALGEBRAS .......................................... 261 7.1 Differences, discriminators and connections with other algebras ...................... 262 References ...................................................................................................................................... 272 Bibliography ......................................................................................................................... 275 Addendum 2020 ................................................................................................................. 281 6 Foreword The last 30 years have witnessed sustained research by a number of individuals in skew lattices, a class of noncommutative generalizations of lattices. (A partial list of published work is given in the Bibliography at the end of this monograph.) Papers on noncommutative lattices in general have appeared since the late 1940s. It would not be unfair to say that the more recent research has been both deeper and more fruitful than the earlier work, for several reasons. To begin, by restricting attention to a particular class of algebras, one is more focused. Indeed, once commutativity is dropped, the possibilities for differing absorption identities that reduce to the familiar identities in the commutative case becomes quite large. Thus, by working within the boundaries of a fixed set of identities, one becomes more concentrated in one’s efforts. Secondly, in more recent times advantage has been made of results in semigroup theory about bands, that is, semigroups consisting entirely of idempotents. Indeed the newer research began by studying multiplicative bands of idempotents in rings, and realizing that under certain conditions such bands would also be closed under an “upward multiplication” to yield a skew lattice. Parallel to this was an expanding role of universal algebra, both due to results of a fairly general scope (basic universal algebra) and also results related to structures that were weakened or modified forms of Boolean algebras. This was especially important in the study of skew Boolean algebras. Summing up: there has been a greater awareness of relevant information. Thirdly, as indicated above, the newer research began with a rich source of motivating examples – bands of idempotents in rings. In particular for bands that were left regular ( xyx = xy), any maximal such band in a ring was also closed under the circle operation x¡ y = x + y – xy. And any band closed under both operations satisfied certain absorption identities, e.g., e( e¡ f) = e = e¡( ef). These observations, along with others related to normal bands of idempotents (that were middle commutative: xyzw = xzyw) indicated the presence of structurally enhanced bands with a roughly lattice-like structure. Thus skew lattices arose, along with a number of potential properties first observed in the setting of rings. To this was added a second class of motivating examples, algebras of partial functions P(A, B) between pairs of sets, A and B. These provided examples of skew Boolean algebras and related structures, much as “partial sets” (that is, subsets) led to basic examples of Boolean algebras and distributive lattices. In addition, there was the effect of computer technology, especially beginning in the late 90s. The internet provided a quick, efficient means of communication, making it easier for like- minded individuals to connect. And computer software made it easier to find examples and initial proofs for some theorems. This continued to impact the development of skew lattice theory in the 21st century. And finally and most fortunately, skew lattices have attracted the attention of a number very fine mathematicians from around the world, including (to my current awareness) individuals from America, Australia, China, Ethiopia, Europe (including Great Britain J), India and Iran. This present volume is an organized presentation of much that has been published on the subject up through 2017. It is divided into seven chapters, the first four of which form the core of this survey. 7 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond The first chapter begins with a review of basic information about lattices and universal algebra. This is followed by a review of results about bands. The reader already familiar with these areas can move to the third section. (Caveat: it is imperative that one be grounded in the basic theory of bands, and especially the theory of regular bands, to be comfortable reading this monograph.) The third and final section provides introductory definitions and a few general results about noncommutative lattices. The latter are always given as at algebras (S; ∧, ∨) where ∧ and ∨ are associative, idempotent binary operations that jointly satisfy a set of absorption laws. Two varieties in particular are introduced: the variety of quasilattices and the variety of paralattices. Their intersection, the variety of refined quasilattices, contains the variety of skew lattices, our main topic. Placing skew lattices in a larger context provides a better sense of their place within the pantheon of generalizations of lattices. In Chapter 2, Skew Lattices, we proceed to study skew lattices in earnest, starting with a selection of basic concepts and results in the first two sections. These include two decomposition theorems (Theorems 2.1.2 and 2.1.5.) and initial results about skew lattices of idempotents in rings. In Section 2.3 we study the important class of normal skew lattices for which x∧y∧z∧w = x∧z∧y∧w. (If the idempotents of a ring are closed under multiplication, then they form a normal skew lattice; indeed they form a skew Boolean algebra.) The remaining sections provide a deeper general analysis of the structure of a skew lattice. Sections 2.1 – 2.3 of this chapter are all that is required to read the remaining chapters, except for Sections 5.3 – 5.6. Chapter 3 is entitled Quasilattices, Paralattices and Their Congruences. Sections 3.1 – 3.3 study congruences on quasilattices, thus obtaining results applicable to skew lattices. Section 4 looks at paralattices and especially refined quasilattices. We show the latter to be roughly skew lattices in the sense given in Theorem 3.4.14. Section 5 discusses the effects of distributivity, the main results being Theorems 3.5.1 and 3.5.2. Section 6 is recreational. Skew Boolean algebras are studied in Chapter 4. Skew Boolean algebras are algebras (S; ∧, ∨, \. 0) of type (2, 2, 2, 0) that look and behave in many ways like Boolean algebras, except that they need not be commutative. Boolean algebras decompose at will, and so do these algebras. This leads to a crisp description of the finitely generated (and thus finite) algebras, and of finite free algebras in particular. The final two sections are about skew Boolean algebras for which the natural partial order ≥ has a meet called the intersection and denoted by ∩. Many skew Boolean algebras have intersections, e.g., all free algebras do. For partial function algebras, ∩ is the standard set-theoretic intersection of the involved partial functions. As noted above, the core of this monograph is these four chapters. More specialized topics are studied in the last three chapters. Chapter 5 is entitled Further Topics in Skew Lattices, Chapter 6 is entitled Skew Lattices in Rings and the final chapter is entitled Further Topics in Skew Boolean Algebras. As the reader will see, skew Boolean algebras understandably get a good bit of attention in this monograph. There is other research on these algebras that is not in this monograph. It is typically of more recent vintage, with much being quite good. Hopefully, before too long, some motivated individual or group will produce a monograph devoted to skew Boolean algebras and related topics. 8 Foreword All chapters begin with a fairly detailed introduction and conclude with a list of relevant references, sometimes preceded by historical remarks. Following Chapter 7 is a Bibliography of publications that to my knowledge are either on or are closely related to skew lattices, appearing up through 2017, plus a few beyond. Although some material is not included in this monograph due to limitations on subject matter, this monograph should give the reader a good idea of the extent of activity within the area. Following the bibliography there is a brief Addendum intended to give the reader a sense of ongoing further research of relevance from the last two years involving newer topics not covered in this monograph. 9 I: PRELIMINARIES Noncommutative variants of lattices have been studied for over sixty-five years. The first person, to our knowledge, to engage in their extended study was Pascual Jordan who published numerous articles over a span of seventeen years. Since then papers on this subject have been written by various authors from a variety of perspectives. Why study noncommutative lattices? One reason comes from an interest in axioms. Clearly many important algebraic structures are characterized by axioms expressed as algebraic identities. In particular, lattices are defined as algebras ( L; ∨, ∧) where ∨ and ∧ are binary operations on a set L satisfying the following pairs of associative, absorption and commutative identities. a ∧ ( b ∧ c) = ( a ∧ b) ∧ c. a ∨ ( b ∨ c) = ( a ∨ b) ∨ c. a ∧ ( a ∨ b) = a. a ∨ ( a ∨ b) = a. a ∧ b = b ∧ a. a ∨ b = b ∨ a. An initial result in lattice theory is that the idempotent identities ( a ∧ a = a = a ∨ a) follow from the two absorption identities above without recourse to either the associative or commutative identities. If one deletes both commutative identities, then the four remaining identities are satisfied by genuinely noncommutative structures. (Consider, e.g., any set A of size greater than 1. Define ∨ and ∧ on A by setting a ∨ b = a = a ∧ b.) On the other hand, if one deletes the commutative laws, and combines instead the associative identities with a modified and expanded set of absorption identities a ∧ ( a ∨ b) = a, ( b ∨ a) ∧ a = a, a ∨ ( b ∧ a) = a, ( a ∧ b) ∨ a = a, then these identities also characterize lattices. (See Theorem 1.3.2 below.) The point is that axiomatic studies of lattices opened the door to considering noncommutative variants of lattices. Indeed, during the period when Jordan studied noncommutative lattices, others were studying axiomatic issues of lattices with an awareness of noncommutative possibilities. Thus an interest in axioms combined with a curiosity about possible noncommutative variations of lattices virtually insured that such variants would appear and then studied to some degree. 11 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond A second source of motivation arises from studying the multiplicative semigroups of rings. In the study of rings, idempotents play an important role. In general, given idempotents e and f in a ring, their product ef need not be idempotent (unless, e.g., the ring is commutative). Nonetheless ef is “below and to the right” of e in that e( ef) = ef, and at the same time “below and to the left” of f in that ( ef) f = ef. Dually, e is “below and to the left” of the circle product e¡ f = e + f – ef in that e( e¡ f) = e, while f is “below and to the right” of e¡ f in that ( e¡ f) f = f. On thus has the following picture: e¡ f   e f   ef. In general, given an element x in a ring, x 2 = x iff x¡ x = x. What is more, for a set of idempotents in a ring that is closed under both multiplication and ¡, the following four absorption identities are satisfied. a( a ¡ b) = a = ( b ¡ a) a. a ¡ ( ab) = a = ( ba) ¡ a. Noncommutative rings that are well endowed with idempotents are rich in such examples. What can one say about their structure? Such ring-based structures will occupy our attention in much of the second and sixth chapters to follow. Abstracting only slightly, one can think of bands – semigroups of idempotents – that are rich enough in structure to possess an idempotent counter-multiplication. Thus multiplication produces products that are generally “further down” in the band, while the counter-products would be generally “further up” in the band. Such bands exist. What can be said about them? A third source of motivation comes from universal algebra, especially the study of what may be loosely termed “generalized Boolean phenomena”. Do noncommutative generalizations of (generalized) Boolean lattices and algebras exist? If so, what connections exist between them and other structures related to Boolean algebras? Clearly noncommutative lattice theory have something to say about all this? Questions such as these will occupy our attention in the fourth and seventh chapters to follow. In the meanwhile, we begin this introductory chapter by reviewing a number of concepts about lattices and universal algebra in the first section. In the second section we recall various facts about bands that are pertinent to the rest of the monograph. And then in Section 3 we discuss some “first principals” of noncommutative lattices. All the material in this chapter is foundational to what follows later. The reader well-versed in the material in either of the first two sections can easily skip over one or both of them and then proceed to the third section. We emphasize, however, that a firm grasp of regular bands, and their left and right-sided cases, is crucial to understanding much that will be said about skew lattices. Left regular bands have received increased attention recently due to their role in combinatorial aspects of algebra and geometry. See, e.g., the introductory remarks in the 12 I: Preliminaries monograph, Cell complexes, poset tolopogy and the representation theory of algebras arising in algebraic combinatorics and discrete geometry by Stuart Margolis, Franco Saliola and Benjamin Steinberg. 1.1 Lattices Recall that a partially ordered set or poset is any pair ( L; ≥) where L is a set and ≥ is a partial ordering of L, that is, reflexive, anti-symmetric and transitive relation on L. Given x ≥ y in L, we think of x as lying above y, or equally of y lying below x. Given elements x, y in a poset ( L; ≥), an element m ∈ L such that (1) x ≥ m and y ≥ m and (2) m lies above all other elements lying jointly below x and y is called the meet of x and y and is denoted by x∧ y. Dually, an element j ∈ L such that (3) j ≥ x, j ≥ y and (4) j lies below all other elements lying jointly above x and y is called the join of x and y and is denoted by x∨ y. x ∨ y   x y   x ∧ y When they exist, x∧ y and x∨ y are unique with respect to the given x and y. If all pairs x, y ∈ S have a meet and a join, then ( L; ≥) is a lattice. In this case ( L; ∨, ∧) satisfies the following idempotent, commutative, associative and absorption identities: L0. x∧ x = x = x∨ x. L1. x∧ y = y∧ x and x∨ y = y∨ x. L2. ( x∧ y)∧ z = x∧( y∧ z) and ( x∨ y)∨ z = x∨( y∨ z). L3. x∧( x∨ y) = x = x∨( x∧ y). Conversely, given ( L; ∨, ∧) with binary operations ∧ and ∨ satisfying L0 – L3, a partial order ≥ is defined on L by x ≥ y if and only if x ∧ y = y, or equivalently, x ∨ y = y. As a poset, ( L; ≥) is a lattice whose meets and joins are precisely the given ∧ and ∨. Indeed the process of passing from a lattice poset ( L; ≥) to an algebra ( L; ∨, ∧) and the reverse process of passing from an algebra ( L; ∨, ∧) satisfying L0-L3 to a lattice poset are reciprocal processes. Thus lattices may be viewed from either a poset perspective or an algebraic perspective. While not verifying all details, we offer the following remarks. To begin, in any lattice poset, L0 and L1 are clear. L2 refers to the unique elements x∧ y∧ z lying maximally below x, y and z and x∨ y∨ z lying minimally above x, y and z. Likewise, the absorption identities in L3 refer to the fact that x∨ y ≥ x ≥ x∧ y in ( L; ≥). Conversely, given an algebra ( L; ∨, ∧) satisfying L0 – L3, the derived relation ≥ is reflexive by L0 and anti-symmetric thanks to L1. L2 is instrumental in 13 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond showing that ≥ is transitive. L3 yields the basic duality x ∧ y = y if and only if x ∨ y = y. ( L; ≥) is indeed a poset. Its meets and joins are given by ∧ and ∨. As stated above, L0 is redundant relative to L1-L3. Indeed: Lemma 1.1.1. Given binary operations ∧ and ∨ on a set L, L3 implie s L0. Proof. Given L3, x ∧ x = x ∧ [ x ∨ ( x∧ x)] = x, and thus x ∨ x = x ∨ ( x ∧ x) = x. £ Lattices of small order are easily drawn. A list of all lattices up through order 5 that is complete up to isomorphism follows. The indexing on the totally ordered chains ( C 0, C 1, etc.) corresponds to their length, which is always 1 less than their order. • • ↓ • • ↓ • ↙ ↓ C • ↓ • ↓ × • • 0 C 1 C C C C • 2 ↓ 3 1 1 • ↓ ↙ • ↓ • • • • • • ↓ ↓ ↙ ↓ • ↙ ↓ •↓ • • •  ↓  • • ↙ ↓ ↓ ↙ • • • ↓ • C 4 ↓ • • •  ↓  • • ↓ ↙ ↓ • ↘ ↓ ↓ • • • • (𝐶 M 5 !×𝐶!)! (𝐶!×𝐶!)! 𝑁! Recall that a lattice ( L; ∨, ∧) is distributive when for all x, y, z ∈ L the following identities hold: D1. x ∧ ( y ∨ z) = ( x ∧ y) ∨ ( x ∧ z). D2. x ∨ ( y ∧ z) = ( x ∨ y) ∧ ( x ∨ z). All chains are distributive lattices as are C I 0 1 × C1, C1 × C1 and C1 × C1 . In general, a lattice is distributive if and only if it has no sublattice that is a copy of either M5 or N5. Distributivity leads us to another fundamental redundancy, whose proof is easily accessible in the literature. Theorem 1.1.2. For any lattice ( L, ∧, ∨), D1 holds if and only if D2 holds. £ The shaping of distributive identities in noncommutative contexts is an important concern in generalized lattice theory. An important characterizing property of distributivity is: Theorem 1.1.3. Distributive lattices are cancellative in that x∧ z = y∧ z and x∨ z = y∨ z together imply x = y. Conversely, cancellative skew lattices are distributive. 14 I: Preliminaries Proof. Given x∧ z = y∧ z and x∨ z = y∨ z, then x = x ∨ ( x ∧ z) = x ∨ ( y ∧ z) = ( x ∨ y) ∧ ( x ∨ z) = ( x ∨ y) ∧ ( y ∨ z) = ( x ∨ z) ∧ y ≤ y and similarly, y ≤ x, so that x = y. Conversely, neither M3 nor N5 can be subalgebras of a cancellative slew lattice. £ A lattice ( L; ∨, ∧) is complete if every subset X of L has a supremum (an element u ≥ x for all x in X, with u being the least such element in L) denoted by sup( X) and an infimum (an element v ≤ x for all x in X, with v being the greatest such element in L) denoted by inf( X). In particular, a complete lattice has a greatest element 1 and a least element 0. Conversely, a lattice with both least and greatest elements 0 and 1 is complete if all subsets have suprema, or equivalently, if all subsets have infima. Finally, in any complete lattice, we let 0 = sup(∅) and 1 = inf(∅). Lattices and universal algebra An algebra is any system, A = ( A: f 1, f 2, …, fr), where A is a set and each fi is an ni-ary operation on A. If B ⊆ A is such that for all i ≤ r, fi( b 1, b 2, …, bn ) ∈ B for all b 1, …, b in B, i ni n then the system B = ( B: f i 1ʹ, f 2ʹ, …, frʹ) where fiʹ= fi ⎢ B is a subalgebra of A. (When confusion occurs, subalgebras may be indicated by their underlying sets.) Under inclusion, ⊆, the subalgebras of an algebra A form a complete lattice Sub(A) with greatest element A, least element the smallest subalgebra containing ∅ and meets given by intersection. If none of the operations are nullary, then the least subalgebra is the empty subalgebra, ∅. If there are no operations, then Sub(A) is the lattice 2 A. Recall that a congruence on A = (A: f 1, f 2, …, fr) is an equivalence relation θ on A such that given i ≤ r with a 1θ b 1, a 2θ b 2, …, an θ b in A, then i ni fi( a 1, a 2, …, an ) θ fi( b 1, b 2, …, b ). i ni Under inclusion, ⊆, the congruences on A form a complete lattice Con(A). Its greatest element is the universal relation ∇ = A×A relating all elements in A. Its least element is the identity relation Δ. Suprema and infima in Con(A) are calculated as in the lattice Equ( A) of all equivalences on A. In particular, infima in Con(A) are given by intersection. £ Recall that an element c in a lattice ( L; ∨, ∧) is compact if for any subset X of L, c ≤ supX implies that c ≤ supY for some finite subset Y of X. (Every cover can be reduced to a finite cover.) An algebraic lattice is a complete lattice for which every element is a supremum of compact elements. The proof of the following result is easily accessible in the literature Theorem 1.1.4. Given an algebra A = (A: f 1, f 2, …, fr), both Sub(A) and Con(A) are algebraic lattices. 15 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Of particular interest is the next result. It’s proof may be obtained in any standard text on lattice theory. Theorem 1.1.5. Congruence lattices of lattices are distributive. £ A subset U of a poset ( L; ≥) is directed upward if given any two elements x, y in U, a third element z exists in U such that x, y ≤ z. The proof of the next result is also easily accessible. Theorem 1.1.6. Given an algebraic lattice ( L; ∨, ∧), a ∧ sup( U) = sup{ a∧ x ⎢ x ∈ U} holds if U is directed upward. This equality holds unconditionally when ( L; ∨, ∧) is also distributive. Recall that two algebras A = ( A; f 1, f 2, …, fr) and B = ( B; g 1, g 2, …, gs) have the same type if r = s and for all i ≤ r, both fi and gi have the same number of variables, that is, both are say ni-ary operations. Recall also that a class V of algebras of the same type is a variety if it is closed under direct products, subalgebras and homomorphic images. A classic result of Birkhoff is as follows: Theorem 1.1.7. Among algebras of the same type, each variety is determined by the set of all identities satisfied by all algebras in that variety. That is, all varieties are equationally determined in the class of all algebras of the same type. £ Let B = {Bi ⎢ i ∈ I} be a set of algebras of the same type. An algebra A of the same type is a subdirect product of the B i if a monomorphism χ: A → ∏ i∈ i B i exists such that for each projection π i: ∏ i∈IBi → Bi, the composite π i oχ: A → Bi goes onto Bi. A is subdirectly irreducible if for any subdirect factorization χ: A → ∏ i∈IB i one of the composites π i o χ: A → B i is an isomorphism. A second classic result of Birkhoff is as follows: Theorem 1.1.8. I n a given variety of algebras V , every algebra A in V is a subdirect product of subdirectly irreducible algebras. £ We apply Theorem 1.1.8 to the variety of distributive lattices. But first recall that an ideal in a lattice L is any subset I of L that is closed joins and given any x ∈ I and y ∈ L, x∧ y ∈ I also. Recall also that a filter (or dual-ideal) in a lattice L is any subset F of L that is closed under meets and given any x ∈ F and y ∈ L, x∨ y ∈ F also. Given any element x ∈ L, the principal ideal x↓ = { y ∈ L⎜ x ≥ y} is the smallest ideal of L containing x. Dually, the smallest filter of L containing x is the principal filter x↑ = { y ∈ L⎜ x ≤ y}. Theorem 1.1.9. Given a distributive lattice ( L; ∨, ∧) and an element a ∈ L, χ: L → a↓ × a↑ defined by χ( x) = ( x∧ a, x∨ a) is a subdirect decomposition of ( L; ∨, ∧) . Thus a distributive lattice is subdirectly irreducible if and only if it is a copy of either C 0 or C 1 . 16 I: Preliminaries Proof. That χ is a homomorphism follows easily from the associative, commutative and distributive laws. By cancellation, χ is one-to-one. Upon composing with either coordinate projections, it clearly it mapped onto each factor. £ Corollary 1.1.10. Every nontrivial distributive lattice is a subdirect product of C 1. £ We return to the variety of all lattices. On any lattice, consider the polynomial M( x, y, z) = ( x∨ y) ∧ ( x∨ z) ∧ ( y∨ z) that was implicit in the proof of Theorem 1.5. M satisfies the identities M( x, x, y) = M( x, y, x) = M( y, x, x) = x. Given an algebra A = ( A; f 1, …, fr) on which a ternary operation M( x, y, z) satisfying these identities is polynomial-defined using the operations of A, then Con(A) is distributive. In general, if a ternary function M can be defined from the functions symbols of a variety V such that M satisfied these identities on all algebras in V, then the congruence lattices of all algebras in that variety are distributive and V is said to be congruence distributive. Boolean lattices and Boolean algebras Given a lattice ( L; ∧, ∨) with maximal and minimal elements 1 and 0, elements x and xʹ are complements in L if x∨ xʹ = 1 and x∧ xʹ = 0. If L is distributive, then the complement xʹ of any element x is unique. Indeed, let xʺ be a second complement of x. Then xʺ = xʺ ∧ 1 = xʺ ∧ ( x ∨ xʹ) = ( xʺ ∧ x) ∨ ( xʺ ∧ xʹ) = 0 ∨ ( xʺ ∧ xʹ) = xʺ ∧ xʹ. Similarly, xʹ = xʹ ∧ xʺ and xʹ = xʺ follows. Clearly 0 and 1 are mutual complements. Recall that Boolean lattice is a distributive lattice with maximal and minimal elements 1 and 0, ( L; ∧, ∨, 1, 0), such that every x in L has a (necessarily unique) complement xʹ in L. If the operation ʹ is built into the signature, then ( L; ∧, ∨, ʹ, 1, 0) is a Boolean algebra. Boolean algebras are characterized by the identities for a distributive lattice augmented by the identities for maximal and minimal elements and the identities for complementation. They also satisfy the DeMorgan identities: ( x ∨ y)ʹ = xʹ ∧ yʹ and ( x ∧ y)ʹ = xʹ ∨ yʹ. Given a Boolean algebra, the difference (or relative complement) of elements x and y is defined by x \ y = x ∧ yʹ. This operation satisfies the relative DeMorgan identities: x \ ( y ∨ z) = ( x\ y) ∧ ( x\ z) and x \ ( y ∧ z) = ( x\ y) ∨ ( x\ z). More generally, given any distributive lattice with a maximum 1 and minimum 0, if x and y have complements, then so do x ∨ y and x ∧ y with ( x ∨ y)ʹ = xʹ ∧ yʹ and ( x ∧ y)ʹ = xʹ ∨ yʹ. 17 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond The classic example of a Boolean algebra is the power set algebra 2 X of all subsets of a given set X with ∨ and ∧ being ∪ and ∩ respectively, and complementation being ordinary set complementation. More generally recall that a ring of sets is any family R of subsets of a given set X that is closed under finite unions and finite intersections. R is a field of sets if it is also closed under complementation. Before stating the next lemma, recall that an ideal P in a lattice is a prime ideal if x∧ y ∈ P implies that either x ∈ P or y ∈ P. Lemma 1.1.11. Let I be an ideal and F be a filter that are disjoint in a distributive lattice. Then a prime ideal P exists such that I ⊆ P but F ∩ P = ∅. Proof. Let P be an ideal that is maximal subject to the stated conditions. Suppose that a∧ b ∈ P for some a ∉ P and b ∉ P. Let P 1 and P 2 be the ideals generated respectively from P ∪ { a} and P ∪ { b}. Since they are properly larger than I, P 1 contains an element p 1∨ a ∈ F and P 2 contains an element p 2∨ b ∈ F where p 1, p 2 ∈ P. But then F contains ( p 1∨ a)∧( p 2∨ b) = ( p 1∧ p 2)∨( p 1∧ b)∨( p 2∧ a)∨( a∧ b) which is also in P, a contradiction. Thus a and b do not exist and P is indeed prime. £ This leads us to the first of several fundamental results about Boolean lattices: Theorem 1.1.12. (M. H. Stone) A lattice is (distributive) Boolean if and only if it is isomorphic to a (ring) field of sets. Proof. The “if” direction is clear. So suppose that a lattice L is distributive. Let P denote the set of all nonempty prime ideals of L. To each x ∈ L, set π( x) = { P ∈ P ⎢ x ∉ P}. It is easily seen that π( x∨ y) = π( x) ∪ π( y) and π( x∧ y) = π( x) ∩ π( y). Moreover, if x ≠ y then either x↓ ∩ y↑ = ∅ or else y↓ ∩ x↑ = ∅. In either case, by the lemma, a prime ideal P exists containing exactly one of x and y. Hence x ≠ y implies π( x) ≠ π( y). Thus π: L → 2P is an embedding of distributive lattices. If L is Boolean lattice, then first π(0) = ∅. Next, in the Boolean case we consider only proper prime ideals. We still have π( x∨ y) = π( x) ∪ π( y) and π( x∧ y) = π( x) ∩ π( y), but now π(1) = P. Moreover, for all x, π( x) ∩ π( xʹ) = π( x ∧ xʹ) = π(0) = ∅. Likewise, π( x) ∪ π( xʹ) = π( x ∨ xʹ) = π(1) = P. Thus π( x) and π( xʹ) are complements in 2P and so π( xʹ) = P \ π( x). Thus π is an embedding of Boolean algebras. £ Recall that an atom in a lattice with 0 is an element a > 0 such that no element x exists properly between 0 and a. A lattice is atomic if every element is a supremum of atomic elements. In particular, each x is the supremum of the set α( x) of all atoms lying beneath x. Again, the proofs of the following three results are easily accessible. We give the proof of the third. 18 I: Preliminaries Theorem 1.1.13. A Boolean lattice L is isomorphic the power lattice of some set if and only if it is complete and atomic. If the latter holds, then upon denoting the set of all atoms in L by AL, one has L ≅ 2AL under the map x → α( x). £ Proposition 1.1.14. For all x in a complete Boolean lattice L and all subsets Y of L: (i) x ∧ sup(Y) = sup({ x∧ y ⎜ y ∈ Y}) and x ∨ inf(Y) = inf({ x∨ y ⎜ y ∈ Y}). (ii) ( sup Y)ʹ = inf({ yʹ ⎜ y ∈ Y}) and ( inf Y)ʹ = sup({ yʹ ⎜ y ∈Y}). (iii) x \ sup({ x \ y ⎜ y ∈ Y}) = inf( x \ Y) and x \ inf(Y) = sup({ x \ y ⎜ y ∈ Y}). (iv) sup(Y) \ x = sup({ y \ x ⎜ y ∈ Y}) and inf(Y) \ x = inf({ y \ x ⎜ y ∈ Y}). £ Theorem 1.1.15. Given a Boolean algebra (L, ∧, ∨, 1, 0, ʹ), let a ∈ L be given. Then both a↓ and a↑ are Boolean lattices and χ: L → a↓ × a↑ defined by χ( x) = ( x∧ a, x∨ a) is an isomorphism of Boolean lattices. Proof. χ is at least a lattice embedding by Theorem 1.1.9. Next, let ( u, v) ∈ a↓ × a↑ be given. If x = ( v \ a) ∨ u, then a∧ x = a∧[( v \ a) ∨ u] = ( a ∧ ( v \ a)) ∨ ( a∧ u) = 0 ∨ u = u and a∨ x = a∨[( v \ a) ∨ u] = a ∨ ( v ∧ aʹ) ∨ a = ( a ∨ v) ∧ ( a ∨ aʹ) = v∧1 = v. Thus χ is also surjective and the theorem follows. £ 2 denotes the Boolean lattice {1 > 0}. By mild abuse of notation, 2 also denotes the Boolean algebra ({1, 0}; ∨, ∧, 1, 0, ʹ) again with 1 > 0. Put otherwise, 2 is the chain C 1 reset in a Boolean context. We have the following sequence of easy corollaries of Theorem 1.1.15 and Corollary 1.10. Corollary 1.1.16. Every finite Boolean lattice factors as a finite power of 2. £ Corollary1.1.17. The only nontrivial subdirectly irreducible Boolean algebra is 2. £ Corollary1.1.18. Every distributive lattice can be embedded into a Boolean lattice. £ By a generalized Boolean lattice is meant a lattice L with a minimal element 0 such that each principal ideal x↓ of L is a Boolean lattice. Such a lattice is necessarily distributive; moreover a difference operation on L is given by setting x \ y = x \ ( y∧ x) in the Boolean lattice x↓. For Boolean lattices both differences agree. The relative DeMorgan identities also hold for generalized Boolean lattices. Upon including \ in the signature, one has a generalized Boolean algebra (L, ∧, ∨, \, 0). It is characterized by the identities for a distributive lattice with a minimal element 0 together with the pair: ( x∧ y) ∧ ( x\ y) = 0 and ( x∧ y) ∨ ( x\ y) = x. Every Boolean algebra (L, ∧, ∨, 1, 0, ʹ) possesses a generalized Boolean algebra reduct (L, ∧, ∨, \, 0) with x \ y given as x∧ yʹ. Conversely, any generalized Boolean algebra (L, ∧, ∨, \, 0) 19 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond possessing a maximal element 1 becomes a Boolean algebra upon setting xʹ = 1 \ x. In particular, every generalized Boolean algebra that is complete as a lattice forms a complete Boolean algebra and thus satisfies all the identities of Theorem 1.1.15. Generalized Boolean algebras play a basic role in the study of skew Boolean algebras in Chapter 5, being both the commutative cases of the latter as well their maximal lattice images. 1.2 Bands A band is a semigroup S whose elements are idempotent. Thus x 2 = x (in multiplicative notation) for all x in S. A band that is also commutative is called a semilattice. Clearly: Lemma 1.2.1. If S is a commutative semigroup, then the set E(S) of all idempotents in S forms a semilattice under the semigroup operation. £ When a semigroup S is not commutative, E(S) need not be closed under multiplication. Closure of E(S) is obtained, however, with a weakened version of commutativity. A semigroup is mid-commutative (or weakly commutative) if it satisfies the identity uxyv = uyxv. A mid-commutative band is called a normal band. The next result is trivial. Lemma 1.2.2. Given a mid-commutative semigroup S, the set of idempotents E(S) forms a normal band under the given multiplication. £ Given a band S, several quasi-orders can be defined on S. To begin, the natural partial order ≥ is defined on S by e ≥ f if ef = f = fe. The natural partial order refines the natural quasi- order ≻ on S defined by e ≻ f if fef = f. Between ≥ and ≻ lie the left and right quasi-orders, ≻L and ≻R defined by respectively by: e ≻L f if fe = f and e ≻R f if ef = f. In the lattice of all quasi-orders on the underlying set of S, ≥ is the meet (intersection) of ≻L and ≻R, and ≻ is the join of ≻L and ≻R. That ≥ is a partial order, and that ≻L and ≻R are quasi-orders meeting at ≥ are easily verified. To see that of ≻ is a quasi-order that is the join of ≻ op op L and ≻R we will need the equivalences L = ≻L ∩ ≻L and R = ≻R ∩ ≻R . Alternatively, L and R are defined by: e L f if both ef = e and fe = f, and e R f if both ef = f and fe = e. L is a right congruence ( e L f ⇒ eg L fg for all g ∈ S) and R is a left congruence ( e R f ⇒ ge L gf for all g ∈ S). E.g., e L f implies egfg = efgfg = efg = eg and likewise fgeg = fg. We now state: Lemma 1.2.3. For any band S , ≻L o ≻R = ≻R o ≻L = ≻ and the result is a quasi-order. Proof. If e ≻L o ≻R f, then for some g, ge = g and gf = f. From this e ≻R ef ≻L f follows. That e ≻R ef is clear. That ef ≻L f follows from ( ef) f = ef and g( ef) = f. Thus ≻L o ≻R ⊆ ≻R o ≻L. The reverse inclusion is shown in similar fashion. This commuting composition is thus a quasi- order; moreover, it contains ≻. Indeed, from e ≻ f we obtain e ≻R ef ≻L fef = f, so that ≻ lies in ≻R 20 I: Preliminaries o ≻L. On the other hand ≻R o ≻L ⊆ ≻. For suppose e ≻R f ≻L g. But then f R fe and thus g = gf R gfe = ge. It follows that geg = g so that e ≻ g. £ L and R were originally defined for arbitrary semigroups by J. A. Green, who showed that LoR = RoL, with the resulting composition also being an equivalence denoted by D. (The proof in the case of bands is similar to that for ≻L and ≻R.) Alternatively, D = ≻ ∩ ≻op. In the lattice of all equivalences on the underlying set of S, D is the join of R and L, while the meet R∩L is the equality relation Δ. To understand the role of these equivalences in the structure of bands we turn to the class of bands that form the “anti-semilattices” amongst bands. A rectangular band is a band S satisfying the identity xyx = x, or equivalently, the identity xyz = xz. (Given the former identity, xyz = xyzxz = xz.) Rectangular bands thus form a variety of bands for which xy = yx iff x = y. Examples are left zero bands having the multiplication xy = x and right zero bands having the dual multiplication xy = y. Given a left zero band L and a right zero band R, their direct product L × R is also a rectangular band. The generality of such an example is demonstrated as follows. Theorem 1.2.4. Given a rectangular band S with e ∈ S, L = { se ⎜ s ∈ S} is a maximal left zero band in S and R = { es ⎜ s ∈ S} is a maximal right zero band in S; moreover the map µ: L × R → S given by µ( x, y) = xy is an isomorphism. Finally, L is the L- class of e, R is the R-class of e and S forms a single D- class. Proof. Fixing e, since S is rectangular, sete = se so that L is at least a left zero semigroup in S. If Lʹ were any left zero band in S such that e ∈ Lʹ, then for all s ∈ Lʹ, s = se so that s ∈L. Thus Lʹ ⊆ L and L is indeed a maximal left zero band in S. Clearly such a subset must be an L-class of S. Similarly, R is a stated. Finally, consider function µ: L × R → S. Viewing L × R as a direct product of bands and applying the defining identity of a rectangular band we get: µ(( re, es)( te, eu)) = µ(( rete, eseu)) = µ( re, eu) = reu = ru = resteu = µ( re, es)µ( te, eu) so that µ is a homomorphism. Since s = ( se)( es) for all s ∈S, µ maps L × R onto S. Finally, let s = µ( ue, ev) = uev. Then se = ueve = ue and es = euev = ev so that ( ue, ev) = ( se, es). Thus µ one-to-one also and an isomorphism. £ A rectangular band can be represented as a rectangular array with R-equivalent elements comprising the rows and L-equivalent elements comprising the columns. ⎡ e f g ⎤ ⎢ ⎥ ⎢ i j k ⎣ ⎦⎥ 21 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Given elements x and y, the product xy is the unique element in the row (R-class) of x and the column (L-class) of y. Thus ek = g while ke = i in this array. Returning to the broader context of arbitrary bands, we have Corollary 1.2.5. Given any band S, its L- classes are the maximal left zero sub-bands in S and its R- classes are the maximal right zero sub-bands in S. £ Theorem 1.2.6. (Clifford-McLean) Given a band S, the equivalence D is a congruence on S. Its congruence classes form maximal rectangular sub-bands of S and the quotient algebra S/D is the maximal semilattice image of S. Proof. If e L f in S, then in general, ev L fv for all v in S. Thus for all u, v in S, ( uev)( ufv)( uev) = uevufvuev = uev( fv) ufvuev = uevfvuev = uevuev = uev and similarly ( ufv)( uev)( ufv) = ufv. Thus eL f implies uev D efv for all u, v in S. In like fashion, e R f implies uev D efv for all u, v in S. Suppose that e D f in S. Then e R ef L f and thus for all u, v in S, uev D uefv D ufv. D is thus a congruence. Since ef D fe for all e, f in S, S/D is commutative and hence a semilattice. Since each D-class is a maximal subset of S satisfying xyx = x, it is a maximal rectangular subalgebra of S. £ In brief, every band is a semilattice of rectangular bands. Thus a band has the appearance of a semilattice diagram with each node filled in by a rectangular band. £ £   £ £   £ While multiplication is performed in rectangular fashionwithin D-classes, multiplication between elements from distinct D-classes is another matter. We can, however, be more specific in this regard for the two subvarieties that we consider next. Unlike D, the relations L and R need not be congruences. A band for which L and R are full congruences is called regular. Since D is a congruence, whenever D = L (so that R = Δ) or D = R (so that L = Δ) the band must be regular. When D = L the band is called left regular and when D = R it is called right regular. It is both precisely when it is a semilattice. A normal, left (right) regular band is called left ( right) normal band. 22 I: Preliminaries Theorem 1.2.7. A band is left regular [ right regular] if and only if xyx = xy [ xyx = yx]. In general, a band is regular, if and only if it satisfies the identity xyxzx = xyzx. Proof. Given x, y in a band S, xyx R xy. Conversely, given x R y in S, x = xyx and y = xy. Thus S is left regular (R = Δ and D = L) precisely when xyx = xy holds on S. Dually S is right regular (L = Δ and D = R) precisely when xyx = yx holds on S. If S is regular, then consider the canonical epimorphisms S → S/L and S → S/R. From the rectangular structure of D-classes of S, it is follows that S/L is right regular and S/R is left regular. Thus both bands satisfy either xyx = xy or xyx = yx and hence the identity xyxzx = xyzx. Since the two epimorphisms induce an embedding of S into S/L × S/R which satisfies xyxzx = xyzx, so does S. Conversely, let S satisfy xyxzx = xyzx. Suppose u L v in S, so that uv = v and vu = v. Then for all w, ( uw)( vw) = uvwuv = uvw = uw and likewise ( vw)( uw) = vw. The assumed identity then gives us ( wu)( wv) = wuwvu = wuwuvu = wuwu = wu and likewise ( wv)( wu) = wv showing that L is a indeed congruence. In similar fashion R is seen to be a congruence. £ Corollary 1.2.8. A band S being regular is equivalent to either of the following: (i) Given e ≻ a, b in S, aeb = ab. (ii) S satisfies xyxʹ zxʺ = xyzxʺ, given xʹ, xʺ D x. Proof. If (i) holds, then xyxzx = ( xy) x( zx) = xyzx follows and S is regular. Conversely, if S is regular and e ≻ a, b in S, xyxzx = xyzx gives us (i): aeb = aebaeb = aeababeb = aababb = ab. Clearly S is regular if (ii) holds. Conversely, (i) implies xyxʹ zxʺ = ( xy) xʹ( zxʺ) = xyzxʺ. £ The function ζ: S → S/L × S/R defined by ζ( x) = (L x, R x) is always 1-1 for any band. The product S/L × S/R is naturally a band and ζ is a homomorphism (and thus a monomorphism) precisely when S is regular. In this case, the image ζ[S] is the fibered product S/L ×S/D S/R of S/L with S/R over the common maximal semilattice image, S/D. Thus the following commuting diagram of natural epimorphisms is a pullback. The isomorphism S ≅ S/R ×S/D S/L is called the Kimura factorization, after its discoverer, Naoki Kimura. S ⎯⎯⎯⎯⎯⎯⎯⎯ → ⎯ S/L ⎜ ⎜ ↓ ↓ S/R ⎯⎯⎯⎯⎯⎯⎯⎯ → ⎯ S/D Corollary 1.2.9. Every normal band is regular. Normal bands are also characterized by the identity xyzx = xzyx. In particular, normal left [right] regular bands are characterized by the identity uxy = uyx [ xyu = yxu]. Proof. Normal bands clearly satisfy this identity, and any band S satisfying this identity is regular: xyxzx = xxzyx = xzyx = xyzx. But a left regular band satisfies this identity it and only if it satisfies xyz = xzy, thus making it normal. Likewise a right regular band satisfies this identity if 23 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond and only if it satisfies yzx = zyx, also making it normal. Since any regular band S can be embedded in S/R × S/L, with the latter being normal if and only if S is, the corollary follows. The remainder of this section is devoted to assorted further remarks about bands. We begin by defining four canonical sub-bands arising for each e in a band S. e↓ ={ f ∈ S ⎢ f ≤ e}, e⇓ = { f ∈ S ⎢ f ≺ e}, L⇓ e = { f ∈ S ⎢ f L≺ e} and e⇓R = { f ∈ S ⎢ f R≺ e}. Lemma 1.2.10. Given a normal band S, for each e ∈S, under the given operation e↓ is a semilattice; conversely, every band satisfying this property is normal. Proof. If S is normal and f, g ≤ e in S, then fg = efge = egfe = gf. Thus e↓ is a commutative sub-band of S, that is, a semilattice in S. Conversely, if each e↓ is commutative, then S at least satisfies the identity xyxzx = xzxyx. From this we derive the identity xyzyxzyzx = xzyzxyzyx. But since xyzy, zyzx and yxz lie in the same D-class, xyzyxzyzx reduces to xyzyzx = xyzx. Similarly, xzyzxyzyx reduces to xzyx and xyzx = xzyx follows. £ Given D-classes A, B in a band S we write A ≥ B if a ≻ b for any (and hence all) pairs a ∈ A and b ∈ B. When A ≥ B but A ≠ B, we write A > B. This reflects, of course, what occurs between the corresponding elements in the underlying lattice S/D. When A ≥ B we say that A and B are comparable D-classes. The following results provide an explicit description of the architecture of a normal band. Lemma 1.2.11. Given D- classes A ≥ B in a normal band S, for each a ∈A exactly one b ∈B exists such that a ≥ b. The function α: A → B determined by α( a) = b if a ≥ b is a homomorphism of rectangular bands; moreover α( a) = aba for all b ∈ B . Conversely, if ≥ induces functions in this manner between all pairs of comparable D- classes of a band S, then S is normal. Proof. Given a ∈ A and b ∈ B for comparable D-classes A ≥ B in a normal band S, observe that a ≥ aba in B. Given bʹ ∈ B also, then bbʹ b = b and bʹ bbʹ = bʹ in B. Normality give us aba = abbʹ ba = abʹ ba = abʹ bʹ ba = abʹ bbʹ a = abʹ bʹ a = abʹ a. Thus the procedure a → aba induces a well-defined map from A to B such that a ≥ aba with aba being independent of b ∈ B. Since b = aba whenever a ≥ b, the first statement of the lemma follows. That α is a homomorphism follows immediately from S being normal: if a, aʹ ∈ A and b ∈ B, then abaaʹ baʹ = aaʹ bbaaʹ aaʹ baaʹ. Conversely, suppose that ≥ always induces a function between comparable pairs of D-classes of a band S in the above manner. Since xyzx D xzyx with x ≥ xyzx and x ≥ xzyx for all x, y, z ∈ S, this assumption gives xyzx = xzyx in S. S is thus normal by Corollary 1.2.9. £ 24 I: Preliminaries Lemma 1.2.12. For all comparable D- classes A ≥ B in a normal band S, if αA : A B → B is the homomorphism determined for all a A ∈ A by a ≥ α ( a) B ∈ B, then: i) αA = id A A for all D- classes A in S, and ii) αB A = αA for all comparable D- classes A ≥ B ≥ C in S. C � αB C iii) Given a A B ∈ A and b ∈ B, ab = α ( ) α ( ) in the meet-class M of A and B. M a M b Proof. (i) is just aaʹ a = a for all aʹ ∈ A. (ii) states that ( aba) c( aba) = aca for all a ∈ A, b ∈ B and c ∈ C, a trivial consequence of normality: abacaba = acbcacacbca = accccca = aca. Finally, (iii) follows from the identity ab = a( ab) ab( ba) b holding in all bands. £ This leads to the following result of Naoki Kimura and Miyuki Yamada. Theorem 1.2.13. Let T be a meet semilattice poset (T; ≥). Assign a rectangular band N a to each a a ∈ T and a homomorphism Ν : N b a → N b to each pair a ≥ b in T, such that: i) Given a ≠ b in T, N a and N b are disjoint. ii) Ν aa is the identity map on N a for all a ∈ T . iii) Ν b a a c � Ν = Ν for all a ≥ b ≥ c in T. b c Given x a b ∈ N a and y ∈ N b, set xy = Ν ( x) Ν ( y) in N ab ab ab. Then S =∪ a ∈T N a is a normal band with maximal rectangular sub-bands being the N a, with maximal semilattice image being T and the canonical homomorphism τ: S → T defined by τ ( x) = a if x ∈N a. Conversely, every normal band arises in this fashion. Proof. It is easily verified that the above construction produces a band such that x ≥ y for x ∈ N a and y a ∈ N b holds precisely when a ≥ b in T and Ν ( x) = y. It follows from Lemma 1.2.11 that ab S is indeed a normal band. The converse assertion follows from Lemma 1.2.12. £ What is the picture in the case of regular bands? Or is there one? To begin, given comparable D-classes A ≥ B in a band S and b ∈B, the set A b A = { abaʹ ⎜ a, aʹ ∈A} is a coset of A in B. Similarly the left coset is A b = { ab ⎜ a ∈A} and the right coset is b A = { ba ⎜ a ∈ A}. Proposition 1.2.14. Let S be regular band, with comparable D- classes A > B. Then: i) For all b, c ∈ B , either A b A = A c A or A b A∩A c A = ∅ . ii) In particular, if A b A = A c A, then abaʹ = acaʹ for all a, aʹ in A. iii) B is partitioned into a disjoint union of nonempty subsets B i, each consisting of all b ∈B inducing the same coset A b A of A in B. In particular, all c ∈ A b A induce the coset A b A, so that A b A ⊆ B i ={ c ∈ B⎪A c A = A b A}. iv) For each a ∈ A and each partition cell B i, a unique bi ∈ B i exists such that a ≥ bi. For any b ∈ B i, aba is this bi and bab = b. Proof. To begin, suppose A b A ∩ A c A ≠ ∅ for some b, c ∈ C. Thus a 1 ba 2 = a 3 ca 4 for some a 1, a 2, a 3, a 4 ∈A. Corollary 1.2.8 then implies that for all a, aʹ in A, 25 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond a( a 1 ba 2) aʹ = ( aa 1 a) b( aʹ a 2 aʹ) = abaʹ and likewise a( a 3 ca 4) aʹ = acaʹ, so that abaʹ = acaʹ. From this (i) - (iii) follow. To see (iv), clearly a ≥ aba in the cell containing A b A. Assuming b ∈ B i, then aca = aba for all c ∈ B i, so that aba is this unique bi ≤ a in B i. That bab = b follows from a ≻ b. £ Given A > B and cell B i in B, B i is called an A- cell in B and α i: A → B i is defined by α i( a) = aba for all a ∈A and any b ∈B i is the cell-map of A into B i. Its image in B is the coset A b A, since by Corollary 1.2.8 again, abaʹ = aaʹ baaʹ for all a, aʹ ∈ A and b ∈ B. Application. All left regular bands with just two D-classes A > B are constructed as follows. (1) Given sets A and B, partition B into disjoint nonempty subsets {B i ⊆ B ⎜ i ∈ I}. (2) For each cell B i of the partition, choose a function α i: A → B i. (3) Define multiplication on S = A ∪ B by first imposing left zero multiplications on A and B separately, setting ba = b for all a ∈ A and b ∈ B, and finally by setting ab = α ( a) i if b ∈ B i. X In the resulting band S = A ∪ B, the cell decomposition of B is the given partition B =∪B i and the cell-maps from A to B are precisely the α i: A → B i ⊆ B for each i ∈ I. Cell decompositions and cell-maps also determine the multiplication of elements in incomparable D-classes in a left regular band. But first a definition: given D-classes A > B in a band with a ∈A, the image of a in B is the set { b ∈ B ⎜ a ≥ b}. Put otherwise, this image is a B a = { aba ⎜ b ∈ B}, or when S is left regular, a B = { ab ⎜ b ∈ B}. In the next situation, both D-classes A and B can be incomparable Proposition 1.2.15. Given a left regular band S and D -classes A and B , with meet D - class M, and with a ∈A and b ∈B, then: (1) The image b M b of b in M lies in a unique cell M i of the A- decomposition of M. (2) If α i: A → M i ⊆ M is the corresponding cell-map, then ab = α i( a). (3) Dual remarks hold for the image class of a in M and the product ba. Conversely, given two left regular bands consisting of D-c lasses A > M and B > M respectively, such that A∩B = ∅, with M the same for both bands with A > M and B > M satisfying (1) – (3), then the multiplications on each band extend uniquely to a left regular multiplication on their union A∪B∪M such that M is the meet class of A and B. (Dual constructions and observations exist for the right regular case. The general case follows using fibered products.) Proof. Given b > bʹ ∈M, abʹ = a( bbʹ) = ( ab) bʹ = ab. Thus all such bʹ < b lie in the same cell of the A-decomposition of M and (1) – (3) are now clear. For the second part, given (1) – (3), one needs to show that all possible multiplications of abm (and bam), amb (and bma), mab (and mba), 26 I: Preliminaries aaʹ b (and bbʹ a), abaʹ (and babʹ) and baaʹ (and abbʹ) are unambiguous. We consider only the non-parenthesized cases, of which the amb, mab and baaʹ cases are trivial. abm: Since bm is in the A-cell in M containing all images of b in M, ( ab) m = ab = a( bm). aaʹ b: Since ab and aʹ b lie in a common A-cell in M, ( aaʹ) b = ab and a( aʹ b) are images of a in this cell and so are equal. abaʹ: Note that baʹ is an image of b in M lying in the unique A-cell of M containing all images of B in M. Thus ( ab) aʹ = ab = a( baʹ). £ We consider next a class of necessarily regular bands for which architectural issues are simplified. A band S satisfies the class covering condition (CCC) if for every comparable pair of D-classes A ≥ B in S, and every b ∈ B an a ∈ A exists such that a ≥ b. Every b ∈ B is thus “covered” by some a in A. The free left regular band on { a, b, c} does not satisfy the CCC. Theorem 1.2.16. In a band S satisfying the class covering condition, the following hold: i) S is regular. ii) An element x lies in the center of S if and only if D x = { x} iii) Given x, y ∈S, D xy is D x D y = { uv⎪ u ∈D x & v ∈D y}. iv) In particular, when D x and D y are finite, so is D xy. Proof. (i) Given x, y, z ∈ S, by the CCC, u, v ∈ D x exists such that u ≥ xy and v ≥ zx. Since uvw = uw holds in D x we obtain: xyzx = ( xyu)( vzx) = xy( uxv) zx = ( xyu) x( vzx) = xyxzx. Thus S is regular. (ii) If x commutes with all elements in S, then clearly D x = { x}. Conversely, let D x = { x} and let y ∈ S be given. By the CCC, x ≥ xy and hence xy = xyx. Similarly, x ≥ yx so that yx = xyx. Thus xy = yx and x lies in the center of S. (iii) Clearly D x Dy ⊆ Dxy. Let z in Dxy be given. Since D x ≥ D x y and Dy ≥ D x y, by the CCC, u ∈D x and v ∈D y exist such that u ≥ z and v ≥ z. Thus z = uvzuv = uv since both z and uv lie in D xy. (iv) is now clear. £ The situation regarding cell-maps in the CCC case is much cleaner. Theorem 1.2.17. In a band S satisfying the class covering condition, the following hold: i) Given D- classes A > B, the A- cells in B are precisely the cosets, A b A. ii) B is partitioned by the cosets of A in B and the cell-maps from A to B are collectively surjective in that B = ∪ b∈BAbA. iii) Given a D -class chain A > B > C , all compositions of cell-maps from A to B with cell-maps from B to C are cell-maps from A to C. iv) Conversely, all cell-maps from A to C are obtained in this fashion. Proof. (i) Given b ∈ B, by the CCC, a ∈ A exists such that a ≥ b. Hence b = aba ∈ A b A and the cell containing b collapses to A b A. (ii) is now clear. (iii) Next, given A > B > C as stated, 27 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond consider the cell-maps ϕ: A → A b 0A ⊆ B and ψ: B → B c 0B ⊆ C defined by ϕ( a) = ab 0 a and ψ( b) = bc 0 b for b 0 ∈ B and c 0 ∈ C. Then for all a ∈ A, ψoϕ( a) = ab 0 ac 0 ab 0 a = = ab 0 c 0 b 0 a by Corollary 1.2.8. Thus ψoϕ: A → C is a cell-map form A to the cell of b 0 c 0 b 0 in C. Finally, given c ∈ C let ζ: A → C be cell-map defined by ζ( a) = aca. By CCC again, b 0 ∈ B exists such that b 0 ≥ c. By Lemma 1.2.8 again, for all a ∈ A, aca = ab 0 cb 0 a = ab 0 acab 0 a. Thus, if ϕ: A → A b 0A ⊆ B and ψ: B → B c B ⊆ C are the cell-maps ϕ( a) = ab 0 a and ψ( b) = bcb, then ψoϕ is precisely ζ: A → C. £ The above need not hold in all regular bands, as is seen in LReg{ a, b, c}. The cell-map from { a} to the bottom D-class in LReg{ a, b, c} sending a to abc does not factor through the intermediate class { ac, ca}. Likewise the cell-map sending a to acb does not factor though the class { ab, ba}. Proposition. 1.2.18. The class of all bands satisfying the class covering condition are closed under products and homomorphic images. Proof. Closure under products is clear. Suppose that band S satisfies the CCC and that f: S → T is a homomorphism. Let a ≻ b in f[S] with a = f( x) and b = f( y) for x, y in S. Then x ≻ yxy in S with f( yxy) = bab = b in T. By CCC, xʹ ∈ D x exists such that xʹ ≥ yxy. Clearly aʹ = f( xʹ) ≥ b and aʹ ∈ D a. £ Thus bands satisfying the class covering condition do not form a variety. This condition holds, however, for both band reducts (S, ∧) and (S, ∨) of any skew lattice (S, ∨, ∧). And skew lattices do form a variety. 1.3 Noncommutative lattices – initial observations In general, a noncommutative lattice is an algebra (N: ∨, ∧) where both ∨ and ∧ are associative, idempotent binary operations satisfying a specified set of absorption identities. We continue to call ∨ the join and ∧ the meet. The adjective “noncommutative” is used here in the inclusive sense of “not-necessarily-commutative”. Thus lattices will play an important role in the general study of noncommutative lattices. Thus far, nearly all types of noncommutative lattices that have been studied assume absorption identities from among the following: B1: a ∧ ( a ∨ b) = a. C1: a ∨ ( a ∧ b) = a. B2: ( b ∨ a) ∧ a = a. C2: ( b ∧ a) ∨ a = a. B3: a ∧ ( b ∨ a) = a. C3: a ∨ ( b ∧ a) = a. B4: ( a ∨ b) ∧ a = a. C4: ( a ∧ b) ∨ a = a. If ∨ and ∧ are commutative, then clearly the B’s merge together as do the C’s. In general we require that ∨ and ∧ satisfy at least a pair of identities that reduce to B1 and C1 when ∨ and ∧ are commutative. We consider several classes of such algebras. 28 I: Preliminaries The noncommutative lattices studied most extensively in recent years are skew lattices that satisfy B1, B2, C1 and C2. We will see that these identities express the dualities: x ∨ y = x if and only if x ∧ y = y and x ∨ y = y if and only if x ∧ y = x. (The equivalence of the absorption identities with the stated dualities will be shown shortly.) Likewise, skew* lattices satisfy the complementary set of identities, B3, B4, C3 and C4, and are characterized by the pair of dualities: x∨ y = x if and only if y∧ x = y and x∨ y = y if and only if y∧ x = x. Either type of algebra is transformed into the other by replacing ∧ by ∧* or by replacing ∨ by ∨* where a ∧* b = b ∧ a and a ∨* b = b ∨ a. A double replacement yields the original type of algebra. Hence we choose to focus on skew lattices. A detailed study of skew lattices will commence in the following chapter. Two further classes of noncommutative lattices are: A quasilattice is a noncommutative lattice satisfying the two-sided absorption identities B5: a ∧ ( b ∨ a ∨ b) ∧ a = a. C5: a ∨ ( b ∧ a ∧ b) ∨ a = a. We will see that these identities expresses the duality, x ∧ y ∧ x = x if and only if y ∨ x ∨ y = y, stating that x ≺ y under ∧ if and only if y ≺ x under ∨. A paralattice is a noncommutative lattice satisfying the absorption identities B6: a ∧ ( a ∨ b ∨ a) = a = ( a ∨ b ∨ a) ∧ a. C6: a ∨ ( a ∧ b ∧ a) = a = ( a ∧ b ∧ a) ∨ a. These identities express the duality, x ∧ y = x = y ∧ x if and only if x ∨ y = y = y ∨ x, stating that x ≤ y under ∧ if and only if y ≤ x under ∨. While B5 – C6 above are not among the previous absorption identities, if combined with flatness (see below), they reduce to identities on the earlier list. (See Theorem 1.3.7 below.) Quasilattices that are also paralattices are called refined quasi-lattices. Skew lattices and skew* lattices are both refined quasi-lattices. Another significant class of examples is as follows. In Section 3.4 we will see that refined quasi-lattices are closely related to skew* lattices. An antilattice is an algebra (N, ∨, ∧) with associative, idempotent binary operations ∨ and ∧ such that both a ∧ b ∧ a = a and a ∨ b ∨ a = a. Lattices and antilattices form antipodal classes of examples having foundational import in the study of noncommutative lattices. Thanks to the simple behavior of rectangular bands, however, antilattices are easily described. A pair of double indexing { x(λ, ρ) ⎢(λ, ρ) ∈L × R} and { x(λ*,ρ*) ⎢(λ*, ρ*) ∈ L* × R*} of the elements of N exist such that: x(λ, ρ) ∨ x(µ, σ) = x(λ, σ) and x(λ*, ρ*) ∧ x(µ*, σ*) = x(λ*, σ*). If N is finite, it can be exhibited as a pair of rectangular arrays. Consider the example: 29 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond a b c b c f (∨) d e f (∧) a e i g h i d g h The result of either x∨ y or x∧ y is the element lying in the row of x and the column of y in the relevant array. Thus a∨ f = c while a∧ f = i. In this example, none of the B i or C i are satisfied for i ≤ 4. In particular, we have neither a skew lattice nor a skew* lattice. If both arrays coincide, then x ∨ y = x ∧ y and N is a skew* lattice. If they are transposes (so that L = R* and R = L*), then x ∨ y = y ∧ x and N is a skew lattice. In both special cases only the ∧-array is needed. (In the transpose case, one has x(λ, ρ) ∧ x(µ, σ) = x(µ, ρ).) In general, both arrays are needed and all we can assert is that N is a refined quasi-lattice. We return to the first eight absorption identities. Initially, each pair (B i, C i) is a dual pair in that either is obtained from the other by switching ∨ with ∧. But other forms of duality exist. Lemma 1.3.1. Given (N, ∨, ∧) where both ∨ and ∧ are associative and idempotent: B1 - a ∧ ( a ∨ b) = a - asserts that for x, y ∈ N, x (∨)≻R y implies x L≺ (∧) y. B2 - ( b ∨ a) ∧ a = a - asserts that for x, y ∈ N, x (∨)≻L y implies x R≺ (∧) y. B3 - a ∧ ( b ∨ a) = a - asserts that for x, y ∈ N, x (∨)≻L y implies x L≺ (∧) y. B4 - ( a ∨ b) ∧ a = a - asserts that for x, y ∈ N, x (∨)≻R y implies x R≺ (∧) y. C1 - a ∨ ( a ∧ b) = a - asserts that for x, y ∈ N, x (∧)≻R y implies x L≺ (∨) y. C2 - ( b ∧ a) ∨ a = a - asserts that for x, y ∈ N, x (∧)≻L y implies x R≺ (∨) y. C3 - a ∨ ( b ∧ a) = a - asserts that for x, y ∈ N, x (∧)≻L y implies x L≺ (∨) y. C4 - ( a ∧ b) ∨ a = a - asserts that for x, y ∈ N, x (∧)≻R y implies x R≺ (∨) y. Thus, we have the following pairs inducing converse implications: B1 and C2, B2 and C1, B3 and C3 and B4 and C4. £ Theorem 1.3.2. (Laslo [1997]) An algebra (N, ∨, ∧) where ∨ and ∧ are associative, idempotent binary operations is a lattice if and only if B1, B2, C3 and C4 all hold or the complementary set B3, B4, C1 and C2 all hold. (Thus absorption can imply commutativity.) Proof. B1, B2, C3 and C4 yield x (∨)≻R y ⇒ x L≺ (∧) y ⇒ x (∨)≻L y ⇒ x R≺ (∧) y ⇒ x (∨)≻R y. Thus, for each operation, both ≻R and ≻L reduce to ≥ and both natural partial orders dualize each other and we have a lattice. The case for B3, B4, C1 and C2 is similar. The converse is clear. £ Similarly we obtain: 30 I: Preliminaries Theorem 1.3.3. An algebra (N, ∨, ∧) where ∨ and ∧ are both associative and idempotent is a skew lattice ( B1, B2, C1 and C2) if and only if ∧≺L dualizes ∨≺R and ∧≺R dualizes ∨≺L, that is: x ∨ y = x iff x ∧ y = y and x ∨ y = y iff x ∧ y = x. Likewise, N is a skew* lattice ( B3, B4, C3 and C4) if a nd only if ∧≺L dualizes ∨≺L and ∧≺R dualizes ∨≺R, that is: x ∨ y = x iff y ∧ x = y and x ∨ y = y iff y ∧ x = x. £ Laslo has shown that no lesser combination than those given in Theorem 1.3.2 suffice to force (N, ∨, ∧) to be a lattice. Moreover, the effect of Theorems 1.3.2 and 1.3.3 is that the combination of any three of B1-B4 with any three of C1-C4 guarantees that (N, ∨, ∧) is at least a skew* lattice, if not a lattice. In the first section on lattices we saw that B1 and C1 together were sufficient to force both ∧ and ∨ to be idempotent operations. In general, we have: Theorem 1.3.4. (Laslo and Cozac [1998]) Let B i and C j where 1 ≤ i, j ≤ 4 be a pair of absorption identities and let (N, ∨, ∧) be an algebra with binary operations ∨ and ∧ \on N that satisfy both B i and C j. Unless B i and C j form a converse pair in the above sense, then both ∨ and ∧ are idempotent. When B i and C j form a converse pair, then even assuming that both operations are associative is not enough to force them to also be idempotent. Proof. We have already seen that B1 and C1 imply that both operations are idempotent. Given B1 and C3 we have a ∧ a = a ∧ [ a ∨ ( b ∧ a)] = a and a ∨ a = a ∨ ( a ∧ a) = a. Next, given B1 and C4 we have a ∨ a = ( a ∧ ( a ∨ a)) ∨ a = a and a ∧ a = a ∧ ( a ∨ a) = a. Lastly, both B1 and C2 are satisfied by the three-element algebra with binary operations given by tables: ∨ a b c ∧ a b c a a b c and a a a a . b c c c b a a b c c c c c a a c Both operations are associative, but clearly not idempotent. We have settled the case for B1 and any of the C j. What about B2 through B4? First, suppose we are given algebra (N, ∨, ∧) where ∨ and ∧ are binary operations. For (N, ∨, ∧) to satisfy B4 it is equivalent that (N, ∨, ∧*) satisfy B1. (Here x ∧* y = y ∧ x as before.) But replacing ∧ by ∧* has the effect of permuting the C j. In particular, (N, ∨, ∧*) satisfying C2, corresponds to (N, ∨, ∧) satisfy C4. Thus, given B4, all C j except C4 induce idempotent operations. Continuing, the B1-C2 pairing for (N, ∨*, ∧) corresponds to the B3-C3 pairing for (N, ∨, ∧) and the B1-C2 pairing for (N, ∨*, ∧*) corresponds to the B2-C1 pairing for (N, ∨, ∧). Thus all B i-Cj pairings insure idempotent operations except for the converse pairings. £ Comment. The proof shows that once one of the operations is known to be idempotent, then the nonconverse pairing is enough to force the other operation to be idempotent also. 31 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Thanks to various dualities, coherent quasi-orderings are defined as follows. 1) If (N, ∨, ∧) is a quasilattice, its natural quasiordering ≺ is just ∧≺, or the dual of ∨≺. 2) If (N, ∨, ∧) is a paralattice, its natural partial ordering ≤ is ≤, or the dual of ≤. ∧ ∨ 3) If (N, ∨, ∧) is a skew(*) lattice, its left and right quasiorderings, ≺L and ≺R, are ∧≺L and ∧≺R respectively, or their duals of the appropriate ∨-quasi-orderings. Congruences A congruence on a noncommutative lattice N is an equivalence relation θ on N such that for all a, b, c ∈ N, a θ b implies a∧ c θ b∧ c, c∧ a θ c∧ b, a∨ c θ b∨ c and c∨ a θ c∨ b. In accord with standard notation, Δ denotes the least congruence (equality) and ∇ denotes the greatest congruence. Besides Δ and ∇, two other congruences of interest are: The least lattice congruence is the smallest congruence λ on N such that N/λ is a lattice. N/λ is thus the maximal lattice image of N. On the interval [λ, ∇], θ∩sup i(θ i) = sup i (θ∩θ i), since [λ, ∇] is lattice isomorphic with Con(N/λ), the congruence lattice of the lattice N/λ. On the other hand, the least rectangular congruence, is the least congruence ρ on N for which N/ρ is rectangular. Clearly ρ is the congruence generated from the relation ≤ ≤. ∨ ∪ ∧ Theorem 1.3.5. If (N, ∨, ∧) is a noncommutative lattice with D and D the ∨ ∧ D -congruences for ∨ and ∧ , then λ is the congruence on N generated from the relation D . ∨ ∪ D∧ Proof. If δ is the congruence on N generated from D , then N/ ∨ ∪ D∧ δ is commutative in both ∨ and ∧. Since N satisfies absorption identities inducing B1 and C1 on any commutative image, N/δ is a lattice. Thus λ ⊆ δ. Since δ ⊆ λ is clear, λ = δ follows. £ Corollary 1.3.6. Let (N, ∨, ∧) be a noncommutative lattice for which D = D . Then ∨ ∧ λ = D, the common D- equivalence for both operations, N is a quasilattice, N/D is the maximal lattice image of N and all the D- classes are the maximal antilattices in N. Conversely, for all quasilattices, D = D . ∨ ∧ Proof. If D = D , then ∨ ∧ λ is clearly the common D-equivalence. By the Clifford-McLean Theorem for bands, N/D is the maximal lattice image of N and the D-classes are maximal anti- lattices in N. Hence a D a ∧ ( b ∨ a ∨ b) ∧ a in N. But since a (∧)≥ a ∧ ( b ∨ a ∨ b) ∧ a in N, equality follows: a = a ∧ ( b ∨ a ∨ b) ∧ a. Similarly, a = a ∨ ( b ∧ a ∧ b) ∨ a so that N is a quasilattice. The converse is clear. £ 32 I: Preliminaries Thus, every quasilattice is a lattice of antilattices. Hence quasilattices are precisely the noncommutative lattices having a direct analogue of the Clifford-McLean Theorem for bands. Flatness A noncommutative lattice is flat if one of the following holds: ( r, l): a∨ b∨ a = b∨ a and a∧ b∧ a = a∧ b. ( l, r): a∨ b∨ a = a∨ b and a∧ b∧ a = b∧ a. ( l, l): a∨ b∨ a = a∨ b and a∧ b∧ a = a∧ b. ( r, r): a∨ b∨ a = b∨ a and a∧ b∧ a = b∧ a. Thus, being ( r, l)-flat means that D(∨) = R(∨) and D(∧) = L(∧), or equivalently, L(∨) = R(∧) = Δ. Modified remarks hold for the other three types of flatness. Clearly: Proposition 1.3.7. Given any variety of noncommutative lattices (e.g., paralattices or quasilattices), the flat algebras of a given type form a subvariety. £ If ( N, ∨, ∧) if flat, then ( N, ∨*, ∧), ( N, ∨, ∧*) and ( N, ∨*, ∧*) are also flat, with all four types of flatness represented. Of particular interest is: Theorem 1.3.8. ( l, l)- flat paralattices are characterized by B1, B4, C1 and C4: a∧( a∨ b) = a = ( a∨ b)∧ a and a∨( a∧ b) = a = ( a∧ b)∨ a. Similarly, ( l, l)- flat quasilattices are characterized by B1, B3, C1 and C3: a∧( a∨ b) = a = a∧( b∨ a) and a∨( a∧ b) = a = a∨( b∧ a). Proof. Assuming ( l, l)-flatness B6, B7, C6 and C7 reduce to B1, B4, C1 and C4. Conversely, from B1, B4, C1 and C4 we get a∧ b∧ a = a∧ b∧[( a∧ b)∨ a]= a∧ b. Switching ∨ and ∧ (which we can since B1, B4, C1 and C4 are operational duals) yields a∨ b∨ a = a∨ b. Thus B6, B7, C6 and C7 can be recovered from B1, B4, C1 and C4. Still assuming ( l, l)-flatness, B5, C5 and their derived identities, a∧( a∨ b∨ a)∧ a = a = a∨( a∧ b∧ a)∨ a, reduce to B1, B3, C1 and C3. From the latter, a∧ b∧ a = a∧ b∧[ a∨( a∧ b)] = a∧ b follows. Similarly, a∨ b∨ a = a∨ b. Thus B5 and C5 are recovered from B1, B3, C1 and C3. £ This illustrates a general rule of thumb: upon assuming flatness, B1 - B4 and C1 - C4 usually suffice to describe the dualities encountered in noncommutative lattice theory. Flatness was encountered in early work on noncommutative lattices. P. Jordan, who wrote numerous articles on the subject in the 1950s and 1960s, often worked with algebras satisfying B2, B3, C1 and C4 which characterize ( r, l)-flat paralattices. M. D. Gerhardts, who 33 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond published in the 1960s and early 1970s, studied algebras satisfying B1, B3, C2 and C4 which characterize ( r, l)-flat quasilattices. In a sequence of papers beginning in the 1980s, Gh. Farcas considered systems combining ( l, l)-flatness with middle commutativity, a∨ b∨ c = a∨ c∨ b and a∧ b∧ c = a∧ c∧ b, so that both operations were left normal. Distributive identities Just as the number of essentially distinct absorption identities proliferates in the absence of commutativity, so do the number of essentially distinct distributive identities. To begin, a noncommutative lattice is fully distributive if it satisfies the identities: D : a ′ : a 1 ∧ ( b ∨ c) = ( a ∧ b) ∨ ( a ∧ c). D1 ∨ ( b ∧ c) = ( a ∨ b) ∧ ( a ∨ c). D : ( a ′ : ( a 2 ∨ b) ∧ c = ( a ∧ c) ∨ ( b ∧ c). D2 ∧ b) ∨ c = ( a ∨ c) ∧ ( b ∨ c). Unlike the case for lattices: Theorem 1.3.9. For skew lattices D , D ′ , D and D ′ are mutually independent. 1 1 2 2 Proof. A skew lattice satisfying precisely D , D ′ and D ′ is given by the tables: 1 1 2 ∨ a b 0 ∧ a b 0 a a a a and a a b 0 . b b b b b a b 0 0 a b 0 0 0 0 0 That D ′ is not satisfied is seen by ( a 2 ∧ 0) ∨ b = b ≠ a = ( a ∨ b) ∧ (0 ∨ b). Replacing (∨, ∧) by (∨*, ∧*) or (∧, ∨) or both (∧*,∨*) gives examples of the other ways that a skew lattice can satisfy just three of the four identities above. £ Being fully distributive is powerful. To see this, observe first that distributive lattices are fully distributive as well as flat antilattices of any of the four types. Hence any direct product of algebras from these five types is also fully distributive. We shall see in Corollary 3.5.3 that for quasilattices (and hence skew lattices) this is all. Thus, any fully distributive quasilattice factors as the product of a distributive lattice and up to four flat antilattices. We next consider two other sets of distributive identities that while less powerful, have a broader scope. A noncommutative lattice is bidistributive if it satisfies the slightly weaker pair of identities: D3 : a ∧ ( b ∨ c) ∧ d = ( a ∧ b∧ d) ∨ ( a ∧ c ∧ d). D3 ʹ: a ∨ ( b ∧ c) ∨ d = ( a ∨ b∨ d) ∧ ( a ∨ c ∨ d). Distributive lattices and arbitrary antilattices are bidistributive, as are their direct products. By Theorem 3.5.2 below, this is all among quasilattices. (In this more general case the antilattice factor need not be completely factored into a product of flat antilattices.) 34 I: Preliminaries Finally, a noncommutative lattice is distributive if it satisfies the even weaker pair of identities: D4 : a ∧ ( b ∨ c) ∧ a = ( a ∧ b∧ a) ∨ ( a ∧ c ∧ a). D4ʹ: a ∨ ( b ∧ c) ∨ a = ( a ∨ b∨ a) ∧ ( a ∨ c ∨ a). Among skew lattices, D4 and D4ʹ play an important role. For instance, skew lattices in rings are distributive, as are skew Boolean algebras. Moreover, as will be seen in Theorem 3.5.1, a distributive, noncommutative lattice is a paralattice if and only if it is a quasilattice. It thus follows that the type of distributivity expressed by D4 and D4ʹ is of maximal general use. Theorem 1.3.10. Among skew lattices identities D3 and D3 ʹ are independent. Proof. The skew lattice example of Theorem 1.3.8 satisfies D3 but not D3 ʹ. Switching the ∧ and ∨ tables provides the complementary example satisfying D3 ʹ but not D3. £ Theorem 1.3.11. (Spinks [2000]) Among skew lattices, D4 and D4ʹ are independent. Proof. Spinks gave the following example of a 9-element skew lattice satisfying D4 but not D4ʹ. 1     d – e f – g (D is indicated by ‘–’;        and  denote ≥) a b – c    0 Both 1 and 0 behave as they ought. Otherwise, the operations are described by the partial tables: ∨ a b c d e f g ∧ a b c d e f g a a d e d e 1 1 a a 0 0 a a 0 0 b d b b d d f f b 0 b c b c b c c e c c e e g g c 0 b c b c b c . d d d d d d 1 1 d a b c d e b c e e e e e e 1 1 e a b c d e b c f 1 f f 1 1 f f f 0 b c b c f g g 1 g g 1 1 g g g 0 b c b c f g In particular, a ∨ ( d ∧ g) ∨ a = a ∨ c ∨ a = e ≠ d = d ∧ 1 = ( a ∨ d ∨ a) ∧ ( a ∨ g ∨ a). Switching the ∧ and ∨ provides a 9-element example satisfying D4ʹ but not D4. £ Spinks’ examples are minimal examples. In all cases of order ≤ 8, D4 is equivalent to D4ʹ . Under the added condition of symmetry ( x∨ y = y∨ x iff x∧ y = y∧ x), Spinks found a machine-generated proof that for skew lattices, D4 is equivalent to D4ʹ. In Section 5.2 we give a short “normal” proof of this fact. 35 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Enriched structures We have already encountered 1 and 0. As is the case for lattices 1, if it exists, is the unique element at the top of the noncommutative lattice and 0, if it exists, is the unique element at the bottom, whatever the (quasi-)ordering may be. This is given expression by the following sets of identities. 1 ∨ a = 1 = a ∨ 1 and 1 ∧ a = a = a ∧ 1. 0 ∧ a = 0 = a ∧ 0 and 0 ∨ a = a = a ∨ 0. For skew lattices (and for paralattices in general) both second pairs of identities is redundant. Paralattices, and particularly skew lattices, have a coherent natural partial order ≤ given as either ≤(∧) or the dual of ≤(∨). It may be that two elements x, y in a paralattice possesses a natural meet that is maximal among all z such that both x, y ≥ z. Dually they may also possess a natural join that is minimal among all z such that both x, y ≤ z. To distinguish natural meets and natural joins from the given operations ∧ and ∨, we shall refer to them as intersections and unions of elements respectively, employing the notation x∩ y and x∪ y. Unless x and y commute, one has x∧ y > x∩ y and x∪ y > x∨ y. Theorem 1.3.12. The natural meet operation in a paralattice is characterized by the identities: NM1: x∩ x = x; NM2: x∩ y = y∩ x; NM3: ( x ∩ y) ∩ z = x ∩ ( y∩ z); NM4: x ∧ ( x∩ y) = x∩ y = ( x∩ y) ∧ x; NM5: x ∩ ( x∧ y∧ x) = x∧ y∧ x. Dual identities characterize a natural joins. £ Theorem 1.3.13. Given a paralattice (P; ∨, ∧) with a natural meet ∩, the enriched algebra (P; ∨, ∧, ∩) has a distributive congruence lattice. Proof. Setting m( x, y, z) = ( x∩ y)∨( y∩ z)∨( x∩ z), one has m( x, x, y) = m( x, y, x) = m( y, x, x) = x. But this implies its congruence lattice is distributive. (Theorem II.12.3 in Burris and Sankappanavar [1981].) £ Rectangular bands and rectangular skew lattices A rectangular skew lattice is a skew lattice (S; ∧, ∨) for which (S; ∧) and (S; ∨) are rectangular bands. Equivalently, its is an antilattice that is also a skew lattice. Given a 36 I: Preliminaries rectangular skew lattice, from x ∧ ( x ∨ y) = x, it follows that x ∨ y = u ∧ x for some u. Similarly, from ( x ∨ y) ∧ y = y, it follows that x ∨ y = y ∧ v for some v. Combining we get x ∨ y = ( x ∨ y) ∧ ( x ∨ y) = y ∧ v ∧ u ∧ x = y ∧ x. Conversely, given an algebra (S; ∧, ∨) for which both (S; ∧) and (S; ∨) are rectangular bands and x∨ y = y∧ x holds on S, it is easily seen that all four relevant absorption identities must hold, making (S; ∧, ∨) a rectangular skew lattice. We thus have: Proposition 1.3.14. The variety of rectangular bands is term equivalent to the variety of rectangular skew lattices. Thus a map between rectangular bands f: A → B is a homomorphism of bands if and only if it is a homomorphism between their induced rectangular skew lattices. Likewise an equivalence on a rectangular band is a band congruence if and only if it is a congruence on the derived skew lattice. £ We continue by considering the case of right zero bands ( xy = y) , and their derived right rectangular skew lattices (where x∧ y = y = y∨ x). The following result is trivial. Proposition 1.3.15. Given right zero bands B and Bʹ , each function f: B → Bʹ is a homomorphism, and thus each equivalence relation on B is a congruence. (Similar remarks hold also for left zero bands where xy = x.) £ We next consider the case of a factored rectangular band B = L × R, where L and R are left zero and right zero bands, respectively. (All rectangular bands are isomorphic to such a factorization.) Theorem 1.3.16. Given a factored rectangular band B = L × R , and a left zero band Lʹ . Then every homomorphism from B to Lʹ has the form f( l, r) = λ( l) where λ is any function from L to Lʹ. Dually, every homomorphism from B to a right zero band Rʹ has the form f( l, r) = ρ( r) where ρ is any function from R to Rʹ. Finally, given a second factored rectangular band Bʹ = Lʹ × Rʹ , all homomorphisms from B to Bʹ are of the form f( l, r) = (λ( l), ρ( r)) where λ is any function from L to Lʹ and ρ is any function from R to Rʹ. Proof. Given a homomorphism from f: L × R → Lʹ, since homomorphisms send R-classes to R- classes, f is constant on each R-class { l} × R and thus must factor through the left factor L, leading to a chain of homomorphisms L × R → L → Lʹ whose composite f must be a homomorphism. Here L × R → L is just the projection onto L. The second map is the induced map λ: L → Lʹ that must be a homomorphism. £ Corollary 1.3.17. Given a factored rectangular band B = L × R , each congruence θ on B has the form ( l, r) θ ( lʹ , rʹ) iff l θL lʹ and r θR rʹ for any pair of equivalences θL and θR on L and R respectively. £ 37 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond The internal perspective. Given rectangular bands B and Bʹ, choose elements b ∈ B and bʹ ∈ Bʹ as base points. Each x ∈ B is of the unique form x = lr where l ∈ L b and r ∈ R b. Indeed, x factors as ( xb)( bx) where xb ∈ L b and bx ∈ R b. Again, this factorization is unique relative to the base point b. Clearly similar remarks hold for Bʹ and bʹ. Next, let λ: L b → L b and ʹ ρ: R b → R bʹ be functions, both of which are trivially homomorphisms between their restricted domains. Finally define f: B → Bʹ by f( x) = λ( xb)ρ( bx), or equivalently, f( x) = λ( xb) bʹρ( bx). Theorem 1.3.18. As defined, f is a homomorphism from B to Bʹ. Conversely, every homomorphism from B to Bʹ arises in this manner. Finally, f is 1-1 or onto if and only if both λ and ρ are. (Of course, the choice of b and bʹ is part of “this manner.”) Proof. We first show that f as defined is a homomorphism from B to Bʹ. f( xy) = λ( xyb) bʹρ( bxy) = λ( xb) bʹρ( by) = λ( xb)ρ( by) whilst f( x) f( y) = λ( xb) bʹρ( bx)λ( yb) bʹρ( by) = λ( xb)ρ( by). Conversely, if f: B → Bʹ is a homomorphism, then upon picking some b in B and setting bʹ = f( b), for any x in B we get f( x) = f( xbbbx) = f( xb) bʹ f( bx) = λ( xb) bʹρ( bx) where the functions λ and ρ are the restrictions of f to L b and R b respectively. £ A primitive skew lattice is a skew lattice consisting of two rectangular algebras A > B, where b∧ a∧ b = b or dually a∨ b∨ a = a, for all a ∈ A and b ∈ B. We are especially interested in the case when B = {0}. We denote such an algebra by A0. Suppose we are given two such algebras, A0 and B0. Then the following assertions follow from the behavior of 0 and the results above Proposition 1.3.19. Given primitive skew lattices A0 and B0, Hom(A0, B0) consists of (i) all constant maps from A0 to B0 and (ii) all maps f from A0 to B0 for which f(0A) = (0B) and the restriction f|A is a homomorphism from A to B. (Thus the nontrivial part of f is governed by the conclusions of the preceding results.) Corollary 1.3.20. Given a primitive skew lattice A0, the nontrivial congruences on A0 ( where not all elements are related), are equivalences for which {0} forms a single equivalence class and their restriction to A is a congruence on the subalgebra A. 38 I: Preliminaries References S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York, 1981. G. Laslo, The classification of noncommutative lattices according to absorption laws, Proc. of the 1997 Annual Meeting of the Romanian Soc. of Math. Sciences, 77 – 84. G. Laslo and I. Cozac, On noncommutative generalizations of lattices, Mathematica 41 (1998), 186 – 196. G. Laslo and J. Leech, Green’s equivalences on noncommutative lattices, Acti Sci. Math (Szeged) 68 (2002), 501-533. (Much in Section 1.3 has its origin in this paper.) M. Spinks, On middle distributivity for skew lattices, Semigroup Forum 61 (2000), 341-345. 39 II: SKEW LATTICES We take a closer look at skew lattices, i.e., at algebras (S; ∨, ∧) with associative binary operations ∨ and ∧ satisfying the absorption identities: B1: a ∧ ( a ∨ b) = a. C1: a ∨ ( a ∧ b) = a. B2: ( b ∨ a) ∧ a = a. C2: ( b ∧ a) ∨ a = a. By Theorem 1.3.4, both ∨ and ∧ are idempotent: x ∨ x = x = x ∧ x. Equivalently, skew lattices are characterized as the double bands (S; ∨, ∧) satisfying the dualities: a ∨ b = a if and only if a ∧ b = b. a ∨ b = b if and only if a ∧ b = a. In this chapter a basic theory for these algebras is developed. For every statement about skew lattices, clearly a parallel statement holds for skew* lattices. Thus, from this point on, the latter will not be mentioned until more general structures are considered in Chapter 3. We begin in Section 1 with some fundamental results about skew lattices. Of particular importance are two core structural results for skew lattices: analogues of the Clifford-McLean Theorem and the Kimura Factorization Theorem (Theorems 2.1.2 and 2.1.5), given originally for bands and regular bands respectively (Theorem 1.2.6 and what follows). We also initiate our study of skew lattices of idempotents in rings, a source of both examples and conceptual motivation, in Theorems 2.1.7 and 2.1.9. (In this case, ∧ and ∨ are given first as e∧ f = ef and e∨ f = e +f – ef.) Due to the latter theorem, a band can be embedded into some skew lattice as a sub-band of its ∧-reduct (or of its ∨-reduct) if and only if the band itself is regular (Theorem 2.1.10). In Section 2 we consider the role of commutativity in skew lattices. The center of a skew lattice S, Z(S) = { e ∈ S⎪ e∨ x = x∨ e and e∧ x = x∧ e for all x ∈ S} is characterized in Theorem 2.2.2 as the sublattice of all elements that form singleton D-classes. We look at the important property of symmetry ( a∨ b = b∨ a iff a∧ b = b∧ a) and some of its consequences. These include Theorem 2.2.10 that asserts that any symmetric skew lattice with a countable maximal lattice image has a lattice section (that is, a sublattice meeting each D-class of S at a single point). Example 2.2.2 41 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond exhibits a nonsymmetric 13-element skew lattice with a generating set of 3 mutually commuting elements (under both ∨ and ∧), that has noncommuting pairs of elements. In the third section we consider normal skew lattices, i.e., skew lattices whose ∧-reducts are normal in that x∧ y∧ z∧ w = x∧ y∧ z∧ w. Of special interest are distributive, symmetric, normal skew lattices characterized in Theorem 2.3.2 by identities a∧( b ∨ c) = ( a∧ b) ∨ ( a∧ c) and ( a ∨ b)∧ c = ( a∧ c) ∨ ( b∧ c). This strengthened form of distributivity is called strong distributivity. Thanks to Theorem 2.3.6, every normal skew lattice of idempotents in a ring is strongly distributive. In this case the operations are given by e∧ f = ef again, but e∨ f = ( e +f – ef)2 = e +f +fe – efe – fef. Of course when e +f – ef is idempotent, both outcomes agree. Strongly distributive skew lattices are also of interest due to their connections to skew Boolean algebras, the subject of Chapter 4. Suffice it to say here that a skew lattice can be embedded into (the skew lattice reduct of) a skew Boolean algebra precisely when it is strongly distributive. In Section 4 we engage in a detailed study of the natural partial order ≥ on a skew lattice. This study is based on the behavior of primitive skew lattices consisting of exactly two D-classes, A > B. Primitive skew lattices have a simple description given in terms of A-cosets arising in B and B-cosets arising in A and the coset bijections between these cosets induced by ≥. (See Theorem 2.4.1) As a consequence, primitive skew lattices of the most general type are easily manufactured. Moreover, the interaction between the various maximal primitive subalgebras of a skew lattice says much about the behavior of the entire algebra. (See, e.g., Theorem 2.4.9 and its consequences.) We also look at skew chains of comparable D-classes A > B > C and consider the case where coset bijections between cosets in the outer classes are compositions of successive intermediate coset bijections (as is the case for skew lattices in rings). When this always occurs in a skew lattice, it is said to be categorical. In Section 5 we continue the analysis of skew lattices by their primitive subalgebras begun in the previous section. But here we pass from coset bijections between comparable pairs of cosets in (usually) distinct D-classes to coset projections from an entire D-class onto a coset in a comparable D-class. The individual projections are obtained by combining all coset bijections from one D-class that share a common coset of outputs in the other class. Even if the skew lattice itself is not categorical in the above sense, these downward (or upward) projections taken collectively along with the involved D-classes, form a category. (See Theorem 2.5.4.) All this is developed in the fifth section and then applied to give a general description of a normal skew lattice in Theorem 2.5.7. In Section 6 we study decompositions of (mostly symmetric) normal skew lattices. After some preliminary cases, theorems of general character are given. For instance, the Reduction Theorem (2.6.9) implies that every symmetric normal skew lattice can be embedded in the product of its maximal lattice image and its maximal distributive image. The Primary Decomposition Theorem (2.6.11) tells how a strongly distributive (hence symmetric and normal) 42 II: Skew Lattices skew lattice with a finite maximal lattice image must factor into a fibered product of primary factors, the latter being algebras of rather simple type. In general, strongly distributive skew lattices can be embedded in powers of a special primitive skew lattice 5, a noncommutative 5-element variant of the lattice 2 for which the latter is its maximal lattice image. (See Theorem 2.6.12.) Finally, the material presented in this chapter first appeared in the papers referenced at the end of this chapter. 2.1 Fundamental results Any noncommutative lattice has two D-equivalences, D(∨) and D(∧), with each being a congruence with respect to its associated operation. For skew lattices we have: Lemma 2.1.1. Given a skew lattice, R(∨) = L(∧), L(∨) = R(∧), and D(∨) = D(∧). Thus the common equivalence D is a congruence of skew lattices. Proof. That R(∨) = L(∧) and L(∨) = R(∧) follow immediately from the above dualities. Hence: D(∨) = L(∨)oR(∨) = R(∧)oL(∧) = D(∧). £ Thus for any skew lattice (S; ∨, ∧) we set R = R(∧) = L(∨) and L = L(∧) = R(∨). A skew lattice is rectangular if either (S, ∨) or (S, ∧) is a rectangular band in which case, thanks to the above dualities, both are rectangular bands with x∨ y = y∧ x. Put otherwise, a rectangular skew lattice is precisely an antilattice for which R(∨) = L(∧) and L(∨) = R(∧). Theorem 2.1.2. (The Clifford-McLean Theorem for skew lattices). Given a skew lattice (S; ∨, ∧), the equivalence D is a congruence, S/D is the maximal lattice image of S and all congruence classes of D are maximal rectangular skew lattices in S. Proof. This follows from the above lemma and Corollary 1.3.6. £ Lemma 1.2.3, Theorem 1.3.3 and the above result give us: Lemma 2.1.3. Given elements a and b in a skew lattice S, a ≥(∧) b iff b ≥(∨) a and a ≻ (∧) b iff b ≻ (∨) a. £ Thus the natural partial order on any skew lattice is given by ≥ = ≥(∧) = ≤(∨) and the natural quasiorder on any skew lattice is given by ≻ = ≻ (∧) = ≺ (∨). 43 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Lemma 2.1.4. On any skew lattice S , both R and L are congruences. In particular, both algebraic reducts (S, ∨) and (S, ∧) are regular bands and given e ≻ a, b ≻ f in S: a ∨ f ∨ b = a ∨ b and a ∧ e ∧ b = a ∧ b. Proof. Given a skew lattice S, we first show that (S, ∧) satisfies the class covering condition. Indeed, given comparable D-classes A > B in S with b an arbitrary element in B, then for any a in A, b ≥(∨) b∨ a∨ b ∈ A. Thus b∨ a∨ b ≥(∧) b and (S, ∧) is seen to satisfy the class covering condition. Thus by Theorem 1.2.16, both L(∧) and R(∧) are ∧-congruences. Similarly, both L(∨) and R(∨) are ∨-congruences. The first statement follows now from Lemma 2.1.1. Thus both band reducts are regular; moreover, by Theorem 1.2.7, the dual conditional identities must hold. £ A skew lattice for which D = R [D = L] is said to be right-handed [ left-handed]. If S is also rectangular, then it is right-rectangular [ left-rectangular]. Remarks following Theorem 1.2.27 give us: Theorem 2.1.5. ( Kimura factorization for skew lattices). Given a skew lattice S, S/R is the maximal left-handed image of S, S/L is the maximal right handed image of S and the commuting diagram S ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎯ S/L ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ↓ ↓ S/R ⎯⎯⎯⎯⎯⎯⎯⎯ → ⎯ S/D is a pullback diagram and S factors as the fibered product S ≅ S/R ×S/D S/L. £ An embedding of a band B into a skew lattice S is a semigroup embedding of B into the band (S; ∧). Clearly for a band to be embedded into a skew lattice, the band must be regular. It turns out that this condition is sufficient. Our easy proof of this fact brings us into the subject of skew lattices in rings. We begin with the following considerations. Given a ring R with set of idempotents E(R), a band in R is any subset of E(R) that is closed under multiplication. Clearly: Proposition 2.1.6. Let R be a ring with identity 1 such that any descending chain e 1 ≥ e 2 ≥ … in E(S) eventually stabilizes: en = en+1 =… for some n ≥ 1. Then the D- classes of any band B in R such that 1 ∈ B are lattice-ordered. (This is the case for matrix rings over fields.) £ 44 II: Skew Lattices {1}    ¢ ¢ ¢     ¢ ¢   ¢ ↓ {0} Such cases naturally raise the question as to whether a counter-operation ∨ exists such that the join class D e ∨ D f of two D-classes, D e and D f, is given as D e∨ f. This leads to a simple but important observation: Theorem 2.1.7. Any multiplicative band B in a ring that is also closed under the circle operation x ○ y = x + y – xy has the following properties: 1. B is also a band under ○. 2. x( x ○ y) = x = ( y ○ x) x and x ○ ( xy) = x = ( yx) ○ x hold on B: Thus (B; ○, •) is a skew lattice. In this case the elements in B also satisfy the identity: 3. xyx + yxy = xy + yx. Proof. To see (1), note that ○ is always associative and x○ x = x + x – x 2 = x if and only if xx = x. Thus if B is closed under ○, then (B, ○) is indeed a band. (2) is straightforward. For instance, x( x ○ y) = xx + xy – xxy = x + xy – xy = x and x ○ ( xy) = x + xy – xxy = x + xy – xy = x. Given the assumptions, x + y – xy = x ○ y = ( x ○ y)2 = ( x + y – xy)2 = x + yx + y – yxy – xyx, and (3) follows. £ The terms in (3) form a rectangular subalgebra for which the diagonal sums are equal. xyx R xy L L yx R yxy Two classes of bands closed under ○ are given in the next theorem. But first a lemma: Lemma 2.1.8. Let B be a right regular band in a ring R and let e, f ∈ B. Then: 1) e + f – ef ∈ E(R), and 2) B  { e + f – ef } generates a right regular band. 45 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. (1) Given e, f ∈ B, since B is a right-regular band, ( e + f – ef)2 = e + ef – ef + fe + f – fef – efe – ef + ef = e + fe + f – ef – fe = e + f – ef. To see (2), observe first that for all g, h ∈B, ( e + f – ef) g( e + f – ef) = ege + egf – egef + fge + fgf – fgef – efge – efgf + efgef = ge + egf – gef + fge + gf – gef – fge – egf + gef = ge – gef + gf = g( e + f – ef) Thus any product g 0( e + f – ef) g 1( e + f – ef) g 2 … gn–1( e + f – ef) gn generated from B  { e + f – ef } reduces to g 0 g 1 g 2… gn–1( e + f – ef) gn. From this observation, (2) follows. £ Theorem 2.1.9. Every maximal right [left] regular band in a ring R is also closed under the circle operation e ○ f = e + f – ef and thus forms a skew lattice in R. £ In general, maximal regular bands in rings need not be closed under ○. We give an example of this in Section 2.3. Nonetheless, what we have seen thus far suffices to prove: Theorem 2.1.10. A band can be ∧- embedded in a skew lattice if and only if it is a regular band. Proof. The condition is clearly necessary. Suppose that B is a regular band. By the Kimura Decomposition B can be embedded in B/R × B/L. In the semigroup ring &[B/R], B/R generates a regular band C1 that is closed under the circle operation. Likewise B/L generates a regular band C2 in &[B/L] that is closed under the circle operation. Thus B/R × B/L is a sub-band of C × 1 C2 in &[B/R] × &[B/L]. Since C × is closed under the circle operation it is a skew lattice and B 1 C2 is embedded in C × . £ 1 C2 Thus, even if a maximal regular band in a ring does not form a skew lattice in that ring, some copy of it will generate a skew lattice under ○ and multiplication in another ring. We complete this section with an important elementary fact about skew lattice in rings. Recall that a skew lattice is distributive if it satisfies identities D3 and D4 in Section 1.3. Theorem 2.1.11. Skew lattices in rings (using multiplication and ○ ) are distributive. Proof. a ∧ ( b ∨ c) ∧ a = a( b + c – bc) a = aba + aca – abca = aba + aca – abaca (regularity) = ( a ∧ b∧ a) ∨ ( a ∧ c ∧ a) . For the dual identity, D4, observe first that a∨ b∨ a = a + b – ab – ba + aba. Thus a ∨ ( b∧ c) ∨ a = a + bc – abc – bca + abca. 46 II: Skew Lattices Expanding ( a ∨ b ∨ a) ∧ ( a ∨ c ∨ a) and then reducing, we get: ( a ∨ b ∨ a) ∧ ( a ∨ c ∨ a) = a + bc – bca – abc – bac +baca + abac. Equating both outcomes and then cancelling corresponding pairs of identical terms we are left with abca = – bac +baca + aba c. Rearranging gives abca + bac = baca + aba c, which is true, since it is a case of Theorem 2.1.7(3) with x = baca and y = abac, so that xy = bac, yx = abca, xyx = baca again and yxy = abac again. £ An alternative proof that D4 holds follows from D3 holding and the fact that skew lattices in rings are symmetric (Theorem 2.2.6). This forces D4 to hold also. (See Section 5.2.) 2.2 Instances of commutative behavior Skew lattices, like many noncommutative structures, can possess abundant instances of commutativity. Indeed, selective instances of commutativity play an important role in their basic theory. We begin a pair of results to this effect. Theorem 2.2.1. Let S be a skew lattice and let A and B be D- classes in S with join class J = A∨B and meet class M = A∧B. Then given v ∈ J with v ≥ a ∈ A and v ≥ b ∈ B, a∨ b = v = b∨ a. Similarly, given m ∈ M such that a ≥ m and b ≥ m for a ∈ A and b ∈ B , a∧ b = m = b∧ a. Thus J = { a∨ b⎮ a ∈ A, b ∈ B & a∨ b = b∨ a} and M = { a∧ b⎮ a ∈ A, b ∈ B & a∧ b = b∧ a}. Moreover, for every a ∈ A there exist b, bʹ ∈ B such that a∨ b = b∨ a in J and a∧ bʹ = bʹ∧ a in M. Proof. Given v ∈ J, let a ∈ A and b ∈ B be such that v ≥ a, b. (Both a and b exist since for any x in A and y in B, v ≥ v∧ x∧ v in A and v ≥ v∧ y∧ v in B.) For this a and b we have a∨ b ∈ J so that a∨ b = a∨ b∨ v∨ a∨ b = a∨ v∨ b = v. Similarly, b∨ a equals v also and the assertion about J is seen. The case for M is similar. For the final assertion, pick a in A and let v ∈ J and m ∈ M be such that v ≥ a ≥ m. Now apply remarks from the first part of the proof. £ This theorem has the important corollary: Theorem 2.2.2. Given an element e in a skew lattice S, the following are equivalent: 1) D e = { e}. 2) For all x ∈ S, e∨ x = x∨ e. 3) For all x ∈ S, e∧ x = x∧ e. The subset of such elements forms a sublattice Z(S) of S (called the center of S). 47 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. First, if e either ∨-commutes with all x ∈ S or else ∧-commutes with all x ∈ S, then in particular it does such with all elements in D e, which forces D e to be trivial. Conversely, if D e is known to be { e}, then by the final assertion of Theorem 2.2.1 e must commute under both operations with all elements of S. £ While an element that join commutes with all elements in a skew lattice also meet commutes with all elements (and conversely), in general two elements commuting under one operation need not commute under the other operation. This is evident in the following example. Example 2.2.1a. Consider the right-handed skew lattice defined by the Cayley tables ∨ − 0 a ∧ j j m′ bn′ jp′ 0 am′ bn′ jp′ 1 2 0 0 am′ bn′ jp′ 0 0 0 0 0     a a 0 a m am am jm jm and m 0 am′ p′ . a − − 1 a 2 b 1 b 2 bn bn jn bn jn bn 0 0 bn′ bp′   jp jp jp jp jp jp 0 am′ bn′ jp′ 0 This skew lattice is jointly determined by being right handed with the displayed Clifford-Mclean picture and having ≥ given by jn ≥ both an, bn ≥ 0 for n = 1 or 2. While a 1 and b 2 ∧-commute, they clearly do not R,0 ∨-commute. We denote this example by NS . (See Section 5.2) £ 7 A skew lattice S is symmetric if for all e, f ∈ S, e∨ f = f∨ e iff e∧ f = f∧ e. A symmetric alternative to the above example is as follows: Example 2.2.1b. The direct product of the right-handed algebras { a , , 1 a 2 } and { b 1 b 2} is   {0} {0} symmetric. Its D-class diagram (with “redundant” 0-coordinates suppressed) is as follows. The partial ordering between the top class and the intermediate classes given by coordinate projection. {( a ), 1, b 1), ( a 1, b 2 ( a 2, b 1), ( a 2, b 2 )}   { a , , . £ 1 a 2 } { b 1 b 2}   0 Two elements in a skew lattice commute if they commute under both operations. If they just commute under a single operation, they are said to ∧-commute or ∨-commute, as the case may be. The following statement is easily verified. 48 II: Skew Lattices Proposition 2.2.3. Any set of mutually commuting elements in a symmetric skew lattice must generate a sublattice of the skew lattice. £ This is not the case with nonsymmetric skew lattices. Example 2.2.2. Consider the 13-element right-handed skew lattice generated from a, b and cʹ. In the diagram, “---“ indicates R-equivalence and the slanted lines indicate the natural partial ordering. Thus in particular e > a, d; eʹ > aʹ, dʹ; f > d, c; and f ʹ > dʹ, cʹ. 1   e - - - e′ f - - - f ′     a - - - a′ d - - - d′ c - - - c′     b     0 Here a∨ b = e = b∨ a, b∨ cʹ = fʹ = cʹ∨ b and a∨ cʹ = 1 = cʹ∨ a, while all three pairs of elements meet at 0. Collectively, a, b and cʹ generate S. But a∨ b = e does not ∧-commute with b∨ cʹ = fʹ. (We will see in Section 2.4 how the natural partial ordering coupled with the D-class diagram can determine the outcome of either operation.) Since b is central and { a, cʹ} generates the sublattice {1, a, cʹ, 0}, we see that a central element and a sublattice of a nonsymmetric skew lattice neednot generate a (possibly larger) sublattice. £ Theorem 2.2.4. Symmetric skew lattices form a subvariety of skew lattices characterized by the identities: x∧ y∧( x∨ y∨ x) = ( x∨ y∨ x)∧ y∧ x and x∨ y∨( x∧ y∧ x) = ( x∧ y∧ x)∨ y∨ x. Proof. Given elements x and y in a skew lattice, x will ∨-commutes with ( x∨ y∨ x)∧ y∧( x∨ y∨ x), the join being x∨ y∨ x. Moreover, if x ∨-commutes with y then ( x∨ y∨ x)∧ y∧( x∨ y∨ x) = y. Thus all elements in S that ∨-commute with x have the form ( x∨ y∨ x)∧ y∧( x∨ y∨ x). Likewise, all elements in S that ∧-commute with x are of the form ( x∧ y∧ x)∨ y∨( x∧ y∧ x). Hence symmetric skew lattices are characterized by the identity x∧( x∨ y∨ x)∧ y∧( x∨ y∨ x) = ( x∨ y∨ x)∧ y∧( x∨ y∨ x)∧ x and its dual. Thus the displayed identities follow from absorption. £ Corollary 2.2.5 . Symmetric right-handed skew lattices are characterized by the two identities: x∧ y∧( x∨ y) = y∧ x and x∨ y = ( y∧ x)∨ y∨ x. 49 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. If S is a right-handed symmetric skew lattice, then it satisfies the identities of the above theorem plus the identities x∨ y∨ x = x∨ y and x∧ y∧ x = y∧ x from which the displayed identities follow. Conversely, if the displayed identities hold on a skew lattice S, then one has x∧ y∧ x = x∧( y∧ x) = x∧[ x∧ y∧( x∨ y)] = x∧ y∧( x∨ y) = y∧ x and similarly x∨ y∨ x = x∨ y so that S is right-handed. But using x∨ y∨ x = x∨ y and x∧ y∧ x = y∧ x, the displayed identities can be transformed back into the identities of the previous theorem. £ Our interest in symmetry is due further to the following three theorems. Theorem 2.2.6. All skew lattices in rings (using multiplication and the circle operation) are symmetric. Proof. Obviously a + b – ab = b + a – ba if and only if ab = ba. £ A lattice section in a skew lattice S is any sublattice T of S having nonempty intersection with each D-class of S, in which case, T ≅ S/D. Theorem 2.2.7. If S is a symmetric skew lattice for which S/D is countable, then S has a lattice section. Proof. Let D1, D2, … be a listing of all D-classes. Pick x 1 ∈ D1. By Theorem 2.2.1 we know that in every D-class of S elements exist that commute with x. By symmetry we know that such commuting is under both operations and, moreover, that the set of all such elements is closed under both operations. Thus S1 = { y ∈ S⎮ y commutes with x} is a sub-skew lattice of S that has nonempty intersection with each D-class of S and for which D = { x x 1} as a D-class in S1. If S1 1 is not a lattice (and hence not a lattice section of S) we find an element x 2 in D2∩S1 and thus not in D in S x 1. The argument repeats. We thus obtain a descending chain of sub-skew lattices 1 S1 ⊇ S2 ⊇ S3 ⊇ … each having nonempty intersection with each D-class of S, and such that D j ∩ S i = { xj} for all i ≥ j where each xj commutes with all xi that precede it. Clearly, the full intersection  is a n≥1S n lattice section of S. £ A left-handed section of S is any left-handed sub-skew lattice SL of S such that its intersection with each D-class in S is an L-class. A right-handed section SR of S is similarly defined. An internal (subobject) variant of the Kimura decomposition is the following result: 50 II: Skew Lattices Theorem 2.2.8. (Cvetko-Vah) Given a skew lattice the following are equivalent: 1) S has both a left-handed section and a right-handed section. 2) Sub-skew lattices SL and SR exist whose intersection with any D -class is an L -class, or respectively, an R -class of S . 3) S has a lattice section S0. When these conditions hold, then: 4) The natural epimorphisms S → S/R and S → S/L induce isomorphisms of SL with S/R and SR with S/L. 5) Every x ∈ S factors uniquely as a = aʹ∧ aʺ with aʹ ∈ SL ∩ D x, aʺ ∈ SR ∩ D x. Under this decomposition ( aʹ∧ aʺ)∧( bʹ∧ bʺ) = ( aʹ∧ bʹ)∧( aʺ∧ bʺ) and ( aʹ∧ aʺ)∨( bʹ∧ bʺ) = ( aʹ∨ bʹ)∧( aʺ∨ bʺ). 6) The functions πL: S → SL and πR: S → SR defined by πL( a) = aʹ and πR( a) = aʺ are retractions of S upon SL and SR respectively and the commuting composite πLπR is a retraction of S upon S0; moreover R = ker(πL), L = ker(πR) and D = ker(πLπR). Proof. (2) clearly implies (1). Given (1), expand Sʹ to the subset SL = ∪{L a ⎜ a ∈ Sʹ} and expand Sʺ to the subset SR = ∪{R a ⎜ a ∈ Sʺ}. Since L and R are congruences, SL and SR are as stated in (2). Given (2) again, since every pair of L and R-classes in a D-class must meet at a single element in that D-class, (3) follows from (2). Conversely, given (3) one has a minimal case of (1). Thus (1) – (3) are equivalent. Since SL is an R-class cross-section, the map S → S/R restricts to an isomorphism of SL upon S/R. Similar remarks apply to SR → S/L and (4) follows. (5) follows from (4) and the embedding S ≅ S/R ×S/D S/L ⊆ S/R × S/L ≅ SL × SR given by a = aʹ∧ aʺ → ( aʹ, aʺ). To see (6), first set a L = a∧ a 0 and a R = a 0∧ a. This gives a∧ b = ( a∧ a 0∧ a)∧( b∧ b 0∧ b) while ( a L∧ b L)∧( a R∧ b R) = ( a∧ a 0∧ b∧ b 0)∧( a 0∧ a∧ b 0∧ b) = ( a∧ a 0∧ b)∧( a∧ b 0∧ b). But both “middles” are equal since a 0∧ a∧ b∧ b 0 = a 0∧ a∧ b∧ a 0∧ b 0∧ a∧ b∧ b 0 = a 0∧ y∧ a 0∧ b 0∧ a∧ b 0 = a 0∧ b∧ a∧ b 0. Thus a∧ b = ( a L∧ b L)∧( a R∧ b R) follows. Recalling that R(∧) = L(∨) and L(∧) = R(∨), so that a = a R∨ a L and b = b R∨ b L gives: a∨ b = ( a R∨ b R) ∨ ( a L∨ b L) = ( a L∨ b L) ∧ ( a R∨ b R). £ In the case where S has a lattice section, Theorem 2.2.8 (4)-(6) describes an internal (monic) Kimura decomposition of S in contrast to the external (epic) Kimura decomposition of 51 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 2.15. Both decompositions are similar in that they both involve copies of the maximal lattice, left-handed and right-handed images of S. While external decompositions always occur and use quotient objects, internal decompositions that use subobjects need not always occur. One is led to ask the following question. Is symmetry really needed for the all this? Otherwise put, does a skew lattice S exist for which S/D is countable, or even finite, but S has no lattice section? Example 2.2.2 continued. Returning to the non-symmetric Example 2.2.2, a lattice section is given by the subset {0, a, b, c, d, f, 1}. Some elementary, albeit useful, cases where symmetry is not needed are as follows. Theorem 2.2.9. A skew lattice S will have a lattice section T if its maximal lattice image S/D is a copy of one of the following lattices: 1 1 1        • (i) 2 × 2 • • (ii) M3 • • • (iii) N5 • •        0 0 0 In these cases, S also has internal copies of S/R and S/L, both occurring as maximal left and right-handed subalgebras of S. Proof. First, suppose S/D is a copy of 2 × 2. Let a, b, e, f in S be such that e is in the top D-class, f lies in the bottom D-class, and a and b are separately in the two incomparable D-classes. We create a lattice section as follows. 1. Reset a to be e∧ a∧ e, b to be e∧ b∧ e and f to be e∧ f∧ e so that a, b, f < e. 2. Reset a to be f∨ a∨ f and b to be f∨ b∨ f so that f < a, b < e is a lattice section. Next suppose S/D is a copy of M3. Let a, b, c, e, f in S be such that e is in the top D-class, f is in the bottom D-class, and a, b and c lie separately in the three incomparable D-classes. We create a lattice section as follows. 1. Reset all x among a, b, c, f to be e∧ x∧ e so that now a, b, c, f < e. 2. Reset all y among a, b, c to be f∨ y∨ f so that f < a, b, c < e is a lattice section. Finally suppose S/D is a copy of N5. Let a, b, c, e, f in S be such that e and f are as before, a, b and c lie separately in the three middle D-classes, where the D-class of a is incomparable with the D-classes of b and c, but the D-class of b lies above the D-class of c. We create a lattice section by first repeating the first two steps so that f < a, b, c < e and then: 3. Reset c to be b∧ c∧ b so that we also have c < b. The final assertion follows from Theorem 2.2.8. £ 52 II: Skew Lattices A skew lattice S is quasi-distributive if its lattice image S/D is distributive. Quasi-distributive skew lattices are a subvariety of skew lattices. Indeed, since ( x∧ y) ∨ ( x∧ z) ≺ x ∧ ( y∨ z) holds for all lattices, these skew lattices are characterized by the identity: [ x∧( y∨ z)] ∧ [( x∧ y) ∨ ( x∧ z)] ∧ [ x∧( y∨ z)] = x∧( y∨ z). We have the following corollary to Theorem 2.2.9. Theorem 2.2.10. A skew lattice S is quasi-distributive if and only if neither M3 nor N5 is a subalgebra of S. 2.3 Normal skew lattices Recall that a band S is normal if it satisfies any and hence all of: a) ∀ a, b, c ∈ S: abca = acba. aʹ) ∀ a, b, c, d ∈ S: abcd = acbd. b) ∀ a ∈S, a S a = { axa⎮ x ∈ S} is a semilattice in S. bʹ) ∀ a, b ∈S, a S b = { axb⎮ x ∈ S} is a semilattice in S. c) Given D-classes A ≥ B in S, ∀ a ∈ A, ∃! b ∈ B, a ≥ b. Likewise, a skew lattice S is normal if either (and thus both) of the following hold: a) (S: ∧) is a normal band. In particular, abcd = acbd holds on (S: ∧). b) ∀ a ∈ S, a∧S∧ a = { a∧ x∧ a⎮ x ∈ S} = { b ∈ S⎮ a ≥ b} is a sublattice of S. Clearly: Normal skew lattices form a subvariety of skew lattices. Applying the Kimura decomposition we get the elementary but important: Theorem 2.3.1. A skew lattice S is normal if and only if its left factor S/R is left normal ( a∧ b∧ c = a∧ c∧ b) and its right factor S/L is right normal ( a∧ b∧ c = b∧ a∧ c). £ Being normal has interesting connections with distributivity. Theorem 2.3.2. [Leech, 1992] Given a skew lattice S , the following are equivalent: 1) a ∧ ( b ∨ c) ∧ d = ( a ∧ b∧ d) ∨ ( a ∧ c ∧ d) holds on S. 2) S is distributive and normal. 3) S/D is distributive and S is normal. 53 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. First observe that each of these conditions holds on S if and only if it holds on both its left factor S/R and its right factor S/L. Thus, we may prove the theorem by proving it for all right [left]-handed algebras. So suppose that (1) holds for a right-handed skew lattice, S. We show that if a ≥ b, c in S with b R c, then b = c. Indeed, the following instance of (1) b ∧ ( a ∨ c) ∧ a = ( b ∧ a∧ a) ∨ ( b ∧ c ∧ a) reduces first to b ∧ c ∧ a = b ∨ c, and then to c = b. Hence S is right normal and thus normal. Next, consider the D-equivalent expressions, a∨( b∧ c)∨ a and ( a∨ b∨ a)∧( a∨ c∨ a). Assuming right normality. They are actually the R-equivalent expressions, a∨( b∧ c) and ( a∨ b)∧( a∨ c). Since [( a∨ b)∧( a∨ c)] ∨ ( a∨ c) = a∨ c by absorption, and [ a∨( b∧ c)] ∨ ( a∨ c) = a ∨ ( b∧ c) ∨ c] = a∨ c. Hence a∨ c ≥ both a∨( b∧ c) and ( a∨ b)∧( a∨ c). S being normal with a∨( b∧ c) R ( a∨ b)∧( a∨ c) we get a∨( b∧ c) = ( a∨ b)∧( a∨ c). Thus (1) implies (2). Next, (2) clearly implies (3). Suppose that (3) holds. Then by normality, a ∧ d ≥ a ∧ ( b ∨ c) ∧ d, a ∧ b∧ d, a ∧ c ∧ d and hence a ∧ d ≥ ( a ∧ b∧ d) ∨ ( a ∧ c ∧ d) with a∧( b ∨ c)∧ d D ( a∧ b∧ d) ∨ ( a∧ c∧ d) since S/D is distributive. Whence, a ∧ ( b ∨ c) ∧ d = ( a ∧ b ∧ d) ∨ ( a ∧ c ∧ d) , again by normality. Thus (3) implies (1). £ Lemma 2.3.3. In any normal skew lattice, a∨ b = b∨ a implies a∧ b = b∧ a. Proof. Indeed, a∨ b = b∨ a implies a, b ∈( a∨ b)∧S∧( a∨ b), a sublattice, so that a∧ b = b∧ a. £ A skew lattice is strongly distributive if it satisfies both (4) and (5) below: 4) a ∧ ( b ∨ c) = ( a ∧ b) ∨ ( a ∧ c). 5) ( b ∨ c) ∧ d = ( b ∧ d) ∨ ( c ∧ d). Theorem 2.3.4. A skew lattice S is strongly distributive if and only if it is distributive, normal and symmetric. Proof. A skew lattice satisfying (4) and (5) also satisfies (1), and thus is distributive and normal. Being normal, a∨ b = b∨ a implies a∧ b = b∧ a holds. So let a∧ b = b∧ a. If a∨ b ≠ b∨ a, then either a ∧ ( b ∨ a) ≠ a or ( b ∨ a) ∧ b ≠ b, for otherwise absorption gives a ∨ b ∨ a = b ∨ a = b ∨ a ∨ b so that a∨ b = b∨ a. Suppose that a ∧ ( b ∨ a) ≠ a. But then (4) fails in general since a ∧ ( b ∨ a) ≠ a while ( a∧ b) ∨ ( a∧ a) = ( a∧ b) ∨ a = ( b∧ a) ∨ a = a. 54 II: Skew Lattices Similarly ( b ∨ a) ∧ b ≠ b implies that (5) fails in general. Hence symmetry must also follow from (4) and (5) combined. Conversely, suppose that S is distributive, symmetric and normal. Assume in addition that S is right-handed. Thus both distributive laws a ∨ ( b ∧ c) = ( a ∨ b) ∧ ( a ∨ c) and ( b ∨ c) ∧ d = ( b ∧ d) ∨ ( c ∧ d) hold by the right-handed version of the standard distributive laws. We consider the status of a ∧ ( b ∨ c) = ( a ∧ b) ∨ ( a ∧ c). First, b ∧-commutes with both a∧( b∨ c) and ( a∧ b) ∨ ( a∧ c) with a∧ b being their common meet. b∧ a∧( b∨ c) = b∧ a∧ b∧( b∨ c) = b∧ a∧ b = a∧ b. a∧( b∨ c)∧ b = a∧ b∧( b∨ c)∧ b = a∧ b. b∧[( a∧ b) ∨ ( a∧ c)] =R b∧[ a∧( b∨ c)]∧[( a∧ b) ∨ ( a∧ c)] = b∧( b∨ c)∧ a∧[( a∧ b) ∨ ( a∧ c)] = b∧ a∧[( a∧ b) ∨ ( a∧ c)] = b∧( a∧ b)∧[( a∧ b) ∨ ( a∧ c)] = b∧( a∧ b) = a∧ b. [( a∧ b) ∨ ( a∧ c)] ∧ b = ( a∧ b) ∨ ( a∧ c∧ b) = ( a∧ b) ∨ ( a∧ b∧ c∧ b) = ( a∧ b) Next, the necessarily commuting join of b with either a∧( b∨ c) or ( a∧ b)∨( a∧ c) is ( b∨ a)∧( b∨ c). Indeed b ∨ [ a∧( b∨ c)] = ( b∨ a) ∧ ( b∨ b∨ c) = ( b∨ a) ∧ ( b∨ c) and b ∨ [( a∧ b)∨( a∧ c)] = b ∨ ( b∧ a∧ b) ∨ ( a∧ c) = b ∨ ( a∧ c) = ( b∨ a)∧( b∨ c). Hence a∧( b∨ c) and ( a∧ b) ∨ ( a∧ c) are D-equivalent and both are ≤ ( b∨ a) ∧ ( b∨ c). By normality they are equal. We have seen that (4) and (5) above hold when S is distributive, symmetric, normal and right-handed. This must also true in the left-handed case. It follows that they must hold when S is distributive, symmetric and normal. £ Corollary 2.3.5. A normal skew lattice (S; ○, •) in a ring is strongly distributive. £ Proof. Indeed, any skew lattice (S; ○, •) is already distributive and symmetric. The simplest class of normal bands are the rectangular bands that satisfy abc = ac. ⎧ ⎡ 0 A AB ⎤ ⎫ ⎪ ⎪ Example 2.3.1. The set of matrices ⎢ j× j ⎥ ⎨ n 0 I B ⎢ ⎥ A ∈ F i× j, B ∈ F j× k ⎬ in F n× ⎪ 0 0 0 ⎪ ⎩ ⎣⎢ ⎦⎥ ⎭ with i + j + k = n and i + j < n, for i, j, k fixed is a rectangular band . For such bands 55 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond ⎡ 0 A AB ⎤ ⎡0 C CD⎤ ⎡0 C AB + CD − AD⎤ ⎡0 C CB⎤ ⎢ 0 I B ⎥ ○ ⎢ ⎥ = ⎢ ⎥ ≠ ⎢ ⎥ ⎢ 0 I D ⎢ ⎥ 0 I B ⎢ ⎥ 0 I B ⎢ ⎥ 0 0 0 ⎣ ⎦⎥ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ ⎡ 2 0 C AB + CD − AD⎤ ⎡0 C CB⎤ except in special cases. However, ⎢0 I B ⎥ = ⎢ ⎥ . £ ⎢ ⎥ 0 I B ⎢ ⎥ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ This leads us to define a cubic join e∇ f by e∇ f = e + f + fe – efe – fef. The cubic join extends the quadratic join given by ○ in the following sense: 1) If e, f, ef, fe ∈ E(R), then ( e ○ f)2 = e + f + fe – efe – fef = e∇ f ∈ E(R) also. 2) Every skew lattice (S; ○, •) in a ring is trivially a skew lattice under ∇ and •, since e∇ f reduces to e ○ f whenever the latter is idempotent. 3) Situations occur where e ○ f is not idempotent but e∇ f is. Indeed, e∇ f ∈ E(S) whenever e, f, ef, fe ∈ E(S). (This combines (1) with the previous example.) 4) Caveat: While ○ is always associative, ∇ needn’t be, even when giving idempotent closure. Example 2.3.2. (Karin Cvetko-Vah) Consider the following matrix band with two D- classes: ⎧⎡0 x + ⎫ 1 x 2 x 1 y 1 x 2 y 2 ⎤ ⎪⎢ ⎪ ⎪ 0 1 0 y ⎥ ⎪ 1 ⎨⎢ ⎥ x ∈F ⎬ 0 0 1 y ⎪⎢ 1, x 2 , y 1, y 2 2 ⎥ ⎪ ⎪⎢0 0 0 0 ⎩⎣ ⎦⎥ ⎭⎪ ↓ ⎧⎡0 u uv ⎫ 1 uv 2 ⎤ ⎪⎢ ⎪ ⎪ 0 1 v ⎥ ⎪ 1 v 2 ⎨⎢ ⎥ u, v ∈F ⎬ 0 0 0 0 ⎪⎢ ⎥ 1, v 2 ⎪ ⎪⎢0 0 0 0 ⎩⎣ ⎦⎥ ⎭⎪ This band is also closed under ∇. However, upon setting 56 II: Skew Lattices ⎡0 0 0 0⎤ ⎡0 1 0 0⎤ ⎡0 0 0 0⎤ ⎢0 1 0 0⎥ ⎢0 1 0 0⎥ ⎢0 1 1 0⎥ A = ⎢ ⎥ , B = ⎢ ⎥ and C = ⎢ ⎥ 0 0 1 0 ⎢ ⎥ 0 0 0 0 ⎢ ⎥ 0 0 0 0 ⎢ ⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥ we get ⎡0 0 0 0⎤ ⎡0 0 −1 0⎤ ⎢0 1 0 0⎥ ⎢0 1 0 0⎥ A∇(B∇C) = ⎢ ⎥ ≠ (A∇B)∇C = ⎢ ⎥ . £ 0 0 1 0 ⎢ ⎥ 0 0 1 0 ⎢ ⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥ A multiplicative band in a ring R that is closed under ∇ is called a ∇- band. Observe that the ∇-band in Example 2.3.2 is not normal. Indeed: ⎡0 0 0 0⎤ ⎡0 0 0 0⎤ ⎢0 1 0 0⎥ ⎢ ⎥ ⎢ ⎥ > 0 1 v 0 ⎢ ⎥ for all v. 0 0 1 0 ⎢ ⎥ 0 0 0 0 ⎢ ⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥ For normal ∇-bands in rings the following results hold. Theorem 2.3.6. Every normal ∇- band in a ring is a strongly distributive skew lattice. Proof. Given e, f, g in a normal ∇-band S, observe that e∇( f∇ g) = e∇( f + g + gf – fgf – gfg) calculates to e + f + g + gf – fgf – gfg + fe + ge + gfe – fgfe – gfge – e( f + g + gf – fgf – gfg) e – ( f + g + gf – fgf – gfg) e( f + g + gf – fgf – gfg) = e + f + g + ( gf + fe + ge) – ( efe + fef +fgf +gfg + ege + geg) – ( fge + gef) + ( efge + fegf + gefg) where in this calculation repeated use is made by the identity xyzw = xzyw. Similarly, ( e∇ f)∇ g = ( e + f + fe – efe – fef )∇ g calculates to the same final expression, showing that ∇is associative. Next, note that e( e∇ f) = e + ef + efe – eefe – efef = e. Likewise, e∇( ef) = e + ef + efe – eefe – efeef = e. Similarly (e∇ f) f and ( ef)∇ f reduce to f. Thus S is a skew lattice. Since e∇ f and f∇ e differ only in the respective terms fe and ef, S is symmetric. Finally, e( f∇ g) e = e( f + g + gf – fgf – gfg) e = efe + ege + egfe – efgfe – egfge = efe + ege + egefe – efegefe – egefege = ( efe)∇( ege). 57 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Applying the identity e∇ f∇ e = e + f – fef (replace g by e in the above calculation of e∇ f∇ g and then simplify) we get ( e∇ f∇ e) ( e∇ g∇ e) = ( e + f – fef)( e + g – geg) = e + eg – eg + fe + fg – fgeg – fe – feg + fefgeg = e + fg – feg = e + fg – fgefg = e∇( fg)∇ e. Thus S is also distributive. £ The above theorem has several consequences of significance for skew lattices. Theorem 2.3.7. If R is a ring for which E(R) is closed under multiplication, then E(R) is a normal skew lattice under ∇ and •. In particular, E(R) is closed under multiplication if R satisfies the identity, abcd = acbd. E(R) is a Boolean lattice when R has an identity, 1. Proof. We prove the last statement first. So let R be a ring with identity 1 for which E(R) is closed under multiplication. Then for all e ∈ E(R), 1 – e ∈ E(R) also with e(1 – e) = 0. Since E(R) is a band, for all f ∈ E(R), ef(1 – e) = 0 also (The Clifford-McLean Theorem). Thus ef = efe holds for all e, f ∈E(R). Likewise, from (1 – e) fe = 0 we get fe = efe. Thus E(R) is commutative under multiplication, forcing (E(R); ○, •, 1, 0) to be a Boolean lattice. Suppose next that R is any ring for which E(R) is a multiplicative band. Given e ∈ E(R), eE(R) e = E( e R e) is a sub-band that is necessarily commutative since e is the identity of e R e. Thus E(R) is a normal band. Moreover, for any e, f ∈ E(R), ( e∇ f)2 = ( e + f + fe – efe – fef)2 = e + ( fe + f – fef) + fe – efe – fe = ( e∇ f) where each term in the third expression is a reduction of x( e∇ f) where x is one of the terms in e∇ f. Thus E(R) is closed under ∇, and being normal forms a skew lattice under ∇ and •. Finally, if R satisfies the identity abcd = acbd, then ( ef)2 = efef = eeff = ef for any pair of idempotents e and f. Thus E(R) is indeed a multiplicative band and the theorem follows. £ Corollary 2.3.8. A normal band in a ring generates a normal skew lattice under ∇ and •. A maximal normal band in a ring R thus forms a normal skew lattice under ∇ and •. Proof. For such a band B, the subring S generated from B satisfy xyzw = xzyw. Thus B ⊆ E(S) which is a normal band forming a skew lattice under ∇ and •. By maximality, B = E(S) . £ Example 2.3.3. If R is the semigroup ring A[B] with A a commutative ring and B a normal band, then E(R) is a normal skew lattice. £ Example 2.3.4. [Karin Cvetko-Vah] Consider the band of all ( n+2)×( n+2) matrices of the form: 58 II: Skew Lattices ⎧⎡ 0 a ⎤ ⎫ 1 a 2 a 3 a 4 ... c ⎪⎢ ⎥ ⎪ 0 δ ⎪⎢ δ = 0, 1 1 0 0 0 ... b 1 ⎥ i ⎪ ⎪⎢ 0 0 δ ⎥ ⎪ 2 0 0 ... b 2 ⎪⎢ ⎥ a = a δ i i i ⎪ 0 0 0 δ ⎨⎢ . 3 0 ... b 3 ⎥ ⎬ = δ ⎪⎢ b b 0 0 0 0 δ ⎥ j j j ⎪ 4 ... b 4 ⎪⎢ ⎥ ⎪       ⎪⎢ ⎥ c = ∑ a b i i ⎪ ⎪⎢ 0 0 0 0 0 ... 0 ⎪ ⎩⎣ ⎦⎥ ⎭ This is the normal skew lattice E(R) for the upper triangular subring R that it ring-generates. £ Query: Are maximal regular bands in rings skew lattices under ∇? Do regular bands in rings generate skew lattices under ∇? Do they even generate ∇-bands? That this is not the case in general is guaranteed by: Example 2.3.5. In the semigroup ring &[Reg a,b,c] of the free regular band on { a, b, c}, the product ( a ∇ b) c = ac + bc + bac – abac – babc fails to be idempotent. £ 2.4 Primitive skew lattices and skew lattice structure A primitive skew lattice is a skew lattice P consisting of exactly two D-classes A > B. Primitive skew lattices and their relation to arbitrary skew lattices were studied in Leech [1993]. We present a number of results from that paper. Given a primitive skew lattice P with D-classes A > B in P/D, a coset of A in B is any subset of B of the form A ∧ b ∧ A = { a ∧ b ∧ aʹ⎮ a, aʹ ∈A} for some fixed b ∈ B. Similarly, a coset of B in A is any subset of the form B ∨ a ∨ B = { b ∨ a ∨ bʹ⎮ b, bʹ ∈ B} for some a ∈ A. Since both operations are regular, an alternative description of both types of cosets is given by A ∧ b ∧ A = { a ∧ b ∧ a⎮ a ∈A} and B ∨ a ∨ B = { b ∨ a ∨ b⎮ b ∈ B}. Indeed, { a ∧ b ∧ a⎮ a ∈A} ⊆ A ∧ b ∧ A as already defined.. But by regularity, a ∧ b ∧ aʹ = a ∧ a′ ∧ a ∧ b ∧ a′ ∧ a ∧ a′ = ( a ∧ aʹ) ∧ b ∧ ( a ∧ aʹ) which is of the form a ∧ b ∧ a. The case for B ∨ a ∨ B is similar. For any a ∈ A, the set a ∧ B ∧ a = { a ∧ b ∧ a⎮ b ∈ B} = { b ∈ B⎮ b ≤ a} is the image set of a in B. Its elements are the images of a in B. Dually, given any b ∈ B, the set b ∨ A∨ b = { a ∈ A; a ≥ b} is the image set of b in A. We have the following fundamental result: 59 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 2.4.1. Let P be a primitive skew lattice with D- classes A > B. Then (1) B is partitioned by the cosets of A in B. In particular, b ∈ A∧ b∧A for all b ∈ B and if x ∈ A∧ b∧A for some x ∈ B, then A∧ x∧A = A∧ b∧A. (2) The image set in B of any a ∈ A is a transversal of the cosets of A in B. (3) Dual remarks hold for cosets and element images of B in A. Furthermore: (4) Given cosets B∨ a∨B in A and A∧ b∧A in B a natural bijection of both cosets is given by the natural partial ordering: x in B∨ a∨B corresponds to y in A∧ b∧A if and only if x ≥ y. (5) The meet and join on P are determined jointly by these coset bijections and the rectangular structure of each D -class. Proof. By absorption, b = ( a∨ b)∧ b∧( b∨ a) for all a in A so that b ∈ A∧ b∧A. Given x ∈ A∧ b∧A, say x = m∧ b∧ n for m, n ∈ A, then for all a ∈ A, a ∧ x ∧ a = a ∧ m ∧ b ∧ n ∧ a = a ∧ b ∧ a where the second identity holds since ∧ is regular and m, n ≻ a, b. Thus (1) is seen and this conditional identity also gives us (2). Condition (3) follows by duality. Given cosets B∨ a∨B and A∧ b∧A, for any x in B∨ a∨B, by (2) x ∧ b ∧ x is the unique element y of A∧ b∧A such that x ≥ y. Dually, for each y in A∧ b∧A, y∨ a∨ y is the unique element x in B∨ a∨B such that x ≥ y. Between these two cosets, the processes x → y ≤ x and y → x ≥ y are reciprocal and (4) is seen. Finally, given x, y ∈ P, whenever x D y then both x∧ y and x∨ y are given by the rectangular structure of the common D-class of x and y. Otherwise, say x ∈ A and y ∈ B, we have x∨ y = x∨( y∨ x∨ y), y∨ x = ( y∨ x∨ y)∨ x, x∧ y = ( x∧ y∧ x)∧ y and y∧ x = y∧( x∧ y∧ x). Since y∨ x∨ y is the image of y in the B-coset in A containing x and x∧ y∧ x is the image of x in the A-coset in B containing y, (5) is seen. £ Given A > B as above, if A is partitioned by cosets {A i⎮ i ∈ I} and B is partitioned by cosets {B j⎮ j ∈ J}, then for each pair of indices i, j let ϕ ji: A i → B j denote the coset bijection given by setting φ ji( x) = y if for x in A i, y is the unique element in B j such that x ≥ y. Then for all x ∈ A i and y ∈ B j x –1 –1 ∨ y = x ∨ ϕ ji ( y), y ∨ x = ϕ ji ( y) ∨ x, x ∧ y = ϕ ji( x) ∧ y and y ∧ x = y ∧ ϕ ji( x) . Thus it seems that any primitive skew lattice should be obtained by a fairly simple construction. To this end we begin by calling a right-handed primitive skew lattice right primitive, with left primitive skew lattices defined in dual fashion. For right primitive skew lattices the description of a coset can be simplified as A ∧ b ∧ A = b ∧ A = { b ∧ a⎮ a ∈ A} and B ∨ a ∨ B = B ∨ a = { b ∨ a⎮ b ∈ B} for any a ∈ A and b ∈ B since a∧ b∧ a = b∧ a and b∨ a∨ b = b∨ a. Dual remarks hold in the left-primitive case. All right [left] primitive skew lattices arise as follows. A P-graph is a pair of partitioned disjoint sets A = ∪ i A B ∈ Ι i and B = ∪ j ∈ ϑ j , where all A i and all B j have a common cardinality, together with a fixed set of bijections φ ji: A i → B j . The P-graph is denoted by the triple (A i, φ ji, B j). A right primitive structure is induced on A∪B in the 60 II: Skew Lattices following way. First, ∧ and ∨ restricted to either A or B must be x ∧ y = y and x ∨ y = x. But given x ∈ A i and y ∈ B j we set x −1 ∧ y = y, y ∧ x = φ ji( x), x ∨ y = x and y ∨ x = φ ji ( y) . The primitive skew lattice thus obtained, denoted by P[A i, φ ji, B j], has D-classes A > B, with cosets Ai and B j, and coset bijections φ ji. A left primitive skew lattice Pʹ[A i, φ ji, B j] is obtained in dual fashion: x −1 ∧ y = φ ji( x), y ∧ x = y, x ∨ y = φ ji ( y) and y ∨ x = x. A partial image of a P-graph is given in the following diagram, where the top row of dots represents an upper D-class A of order 9 and the bottom row represents a lower D-class B of order 6. All cosets in this example have size 3, with the members of each coset linked in the diagram. The four arrows represent four of the six coset bijections, with arrows corresponding to ϕ20 and ϕ02 left out. A: • --- • --- •0 – • --- • --- •1 – • --- • --- •2     ϕ00 ϕ01 ϕ11 ϕ12 B: • --- • --- •0 – • --- • --- •1 In terms of the natural partial order, ≥, the situation looks more like • − − • − − • • − − • − − • • − − • − − •            • − − • − − • • − − • − − • where the dotted lines indicate the partial order relationships. Applying Kimura factorization, every primitive skew lattice is isomorphic to the fibred product of a left primitive skew lattice and a right primitive skew lattice over their maximal lattice image, which here is isomorphic to 2 = {1 > 0}. This factorization coupled with the ideas above yields: Theorem 2.4.2. Every primitive skew lattice P has a fibred product decomposition, P ≅ P[A i, φ ji, B j] ×P/D Pʹ[C k, ψ lk, D l], where (A i, φ ji, B j) and (C k, ψ lk, D l) denote P- graphs and P/D is isomorphic to the primitive lattice 1 > 0. Both P- graphs are unique to within isomorphism of P- graphs. £ The P-graph description of right [left] primitive skew lattices can be refined as follows. A P-graph coordinate system consists of a sextuple (I, J, C, G, µ, θ) where 61 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond (i) I, J and C are nonempty sets. (ii) G is a group and µ: G × C → C is a group action of group G on C. (iii) θ: J × I → G is a map. From this data construct a P-graph by first setting A = I × C and B = J × C. The B-cosets in A are the A i = { i} × C and the A-cosets in B are the B j = { j} × C. Coset bijections are given by φ ji( i, c) = ( j, θ( j, i) c). From this data, a right or left primitive skew lattice is constructed to be denoted by P[I, J, C, G, µ, θ] or Pʹ[ I, J, C, G, µ, θ]. Every right [ left] primitive skew lattice has a coordinatization. Given a P-graph representation, P ≅ P[A i, φ ji, B j], let C = A0 for some common index 0 in I ∩ J. A0 A i A0 ↓ ϕ  ϕ−1 ↓ ϕ  ϕ−1 00 0 i ji 0 j B0 B j Next, for each pair ( j, i), let θ( j, i) be the permutation –1 –1 ϕ j 0 ϕ jiϕ i 0 ϕ00 of A0 and then let G denote the permutation group on A0 generated collectively by the various θ( j, i). Here the coordinatization is normalized in that θ({0} × I ∪ J × {0}) = 1 in G. In the case where G = C and u is group multiplication, P has a coordinatization with group translations. In this case the data reduces to the indexing sets I, J, the group G and map θ: I ×J → G, and the skew lattice is denoted by P[I, J, G, θ]. By the above discussion we may assume that 0 ∈ I ∩ J, and θ({0} × I ∪ J × {0}) = 1 in G (or = 0 in the case of additive notation). An instance of this is given by any maximal right [left] primitive skew lattice in a ring. Example 2.4.1. Given a ring R and an idempotent e ∈ E(R), the R-set of e in R is the set Re = e + e R(1 – e) = { x ∈ R⎪ ex = x and xe = e} It is the maximal right-0 band ( xy = y) in R containing e. Re also forms a left-0 band under ○, and thus is a rectangular skew lattice in R. Let f be a second idempotent in R such that e > f. Then R f forms a second rectangular skew lattice and the union P e> f = R e ∪ R f is a right primitive skew lattice in R with upper D-class R e and lower D-class R f. Its coordinatization is given as follows. To begin, notice that the group A = e R(1 – e) acts simply transitively on R e and that the group B = f R(1 – f) acts simply transitively on R f, both under the operation of addition. A and B share the common subgroup G = A ∩ B = e R(1 – e) ∩ f R(1 – f) = f R(1 – e). A splits as f R(1 – e) ⊕ ( e – f)R(1 – e) and B splits as f R(1 – e) ⊕ f R( e – f). The various summands are naturally arranged in the following array format: 62 II: Skew Lattices ⎡ f f R( e − f ) f R(1 − e) ⎤ ⎢ ⎥ e − f ( e − f )R(1 − e) ⎢ ⎥ . ⎢ 1 ⎣ − e ⎦⎥ B-cosets in A are given as A c = e + G + c. for any c ∈ ( e – f)R(1 – e). Similarly, A-cosets in B are B d = f + G + d for any d ∈ f R( e – f). Coset maps are given by ϕ d,c( e + g + c) = ( f + d)( e + g + c) = f + [ g + dc] + d. Identifying A with R e under a → e + a, and similarly identifying B with R f under b → f + b, yields the coordinatization. R e ∪ R f corresponds to G ⊕ ( e – f)R(1 – e) ∪ G ⊕ f R( e – f). Under this correspondence, for all c ∈ ( e – f)R(1 – e) and all d ∈ f R( e – f), ϕ d,c( g, c) = ( g + dc, d). Clearly θ: f R( e – f) × ( e – f)R(1 – e) → fR(1 – e) = G is given by the ring multiplication. £ Another instance involving coordinatization using group translations is as follows. To begin, a connected graph with each vertex having degree 2 is called a simple circuit in the finite case and an infinite simple path when infinite. By the natural graph of a primitive skew lattice P we mean the graph with vertices being the elements of P and with edges given by the relation for > ∪ >op. That is, e – f is an edge for e, f ∈ P if either e > f or f > e. We state the following result without proof. (See Leech [1993].) Theorem 2.4.3. Let P be a right primitive skew lattice. Then the natural graph of P is a simple circuit precisely when P has a coordinatization by group translations under addition, P[&2, &2, & n, θ] for some n ≥ 1, where θ( j, i) = ji for i, j in {0, 1}. The graph is a simple path precisely when P has a coordinatization by group translations P[&2, &2, & θ] where again θ( j, i) = ji in which case P[&2, &2, &, θ] is an infinite primitive skew lattice on four generators, namely (0,0), (0,1), (1,0) and (1,1). £ Exercise. Show that infinite primitive skew lattices require at least four generators. Primitive skew lattices, connectedness and maximal rectangular inages In general, the natural graph of a skew lattice S is the undirected graph on S generated from the natural partial order on S. Thus the vertices are just the elements of S and two vertices e and f are adjacent precisely when either e > f or f > e. A component of S is any maximally connected subset of its natural graph. In particular, a skew lattice is connected if its natural graph has a single component. Theorem 2.2.1 implies that each component of S has nonempty intersection with each D- class of S. Clearly the components of S are a partition of the underlying set. 63 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond G iven a primitive skew lattice P , each component of P has nonempty intersection with each coset of P. Thus we say that a primitive skew lattice P is maximally disconnected if distinct elements of each coset of P belong to distinct components of P. P is degenerate if it is both connected and maximally disconnected. If P has D-class structure A > B, then P is degenerate precisely when a > b for all a ∈A and all b ∈B. Finally, a coset component of P is any (necessarily nonempty) intersection of a coset in P with a component of P. For primitive skew lattices we have the following results: Lemma 2.4.4. Given a primitive skew lattice P, the cosets of P form a congruence partition of P. If ζ is the corresponding congruence, then the primitive skew lattice P/ζ is the maximal degenerate image of P. Proof. Given our descriptions of primitive skew lattices in Theorems 2.4.1 and 2.4.2 above, pointwise computations yield A i ∨ B j = A i = B j ∨ A i and A i ∧ B j = B j = B j ∧ A i. £ Theorem 2.4.5. If P is a primitive skew lattice, then its components are the maximal connected subalgebras of P. Moreover, they are the congruence classes of the congruence ρ whose quotient algebra P/ρ is the maximal rectangular image of P. Thus the coset components of P form the congruence partition for the congruence ζ∩ρ for which the quotient algebra P/ζ∩ρ is the maximal disconnected image of P. Finally, if P is maximally disconnected, then it factors as the product of a degenerate skew lattice with a rectangular skew lattice. Proof. The components of P clearly induce an equivalence relation ρ on P. To begin, let σ denote the symmetric closure of ≥. Thus a σ b means that either a ≥ b or b ≥ a. Since ρ is the transitive closure of σ, to show that ρ is a congruence we need only show that a σ b and aʹ σ bʹ imply that both a∨ aʹ ρ b∨ bʹ and a∧ aʹ ρ b∧ bʹ. We first show this under the added restriction that P is right primitive. Thus a∨ aʹ ≥ a and b∨ bʹ ≥ b so that a∨ aʹ ρ b∨ bʹ follows. Similarly, a∧ aʹ ≤ aʹ and b∧ bʹ ≤ bʹ so that a∧ aʹ ρ b∧ bʹ also follows and ρ is shown to be a congruence. Similarly, if P is left primitive then again, ρ is a congruence. By the Kimura factorization, ρ is a congruence on any primitive skew lattice. The first part of the theorem follows. Since the coset components arise as congruence classes for the meet congruence ζ∩ρ, the second part of the theorem is seen. Finally the assumption of being maximally disconnected insures that the induced homomorphism from P to P/ζ × P/ρ with kernel congruence ζ∩ρ is an isomorphism. £ A skew lattice S is bounded if it has a maximal class A and a minimal class Z in which case the primitive skew lattice A∪Z forms the boundary Bd(S) of S. A generalization of first part of the above theorem complements the Clifford-McLean Theorem. Theorem 2.4.6. The components of a skew lattice S are its maximal connected subalgebras. Moreover, their partition of S is the congruence class partition into for the congruence ρ for which the induced quotient algebra S /ρ is the maximal rectangular image of S. If S is also bounded with boundary algebra Bd(S), then the inclusion Bd(S) ⊆ S induces an isomorphism of maximal rectangular images. 64 II: Skew Lattices Proof. The theorem holds when S is primitive. Next, assume S is bounded with maximal class A and minimal class Z. Then Bd(S) = A∪Z decomposes into components A i∪Z i. We say that an element x ∈S belongs to component A i∪Z i if the latter is the unique boundary component such that there exists u ∈A and v ∈Z such that u ≥ x ≥ v. For any y ∈ S such that either x ≤ y or x ≥ y it is clear that x and y belong to the same boundary component. Hence the inclusion Bd(S) ⊆ S indices a bijection between the classes of components. Next, let x belong to A i∪Z i and y belong to A j∪Z j. Pick u ∈ A i and w ∈ A j such that u ≥ x∨ y∨ x and w ≥ y∨ x∨ y. By the previous theorem, u∨ w ρ ( x∨ y∨ x)∨( y∨ x∨ y) = x∨ y. Thus x∨ y lies in the component of S containing (A i∪Z i) ∨ (A j∪Z j). Likewise, x∧ y lies in the component containing (A i∪Z i) ∧ (A j∪Z j). The bounded case of the theorem now follows from the primitive case. The general case follows from the fact that every skew lattice is the directed union of its intervals. £ Recall that a noncommutative lattice splits if it factors as the product of a lattice and an antilattice (here a rectangular skew lattice). Since every component of a skew lattice S meets every D-class of S we have the following corollary. Corollary 2.4.7. Given a skew lattice S, S/(D∩ρ) is its maximal split image. £ Orthogonal D -classes and the behavior of skew diamonds Let A, B and C be D-classes in a skew lattice S such that both A and B are comparable with C. By a class component of A [or B] in C is meant the intersection of a component of A∪C [or B∪C] with C. We say that A and B are orthogonal in C if each class component of A in C lies in a unique coset of B in C and likewise each class component of B in C lies in a unique coset of A in C where A and B being orthogonal in C is equivalent to asserting that the image in C of each x in A lies in a unique coset of B in C and likewise the image in C of each y in B lies in a unique coset of A in C. That is, each x in A is covered in C by a unique coset of B in C and dually each y in B is covered in C by a unique coset of A in C. Lemma 2.4.8 . Given D- classes A, B and C in a skew lattice S, if A and B are orthogonal in C, then each coset of A in C has nonempty intersection with each coset of B in C; all such coset intersections, moreover, have a common cardinality. Proof. Indeed let A1 and A2 be cosets of A in C, let B1 and B2 be cosets of B in C, and let ϕ1 and ϕ2 be coset bijections of A1 and A2 into a common coset of A in C. (If C lies above A1 and A2 then −1 ϕ1 and ϕ2 are inverses of the downward bijections.) The bijection ϕ2 ϕ1 and its inverse ϕ −1 1 ϕ2 keep individual elements in the same class component of A in C. Orthogonality implies that both bijections restrict to an inverse pair of bijections of A1∩B1 with A2∩B1. Similarly A2∩B1 is in 1-1 correspondence with A2∩B2, so the assertion is verified. £ 65 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond ∗ • • •∗ • • A-cosets  a − images∗ C: ∗ B-cosets  b − images• ∗ Let the rows in the diagram above represent A-cosets in C and the columns represent B- cosets in C, where A and B are orthogonal in C. Then for any a in A, the images of a in C all lie in different rows, but a single column and for any b in B, the images of b in C all lie in different columns, but a single row. Since the A- and B-cosets all have nonempty intersection by the lemma above, some unique A-C coset intersection contains both an image of a and an image of b. Theorem 2.4.9. Let A and B be D- classes in a skew lattice. Then A and B are orthogonal in both their join class J and their meet class M. For each x ∈ A and y ∈ B, x ∨ y = xʹ ∨ yʹ where xʹ is the image of x in J lying in the unique coset of A in J covering y, and yʹ is the image of y in J lying in the unique coset of B in J covering x. That is, both xʹ and yʹ lie in the unique A-B coset intersection in J containing both an image of x and an image of y, namely xʹ and yʹ. The computation of x ∧ y in M is determined in dual fashion. Proof. Given x ∈ A and y ∈ B, for all u ∈ J such that u ≥ x we have y∨ x∨ y = y∨ x∨ x∨ y = y∨ x∨ u∨ x∨ y = y∨ u∨ y. Thus each x ∈A is covered by a fixed coset of B in J. Likewise each y ∈ B is covered by a unique coset of A in B. Thus both A and B are orthogonal in J, and similarly, they are orthogonal in M. Since x∨ y = ( x∨ y∨ x)∨( y∨ x∨ y) = ( x∨ v∨ x)∨( y∨ u∨ y) for any u, v ∈J such that u ≥ x and v ≥ y, it follows that indeed x ∨ y = xʹ ∨ yʹ where xʹ is the image of x in J lin the unique coset Aʹ of A in J covering y, and yʹ is the image of y in J lying in the unique coset Bʹ of B in J covering x. Clearly both xʹ and yʹ lie in Aʹ∩Bʹ but no other pair of x and y images in J can belong to a common A-B coset intersection. £ The double partition of either J or M by A-cosets and B-cosets is illustrated below where the partition is further refined by the coset partitions that J and M directly induce on each other. Indeed, if say m ∈ M, j ∈ J and j ≥ a ∈ A, then m ∨ j ∨ m = ( m ∨ a) ∨ j ∨ ( a ∨ m) ∈ A ∨ j ∨ A. Thus M ∨ j ∨ M ⊆ A ∨ j ∨ A. Likewise M ∨ j ∨ M ⊆ B ∨ j ∨ B. Similar remarks hold for cosets in M. ⎡ | | | ⎤ ⎢ − + − − + − − + − ⎥ ⎢ ⎥ | | | ⎢ ⎥ A-cosets ↔ ⎢ ⎥ | | | B-cosets  ⎢ − + − − + − − + − ⎥ ⎢ ⎥ | | | ⎣ ⎦ 66 II: Skew Lattices When do these double partitions coincide with the (generally finer) J-M partitions. Theorem 2.4.10. A skew lattice S is symmetric if and only if for many two equivalence classes A and B: (i) the double partition of the join class J by intersections of A- cosets with B- cosets equals the partition by cosets of the meet class M; (ii) the dual assertion holds for the meet class M. Proof. First assume that A-cosets and B-cosets in J and M intersect to J-M cosets. Let a ∈A and b ∈B be given with a∨ b = b∨ a. By Theorem 2.4.9, a∧ b and b∧ a lies in the same A-B coset intersection in M. Thus a∧ b and b∧ a belong to the same J-coset in M. Since a∨ b ≥ both a∧ b and b∧ a, we must have a∧ b = b∧ a. Dually, a∧ b = b∧ a implies a∨ b = b∨ a. Suppose instead that say more than J-coset in M lies inside the intersection I of an A-coset with a B-coset. Thus there exists j ∈ J with at least two distinct images m and mʹ in I. We form a subalgebra T = Jʹ ∪ Aʹ ∪ Bʹ ∪ Mʹ inside the subalgebra Sʹ = j∧S∧ j as follows. Set Jʹ = {j}, Mʹ = j∧I∧ j and set Aʹ equal to the image of Mʹ in A∩Sʹ under a single coset bijection of Sʹ from Mʹ to the intermediate class. Define Bʹ in similar fashion. Thus in T exactly one coset bijection exists from Aʹ to Mʹ and likewise exactly one coset bijection exists from Bʹ to Mʹ. In particular, exactly one aʹ ∈A and exactly one bʹ ∈B exist such that aʹ ≥ m and bʹ ≥ mʹ. Then aʹ∨ bʹ = bʹ∨ aʹ = j and by orthogonality aʹ∧ bʹ = m∧ mʹ and bʹ∧ aʹ = mʹ∧ m. Since m ≠ mʹ, m∧ mʹ ≠ mʹ∧ m follows and S has an instance of anti-symmetry. £ These results have a number of worthwhile corollaries. Coset bijections in skew chains and as morphisms in categories A skew chain is a skew lattice S with finitely many D-classes that is totally quasiordered under ≻ . Thus it can be viewed as a chain of D-classes A > B > C > … > X in that are totally ordered in S/D. We begin with a near-obvious lemma. Lemma 2.4.11. Given a skew chain A > B > C: (1) For each c ∈ C, there is the inclusion of cosets A∧ c∧A ⊆ B∧ c∧B in C. (2) For each a ∈ A, there is the inclusion of cosets C∨ a∨C ⊆ B∨ a∨B in A. (3) Given a > b > c where a ∈ A, b ∈ B and c ∈ C, if ϕ: B∨ a∨B → A∧ b∧A, ψ: C∨ b∨C → B∧ c∧B and χ: C∨ a∨C → A∧ c∧A are coset bijections between the relevant cosets in the respective D -classes taking a to b, b to c and a to c, then ψoϕ ⊆ χ. Proof. Given x = a∧ c∧ a ∈ A∧ c∧A, b ∈ B exists such that b∧ c∧ b = c. Hence, x = a∧ b∧ c∧ b∧ a which us in B∧ c∧B and (1) follows. The proof of (2) is dual. To see (3), first observe that the output set of ϕ and the input set for ψ have the intersection A∧ b∧A ∩ C∨ b∨C within B. Thus (3) follows from the inclusions, 67 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond [A∧ b∧A ∩ C∨ b∨C]∨ a∨[A∧ b∧A ∩ C∨ b∨C] ⊆ C∨ b∨C∨ a∨C∨ b∨C = C∨ a∨C in A and [A∧ b∧A ∩ C∨ b∨C]∧ c∧[A∧ b∧A ∩ C∨ b∨C] ⊆ A∧ b∧A∧ c∧A∧ b∧A = A∧ c∧A in C, where the equalities on the right are due to regularity. Here the expression to the left of ⊆ are the respective input and output sets of ψoϕ. Applying the letter to any x in the input set we get ψoϕ( x) = ( x∧ b∧ x)∧ c∧( x∧ b∧ x) = x∧ b∧ c∧ b∧ x = x∧ c∧ x = χ( x). £ A skew chain A > B > C is categorical if ψoϕ = χ always holds in (3) above, in which case every coset bijection χ between a C-coset in A and an A-coset in C must factor as such. Indeed, given a ∈A and c ∈C such that χ sends a to c, some b ∈B exists such that a > b > c. (Set b = a∧( c∨ y∨ c)∧ a for some y ∈B.) If ϕ and ψ are the A-B and B-C coset bijections sending a to b and b to c, then χ = ψoϕ. A skew lattice S is categorical if every chain of D-classes A > B > C in S is categorical. S is strictly categorical if it is categorical and all such composites ψoϕ of coset bijections between cosets in comparable D-classes A > B > C are nonempty. The outer cosets in a categorical skew chain A > B > C induce virtual versions of themselves in the middle D-class B as follows. Theorem 2.4.12. (Cvetko-Vah [2005c]) A skew chain A > B > C is categorical if and only if for all a ∈ A, b ∈ B and c ∈ C that satisfy a > b > c, (C∨ a∨C) ∧ b ∧ (C∨ a∨C) = (A∧ b∧A) ∩ (C∨ b∨C) = (A∧ c∧A) ∨ b ∨ (A∧ c∧A). Proof. In terms of the coset bijections ϕ: A , and i → B j ψ: ′ B A k → C l χ: ′ m → ′ C n such that ϕ( a) = b, ψ( b) = c and χ( a) = c with ψ○ϕ = χ, the given equalities are what is needed for ψoϕ to occur. £ The term “categorical” comes from the following result: Theorem 2.4.13. If a skew lattice S is strictly categorical, then a category Cat(S) is defined by: (1) Letting the objects of S to be the D- classes of S. (2) For comparable D- classes A ≥ B, Hom(A, B) consists of all coset bijections from all B- cosets in A to A- cosets in B. Otherwise, Hom(A, B) is empty. (3) In particular, each Hom(A, A) consists of the unique identity bijection on A. (4) Letting morphism composition be the usual composition of partial bijections. When S is just categorical, a modified category Cat0(S) is given as above, but in addition, for each comparable pair A ≥ B, Hom(A, B) contains a labeled copy of the empty bijection 0AB. £ 68 II: Skew Lattices Theorem 2.4.14. (1) A skew lattice S is categorical if and only if for all x, p, q, r with x D p ≥ q ≥ r, ( x∧ r∧ x) ∨ q ∨ ( x∧ r∧ x) = [( x∧ r∧ x) ∨ p ∨ ( x∧ r∧ x)] ∧ q ∧ [( x∧ r∧ x) ∨ p ∨ ( x∧ r∧ x)]. (2) Categorical skew lattices form a subvariety of skew lattices. (3) A skew lattice S is categorical if and only if its left and right factors are categorical. Proof. Observe that in any skew lattice S, all p such that x D p arise as ( u∧ x∧ u) ∨ x ∨ ( u∧ x∧ u) for u unrestricted. Observe next that for any p, all q, r such that p ≥ q ≥ r arise q = p∧ v∧ p and r = q∧ w∧ q for v and w unrestricted. Thus the condition stated in (1) can be made unconditional so that (2) follows from (1) and thus (3) follows from (2). To see that (1) holds, consider the case of a nonempty composition of coset bijections ψoϕ, where ϕ is a coset bijection from A to B and ψ is a coset bijection from B to C. Since ψoϕ is nonempty, for all elements p in A in the domain of ψoϕ, ϕ send p to some q in B and ψ sends that q to some r in C yielding p > q > r. Clearly all triples p > q > r arise in this fashion for some ψ and ϕ. For the given ψ and ϕ, letting χ be the coset bijection from A to C sending p to r. Thus at least ψoϕ ⊆ χ by Lemma 4.16. The equation of (1) states that –1 –1 ψ [A ∧ r ∧ A] = ϕ[χ [A ∧ r ∧ A]] with A∧ r∧A being the image of χ. Since all indicated cosets in this equation lie in the domains of the relevant bijections, this equation is equivalent first to [A –1 ∧ r ∧ A] = ψoϕ[χ [A ∧ r ∧ A]] and then to –1 –1 χ[χ [A∧ r∧A] = ψoϕ [χ [A∧ r∧A]] so that ψoϕ must be all of χ. £ While 2.4.16(1) at first may appear to be an obfuscation of the simple implication, if ψoϕ ⊆ χ then ψoϕ = χ, it does unpack ψ○ϕ = χ at the element-wise level, thus setting the stage for (2) and (3). In Chapter 5, (strictly) categorical skew lattices will be studied more closely. We next present several classes of categorical skew lattices. Theorem 2.4.15. Skew lattices in rings are categorical. Proof. Assume first that S is a right-handed skew lattice in a ring R and that x R p ≥ q ≥ r in S. The conditional identity in of the previous theorem thus reduces to ( r∧ x) ∨ q = q ∧ [( r∧ x) ∨ p]. Applying ○ and multiplication, the left side of the equation reduces to rx + q – rxq = rx + q – r, and the right side reduces to q[ rx + p – rxp] = rx + q – r again. Hence S is categorical. Similarly, if S is left handed in R, then S must be categorical. Next suppose that is neither left nor right-handed with either ○ or ∇ for a join. S will be categorical if all countable subalgebras are thus. So let Sʹ be a countable subalgebra of S. By symmetry, copies the maximal left and right-handed images of Sʹ arise as subalgebras of Sʹ in R and thus these are categorical. By the Theorem 2.4.14(3), Sʹ is categorical. Since this holds for all countable subalgebras of S, S is also categorical. £ 69 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 2.4.16. Normal skew lattices are strictly categorical. Proof. Since the lower D-class in any maximal primitive subalgebra of a normal skew lattice has exactly one coset, the composition of adjacent coset bijections is a nonempty coset bijection. £ A third class of categorical skew lattices is as follows. Theorem 2.4.17. Every primitive skew lattice is strictly categorical, distributive and symmetric. Thus all skew lattices in the subvariety generated from the class of primitive skew lattices are categorical, distributive and symmetric. Proof. Any primitive skew lattice S is trivially strictly categorical. The only nontrivial instances of either x∨ y = y∨ x or x∧ y = y∧ x when S is primitive are when x > y or y > x. Thus S is also symmetric. Next, consider the equation x∧( y∨ z)∧ x = ( x∧ y∧ x)∨( x∧ z∧ x). It holds trivially when x is in the lower D-class B. So let x = a in the upper class A and let y = b and z = c in B. Then a∧( b∨ c)∧ a = a∧( c∧ b)∧ a = ( a∧ c∧ a) ∧ ( a∧ b∧ a) = ( a∧ b∧ a) ∨ ( a∧ c∧ a) in B. If a in A and say b in B but c in A then b∨ c in A so that a∧( b∨ c)∧ a = a, while ( a∧ b∧ a) ∨ ( a∧ c∧ a) = ( a∧ b∧ a) ∨ a = a in A. The case where the locations of b and c are switched is similar. Finally, one similarly verifies x∨( y∧ z)∨ x = ( x∨ y∨ x) ∧ ( x∨ z∨ x). £ We eventually show that all skew lattices in this subvariety are in fact strictly categorical and also cancellative. Query: are skew lattices in this subvariety characterized by being cancellative, distributive, strictly categorical and symmetric? Example 2.4.2. [Kinyon and Leech, 2013] A minimal noncategorical skew chain has the following Hasse diagram where a 1 > b 1, b 3; a 2 > b 2, b 4; b 1, b 2 > c 1; b 3, b 2 > c 2; and thus both ai > both cj. A a − 1 a 2     B b − − − ( ) 1 A b 2 C b 3 A b 4 −C b 1   C c − 1 c 2 . Instances of left-handed operations are given by a 1 ∨ c 2 = a 2 = a 1 ∨ a 2, a 1 ∧ b 4 = b 3 ∧ b 4 = b 3 and b 1 ∨ c 2 = b 1 ∨ b 4 = b 4. £ Before moving to the next section we clear up one detail: Proposition 2.4.18. Every coset bijection ϕ: A i → B j is a skew lattice isomorphism. 70 II: Skew Lattices Proof. Pick b ∈ B j. Then for all x ∈ A i, ϕ( x) = x∧ b∧ x. Thus given x, y in A i we have ϕ( x∧ y) = ( x∧ y)∧ b∧( x∧ y) = x∧ b∧ y = x∧ b∧ b∧ y = x∧ bx∧ y∧ b∧ y = ϕ( x) ∧ ϕ( y), due to regularity and Corollary 1.2.8. But since A i and B j are rectangular algebras satisfying x∨ y = y∧ x, the proposition follows. £ In the left [right] rectangular case this proposition is trivially true since it is easily seen that any bijection between left [right] rectangular skew lattices must be an isomorphism. 2.5 Partial skew lattices and coset projections Every nonrectangular skew lattice S is the union of its maximal primitive skew lattices and the latter jointly determine its structure. Indeed one could view primitive skew lattices as “lego pieces” that when appropriately “snapped together” produce entire skew lattices. To pursue this perspective, we need a new concept. A partial skew lattice on a quasi-ordered set (S, ≻) is a 4-tuple where ∨ and ∧ are partial binary operations defined for pairs of elements (S, , ∨, ∧) comparable under ≻ in such a way that the union U of any chain of ≻-equivalence classes forms a skew lattice under ∨ and ∧ whose natural quasi-order coincides with ≻ over U. In particular, each equivalence class A under ∨ and ∧ is a rectangular skew lattice and each comparable pair of such classes, say A > B, forms a primitive skew lattice. That ∨ and ∧ are associative on totally quasi-ordered subsets is called linear associativity. Lemma 2.5.1. Let be a partial skew lattice and let m ≺ x, y ≺ p in S. Then (S, , ∨, ∧) both ( x ∨ p) ∨ y = x ∨ ( p ∨ y) and ( x ∧ m) ∧ y = x ∧ ( m ∧ y). (This form of associativity, where x and y need not be comparable, is called extremal associativity.) Proof. ( x ∨ p) ∨ ( p ∨ y) reduces to both ( x ∨ p) ∨ y and x ∨ ( p ∨ y). Thus the latter must be equal. Similar remarks hold for ( x ∧ m) ∧ y and x ∧ ( m ∧ y). £ Every skew lattice (S; ∨, ∧) induces a canonical partial skew lattice (S; ≻, ∨, ∧) upon restricting the given binary operations, ∨ and ∧, to ≻-related pairs. When is a given partial skew lattice canonical for some skew lattice? Two conditions are obviously necessary. (S, , ∨, ∧) I. The ≻-equivalence classes (proto-D-classes of mutually ≻-related elements) must have both join and meet classes. II. Each pair of equivalence classes must be orthogonal in both their join and meet classes. Given both conditions, both ∨ and ∧ can be extended to full binary operations as follows. III. Set x∨ y = ( x∨ q∨ x)∨( y∨ p∨ y) for any p, q in the join class of x and y such that p ≥ x and q ≥ y. 71 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond IV. Set x∧ y = ( x∧ n∧ x) ∧ ( y∧ m∧ y) for any m, n in the join class of x and y such that m ≤ x and n ≤ y. Since the equivalence classes of x and y are orthogonal in both their join and meet classes, any p, q, m and n satisfying the stated conditions must produce the same values for x∨ q∨ x, y∨ p∨ y, x∧ n∧ x and y∧ m∧ y so that the extended binary operations are uniquely determined. The full algebra (S; ∨, ∧) is called the algebraic closure of (S; ≻, ∨, ∧). Example 2.5.1. Given a ring R, two partial skew lattice structures exist for the entire set of idempotents: (E(R); ≻L, ∨, ∧) and (E(R); ≻R, ∨, ∧). Here e ≻L f iff fe = f and e ≻R f iff ef = f. In both cases e∨ f = e○ f and e∧ f = ef, given that e and f are ≻-related. While Condition I above is met in many classical rings (e.g., full matrix rings over fields), Condition II is rarely met. £ Theorem 2.5.2. A partial skew lattice (S; ≻, ∨, ∧) is the canonical partial algebra of a full skew lattice (S; ∨, ∧) if and only if Conditions I and II above hold, in which case (S; ∨, ∧) is precisely the algebraic closure of (S; ≻, ∨, ∧). Proof. Necessity is clear. Moreover, if such a lattice extension exists, it must be the algebraic closure by Theorem 2.4.9. We show that the algebraic closure must be a skew lattice. So let x and y be given. Then x∨ y is calculated as xʹ∨ yʹ for some xʹ and yʹ in the join class of x and y such that x ≤ xʹ and y ≤ yʹ. Thus in the partial skew lattice we have x ∧ ( x∨ y) = x ∧ ( xʹ∨ yʹ) = x ∧ ( x∨ xʹ∨ yʹ) = x. The other instances of absorption are similarly seen. Suppose next that both x, y ≺ s in S. By extremal associativity, x∨ y = x∨ q∨ p∨ y for any p ≥ x and q ≥ y in the join class of x and y. Choose q such that y ≤ q ≤ y∨ s∨ y. (If need be, replace the given q by ( y∨ s∨ y)∧ q∧( y∨ s∨ y)). Applying linear associativity we get ( x ∨ y) ∨ s = ( x∨ q∨ p∨ y) ∨ s = ( x∨ q) ∨ p ∨ ( y ∨ s ∨ y) ∨ s = ( x∨ q) ∨ ( y ∨ s ∨ y) ∨ s = x ∨ ( q ∨ y ∨ s ∨ y) ∨ s = x ∨ ( y ∨ s). We similarly obtain the three other cases of outer associativity: if r ≤ x, y ≤ s, then also s ∨ ( x ∨ y) = ( s ∨ x) ∨ y, r ∧ ( x ∧ y) = ( r ∧ x) ∧ y and ( x ∧ y) ∧ r = x ∧ ( y ∧ r). With the available conditional associative identities we obtain the unconditional identities. For instance, let s lie in the join class of x, y and z. Then x∧( y∧( z∧ s)) must equal both (( x∧ y)∧ z)∧ s and ( x∧( y∧ z))∧ s. Likewise (( s∧ x)∧ y)∧ z must equal both s∧(( x∧ y)∧ z) and s∧( x∧( y∧ z)). It follows that x∧( y∧( z∧ s))∧((s∧ x)∧ y)∧ z must equal both (( x∧ y)∧ z)∧ s∧ s∧(( x∧ y)∧ z) which reduces to ( x∧ y)∧ z and ( x∧( y∧ z))∧ s∧ s∧( x∧( y∧ z)) which reduces to x∧( y∧ z). Thus ( x∧ y)∧ z = x∧( y∧ z). The dual form of associativity is likewise seen. The algebraic closure is thus indeed a skew lattice and the theorem follows. £ 72 II: Skew Lattices Returning to primitive skew lattices, let (S; ≻) be a quasi-ordered set. A primitive covering P of (S; ≻) consists of (1) an assignment of a rectangular skew lattice structure to each equivalence class A of (S; ≻) and (2) to any comparable pair of equivalence classes A > B a primitive skew lattice structure is assigned that extends the separate rectangular structures on A and B in such a way that A > B as separate D-classes. Given a partial skew lattice , (S; , ∨, ∧) its canonical primitive covering is the class of all maximal primitive subalgebras of (S; ≻, ∨, ∧). Given a primitive covering P of a quasi-ordered set (S; ≻), conditional operations ∨ and ∧ are defined on any ≻-comparable pair of elements e and f in S by letting e∨ f and e∧ f be the join and meet respectively given in any primitive subalgebra of P containing both. (This subalgebra is unique if e and f are not equivalent. Otherwise, e and f lie in a common rectangular subalgebra where ∨ and ∧ are defined.) One can ask: is a partial skew lattice? If “yes”, then P (S; , ∨, ∧) would be its canonical primitive covering. Put otherwise: when is a primitive covering of given quasi-ordered set (S; ≻) the canonical primitive covering of some partial skew lattice on (S; ≻)? Or: when are the induced operations linearly associative? (S; , ∨, ∧) So let a primitive covering P of (S; ≻) be given and consider equivalence classes A > B. If B j is an A-coset in B, then the lower coset projection of A onto B j is the function pj: A → B (note that B is the codomain) projecting each element of A onto its unique image in B j. Clearly p j( a) = a∧ b∧ a for any b in B j. Similarly for each B-coset A i in A, an upper coset projection qi: B → A is given by qi( b) = b∨ a∨ b for any a in A i. When A = B, set p = q = 1A. We let Proj (S; ≻) [respectively, Proj (S; ≻)] denote the family of all lower [upper] coset projections ↓ ↑ between comparable equivalence classes of (S; ≻). If composites of lower [ upper] coset projections are also lower [ upper] coset projections, then Proj (S; ≻) forms the category of lower ↓ coset projections and Proj (S; ≻) forms the category of upper coset projections. ↑ Theorem 2.5.3. The partial algebra induced from a primitive covering P (S; , ∨, ∧) of a quasi-ordered set (S; ≻) is a partial skew lattice precisely when both coset projection families, Proj (S; ≻) and Proj (S; ≻), form categories under ordinary composition of functions. ↓ ↑ Proof. Suppose that Proj (S; ≻) and Proj (S; ≻) are categories under the usual composition of ↓ ↑ functions. Given comparable classes A > B > C, with a ∈ A, b ∈ B and c ∈ C we first show that a∧( b∧ c) = ( a∧ b)∧ c. First set d = b∧ c∧ b in C. Using only primitive operations and projections: a∧( b∧ c) = a∧( d∧ c) = ( a∧ d)∧ c = pd( a)∧ d∧ c = pd( a)∧ c and ( a∧ b)∧ c = ( pb[ a]∧ b)∧ c = pc( pb[ a]∧ b)∧ c = pcpb[ a]∧ pc[ b]∧ c == pcpb[ a]∧ c. where pb: A → B, pc: B → C and pd: A → C denote coset projections with b, c and d respectively in their images. By our assumption about Proj (S; ≻) the composition p ↓ cpb: A → C is either pd or their images are disjoint in C. Given aʹ ∈ A such that aʹ > b we have pcpb[ aʹ] = pc[ b] = d. Thus the images of pcpb and pd overlap and so are equal. Hence a∧( b∧ c) = ( a∧ b)∧ c for all a ∈A. 73 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Next observe that a∧( c∧ b) = a∧( c∧( c∧ b)) = ( a∧ c)∧( c∧ b) in A∪C while ( a∧ c)∧ b = (( a∧ c)∧ c)∧ b = ( a∧ c)∧( c∧ b) in B∪C so that a∧( c∧ b) = ( a∧ c)∧ b. Also c∧( a∧ b) = c∧( a∧ b)∧ b = c∧ b in B∪C, while ( c∧ a)∧ b =c∧( c∧ a)∧ b = c∧ b so that c∧( a∧ b) =( c∧ a)∧ b. The three other cases of potential associativity under ∧ with a, b and c are left-right reflections of cases already considered and must also hold. Finally, the six cases involving ∨ are the duals of the cases considered. Hence, using our assumption about Proj (S; ≻), the dual ↑ arguments in all cases for associativity of the ∨-product involving a, b and c in some order will be successful. Thus is indeed a partial skew lattice. (S; , ∨, ∧) Conversely, let be a partial skew lattice. Conditional regularity yields (S; , ∨, ∧) ( a∧ b∧ a)∧ c∧( a∧ b∧ a) = a∧( b∧ c∧ b)∧ a for all a ≻ b ≻ c in S. Thus, pcpb = pb (S; ∧ c∧ b and Proj ≻) is seen to be a category. In similar ↓ fashion, so is Proj (S; ≻). £ ↑ Combining the above two results we have: Theorem 2.5.4. A quasi-ordered set (S; ≻) with a covering P of primitive skew lattices is the primitive covering of a (necessarily unique) skew lattice (S; ∨, ∧) if and only if i) Both coset projection families, Proj (S; ≻) and Proj (S; ≻), form categories under the ↓ ↑ usual composition of functions. ii) The equivalence classes of (S; ≻) form a lattice under their usual partial ordering. iii) Each pair of equivalence classes is orthogonal in both their join and meet classes. £ The case for normal skew lattices From the perspective of coset bijections and projections, the significant features of normal skew lattices are (1) that they are strictly categorical and (2) that for each primitive sub- algebra A > B there is exactly one A-coset in the lower D-class B, namely B, and thus exactly one projection of A onto B. One thus has a situation like following, where the ϕ i are coset bijections from individual B-cosets A i in A onto B. A A1 A2 A3 ∨ ≅ ↓≅ ≅ p = ϕ ∪ϕ ∪ϕ B 1 2 3 B B Clearly upward projections are just the upward coset bijections. We thus modify our discussion of these matters in a way that is more commensurate to the situation for normal skew lattices. A projective pair is a pair (K, k) where ( i) K: A → B is a regular epimorphism of rectangular skew lattices, that is, a factorization ϕ: J × B ≅ A exists such K o ϕ is the B-coordinate 74 II: Skew Lattices projection of J × B upon B. ( ii) k is a set of monomorphisms kj: B → A called injections such that the compositions –1 ϕ kj are precisely the canonical injections b → ( j, b) of B into the various { j} × B. Clearly the inverse injections k –1 j jointly decompose the projection K. Any factorization ϕ for which ( i) and ( ii) hold is said to be compatible with the projection. • ⎡ • • • • • ⎤ • ⎡ •⎤ Example 2.5.2. A = ⎢ ⎥ → B = • [ • •] with J = • ⎢ • • • • • ⎣ ⎦⎥ • • ⎣ ⎢ ⎦ ⎥ . K is the union of the four obvious isomorphisms of each of the four displayed quadrants of A onto B. The four injections are the four embeddings of B upon each of the four quadrants given by the inverse isomorphisms. £ Rectangular skew lattices and projective pairs form a category. Indeed, if (K, k): A → B and (L, l): B → C are projections, then the composite projection is (L, l)(K, k) = (LK, kl): A → C upon setting kl = { k o l⎮ k ∈ k, l ∈ l}. This composition is well-defined and associative. Moreover if ϕ: J × B ≅ A is a compatible factorization for (K, k): A → B and ψ: Jʹ × C ≅ B is a compatible factorization for (L, l): B → C, then a compatible factorization for (LK, kl): A → C is given by J × J′ × C ⎯⎯ → ⎯ 1 ×ψ J × B ϕ ⎯ → ⎯ A where kjl ( c) = k j , c)) = ϕ( j, ψ( j′, c)) for c ∈C. j′ j (ψ( ′ J Lemma 2.5.5. Let A > B be two D- classes in a normal skew lattice, let K: A → B be the unique projection of A onto B defined implicitly by x ≥ K( x) and let k be the let of all coset bijections from B onto cosets of A . Together the pair (K, k) forms a projective pair. (We call this pair the natural projective pair form A to B.) Proof. For any b in B, a factorization is given by ϕ: b∨A∨ b × B → A, where ϕ( x, y) = y∨ x∨ y. Here b∨A∨ b is the image set { b∨ a∨ b⎪ a ∈A} = { a ∈A⎪ a ≥ b} of b in A that naturally parametrizes the cosets of B in A. Clearly ϕ is a bijection. Given the rectangular situation one need only show that ϕ is a ∨-homomorphism; but ( y∨ x∨ y)∨( yʹ∨ xʹ∨ yʹ) = ( y∨ yʹ)∨( x∨ xʹ)∨( y∨ yʹ) is given by regularity, since x, xʹ ≻ y, yʹ. £ A rectangular functor on a lattice T is a functor K from (T, ≥) to the category RP of rectangular skew lattices and projective pairs. K is separable if K( s) ∩ K( t) = ∅ for all s ≠ t ∈ T. For s ≥ t ∈ T, the projection from K( s) to K( t) is denoted by K( s, t) and its injections (upward coset bijections) by k( s, t, i) with i parametrizing the various injections. Lemma 2.5.6. If S is a normal skew lattice with T = S/D, then K: T → RP defined as in the previous lemma is a rectangular functor. Proof. This follows from the previous lemma and the fact that any normal skew lattice is strictly categorical. £ 75 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Given a lattice T and a separable, rectangular functor K from (T, ≥) to the category RP, a normal ∧-band may be constructed on S = ∪ s∈TK(s) by setting a∧ b = K( s, s∧ t)[ a] ∧ K( t, s∧ t)[ b], for a ∈ K(s) and b ∈ K(t). This is, of course, the Yamada-Kimura construction. One would like to be able to use injections to define a join operation ∨ on S and thus turn this normal band into a normal skew lattice. To do so, requires a key concept from Section 2.4. Given s, t ∈ T with join n ∈ T we say that projections K( n, s) and K( n, t) are orthogonal in K( n) if (i) for each a ∈ K( s), its pre-image K( n, s)–1[ a] in K( n) lies in the image of a unique injection k( n, t, j) from K( t) to K( n) and similarly (ii) for each b ∈ K( t), its pre-image K( n, t)–1[ b] in K( n) lies in the image of a unique injection k( n, s, j) from K( s) to K( n). Our observations above combined with earlier results in this section yield the following extension of the description of normal bands by Yamada and Kimura [1958]: Theorem 2.5.7. Let K be a separable, rectangular functor defined on a lattice T such that for every join situation n = s∨ t in T the projections K( n, s) and K( n, t) are orthogonal in K( n). Then S = ∪ s∈TK(s) becomes a normal skew lattice with K providing the system of natural projections and coprojections, if given a in K( s) and b in K( t), their meet and join are defined by a∧ b = K( s, m)[ a] ∧ K( t, m)[ b] and a∨ b = k( n, s, i)[ a] ∨ k( n, t, j)[ b] where m = s∧ t, n = s∨ t, the image of k( n, s, i) contains K( n, t)–1[ b] and the image of k( n, t, j) contains K( n, s)–1[ a]. Conversely, every normal skew lattice arises in this fashion. £ 2.6 Decompositions of normal, symmetric skew lattices Given rectangular skew lattices I and B, let A be their direct product I × B. A normal primitive skew lattice PI,B with D-class structure A > B is given by letting B be a full A-coset in itself and for each i ∈I, letting { i} × B be a full B-coset in A, and using the coset bijections: ϕ −1 −1 i : { i} × B → B and ϕ i : B → { i} × B given by ϕ i ( i, b) = b and ϕ i ( b) = ( i, b) . Clearly PI,B ≅ I0 × B where I0 is just I with a zero element 0 adjoined, so that I > {0}. Our first decomposition result states that for normal, primitive skew lattices this is essentially all there is. Lemma 2.6.1. Let P be a normal, primitive skew lattice with D- class structure A > B, and let I be a set of indices for the B -cosets A i in A. Then a rectangular skew lattice structure exists on I such that I0 × B ≅ P. Given b ∈B, I can be given as the image set b∨A∨ b of b in A, in which case I0 ≅ I b = I∪{ b} = b∨S∨ b. An isomorphism θ: I b × B ≅ S is given by θ( x, y) = y∨ x∨ y, the unique image of y in B∨ x∨B in S for all ( x, y) ∈ I b × B. 76 II: Skew Lattices Proof. The basic coset structure insures that θ is at least a bijection. To see that θ as defined is an isomorphism, first observe that for x, xʹ in I b and y, yʹ in B, regularity gives: θ( x, y) ∨ θ( xʹ , yʹ) = y∨ x∨ y ∨ yʹ∨ xʹ∨ yʹ = y∨ x∨ xʹ∨ yʹ = y∨ yʹ∨ x∨ xʹ∨ y∨ yʹ = θ( i∨ j, b∨ bʹ). Thus θ is at least a ∨-isomorphism of skew lattices, in which case it is also a full isomorphism between corresponding D-classes, where u∧ v = v∧ u holds. Suppose say x = b. Then we have both θ( b, y) ∧ θ( xʹ , yʹ) = y ∧ yʹ∨ xʹ∨ yʹ = y∧ yʹ since yʹ is the unique image of yʹ∨ xʹ∨ yʹ in B, while θ( b∧ xʹ , y∧ yʹ) = ( y∧ yʹ)∨( b∧ xʹ)∨( y∨ yʹ) = y∨ yʹ since b∧ xʹ = b. The case where xʹ = b is similar. £ This simple result can be extended several ways. We begin with normal skew lattices possessing D-classes that are minimal with respect to the partial ordering of D-classes. Proposition 2.6.2. Let S be a normal skew lattice with a minimal D- class B. Pick b ∈ B and let T be the subalgebra b∨S∨ b of S given as { b∨ x∨ b⎪ x ∈ S} or equivalently { x ∈ S⎪ x ≥ b}. Then b is the zero element of T , and an isomorphism θ: T × B → S is given by θ( x, y) = y∨ x∨ y. Proof. That θ: T × B → S is a bijection, that between corresponding D-classes an isomorphism, follows from the lemma. Given ( x, y), ( xʹ, yʹ) in T × B, that θ( x, y) ∨ θ( xʹ , b) = θ( x∨ xʹ , y∨ b) is seen exactly as above so that θ is at least a ∨-isomorphism. Next, observe that θ( x, y) ∧ θ( xʹ , yʹ) = ( y∨ x∨ y) ∧ ( yʹ∨ xʹ∨ yʹ) ≤ ( yʹ∨ xʹ∨ yʹ) ∨ ( y∨ x∨ y) = yʹ∨ xʹ∨ x∨ y since u∧ v ≤ v∨ u for skew lattices in general. θ( x∧ xʹ , y∧ yʹ) = ( y∧ yʹ)∨( x∧ xʹ)∨( y∧ yʹ), on the other hand, is D-related to x∧ xʹ and thus to ( y∨ x∨ y)∧( yʹ∨ xʹ∨ yʹ). But ( y∧ yʹ)∨( x∧ xʹ)∨( y∧ yʹ) ≤ yʹ∨ xʹ∨ x∨ y also. Indeed: ( y∧ yʹ)∨( x∧ xʹ)∨( y∧ yʹ)∨( yʹ∨ xʹ∨ x∨ y) = ( y∧ yʹ)∨( x∧ xʹ)∨( yʹ∨ xʹ∨ x∨ y) = ( y∧ yʹ)∨( x∧ xʹ)∨( xʹ∨ x∨ y) = ( y∧ yʹ)∨( xʹ∨ x∨ y) = ( y∧ yʹ)∨ yʹ∨( xʹ∨ x∨ y) = yʹ∨ xʹ∨ x∨ y by a combination of absorption and regularity. Likewise: ( yʹ∨ xʹ∨ x∨ y) ∨ ( y∧ yʹ)∨( x∧ xʹ)∨( y∧ yʹ) = yʹ∨ xʹ∨ x∨ y. Since θ( x, y)∧θ( xʹ , yʹ) is D-equivalent to θ( x∧ xʹ , y∧ yʹ) in a normal skew lattice with a common upper bound in (S, ≥), they are equal, i.e., θ( x, y) ∧ θ( xʹ , yʹ) = θ( x∧ xʹ , y∧ yʹ). £ As an application, let A > B > C be a normal skew chain. Then first, this skew chain is isomorphic to the product of a skew chain Aʹ > Bʹ > {0} with C. But Aʹ > Bʹ in turn is isomorphic to the product of a skew chain Aʺ > {0} with Bʹ. Thus A > B > C essentially is Aʺ×Bʹ×C > Bʹ×C > C with all downward projections and upward coset injections being the coordinate-wise functions: p( a, b, c) projects down to ( b, c) or even further down to c, while say –1 ϕ a ( b, c) = ( a, b, c). In general we have: 77 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 2.6.3. For each finite chain T, to within isomorphism every normal skew chain S with maximal lattice image isomorphic to T is obtained as follows. Take a T- indexed family of rectangular skew lattices {X( t) | t ∈T}. For each t ∈T, set D( t) = ∏ s ≤ t X( s). Then for t 1 < t 2, the primitive subalgebra with D -classes D( t 2) = ∏ ( ) > ∏ ( ) is determined s ≤ t X s D( t 1) = X s 2 s ≤ t 1 by letting the projection from D( t 2) onto D( t 1) and the coset bijections from D( t 1) into D( t 2) be the coordinate-wise projection and injections between these products. £ Thus given say t 1 ≤ t 2, ( x ∧ ∧ 1, … , xt ) ∧ ( y ) = ( x y y ) and 1 1, … , yt 2 1 1, … , xt 1 t 1 ( x ) ∨ ( y ) = ( x ∨ y ∨ y , y ,..., y ) . 1, … , xt 1 1, … , yt 2 1 1, … , xt 1 t 1 1+ t t 1 2 Alternatively, such a skew chain S is isomorphic to a fibered product of “near constant” skew chains over a common maximal chain image. In the case of A > B > C above, the skew chain is isomorphic to the fibered product of the following skew chains over 2 > 1 > 0: C ′ B A′ ≅  ≅  C ′ B {0}  ≅   C {0} {0}. We saw in Theorem 2.4.10 that symmetric skew lattices are characterized by the fact that all skew diamonds have some special properties. For symmetric normal skew lattices we have: Theorem 2.6.4. A normal skew lattice S is symmetric if and only if for every incomparable pair of D- classes A and B of S, the skew diamond SA∪B generated from A ∪ B is isomorphic to a normal skew diamond of the form X × M × Y ↙ ↘ A = X × M M × Y = B ↘ ↙ M where the downward projections and upward coset-bijections are all given in coordinate-wise fashion. In this case the skew diamond decomposes as a direct product, M × X0 × Y0, where X0 and Y0 are skew chains X > {0} and Y > {0}. In particular, if A∧B = {0} , then SA∪B ≅ A0 × B0. Proof. By Proposition 2.5.2, the proof reduces to the special case where A∧B = {0}. So what is the situation in the join class? First it consists of all possible outcomes a∨ b. By symmetry, in all cases a∨ b = b∨ a. Suppose that a∨ b = aʹ∨ bʹ. Then since a∨ b > both a and aʹ in A, a = aʹ. Likewise b = bʹ. Thus the a∨ b-outcomes are all distinct, with each pair commuting. Thus the 78 II: Skew Lattices join class factors as A × B with both projections being canonical: a∨ b → a and a∨ b → b. The isomorphism of A0 × B0 with SA∪B is given by θ( x, y) = x∨ y for x in A0 and y in B0. £ All results in this section thus far involve decompositions. We attempt to develop this theme in what follows. We begin by generalizing the construction of Theorem 2.6.3. Let T be a lattice and let P be a prime filter of T. Thus P is a filter and T \ P is an ideal of T. Put otherwise: (1) For all p ∈ P and t ∈ T, p ∨ t ∈ P. (2) P is closed under ∧. (3) If s ∨ t ∈ P, then either s ∈ P or t ∈ P. Given T, P and a rectangular skew lattice X, let T[X⎮P] be the normal skew lattice defined on (P × X) ∪ (T \ P) by extending the operations on the skew lattice s P × X and the lattice ideal T \ P by setting s ∨ ( p, x) = ( s∨ p, x) = ( p, x) ∨ s and s ∧ ( p, x) = s∧ p = ( p, x) ∧ s for ( p, x) ∈ P × X and s ∈ S \ P. Any skew lattice isomorphic to T[X⎮P] is said to be P- primary over T with fiber X. Prime filters arise as inverse images f–1(1) for lattice epimorphisms f: T → 2 where 2 is the lattice 10. Thus T[X⎮P] may be viewed as the fibered product T ×2 X0 obtained by pulling the surjection X0 → 2 back along f: T → 2. More generally, let P(T) be the family of all prime filters of T, including T and let {XP⎮P ∈Pr(T)} be a corresponding family of rectangular algebras. Then the fibered product over T, ∏T{T[XP | P]⎮P ∈P(T)} is both symmetric and normal. To within isomorphism, its rectangular D-classes are given by setting D( t) = ∏{XP⎮P ∈Pr(T) & t ∈P} and using canonical coordinate projections and injections. Any skew lattice S isomorphic to such a fibered product is decomposable with the fibered product being its primary decomposition. The rest of this section is devoted to proving a main result of this section. Theorem 2.6.5. ( The Decomposition Theorem) Every symmetric normal skew lattice with a finitely generated maximal lattice image is decomposable. More generally, a symmetric normal skew lattice with a finite maximal distributive lattice image is decomposable. £ The first major step in proving this result is our next theorem. But we first need several preliminary lemmas, beginning with: Lemma 2.6.6. Let S be a symmetric, normal skew lattice and let A ≥ B be D- classes in S. If ≥ induces an isomorphism between A and B as rectangular skew lattices (making both full cosets of each other), then for any D- class C such that A ≥ C ≥ B, ≥ also induces isomorphisms between A and C and between C and B. 79 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Elements x and y in a normal skew lattice S are reflections of each other if D-classes A ≥ B exist such that (i) x and y lie in intermediate D-classes, (ii) ≥ induces an isomorphism of A with B and (iii) x and y have the same images in A and in B. (This includes the possibility that either x or y lies in either A or B or both.) Lemma 2.6.7. Reflection is an equivalence on symmetric, normal skew lattices. Proof. Let x, y and z be given where a ≥ x ≥ b, a ≥ y ≥ b, c ≥ y ≥ d and c ≥ y ≥ d with a, b, c and d lying in respective D-classes A, B, C and D such that both A ≅ B and C ≅ D under ≥. It follows that A ≅ Y and C ≅ Y under ≥ where Y = D y. Thus A ≅ A∧C ≅ C under ≥ also. Setting J = A∨C, J is isomorphic to both A and C under ≥ by Theorem 2.6.4. In similar fashion if M = B∧D, then B and D are isomorphic to M. Hence x, y and z all lie between the D-classes J and M that are isomorphic under ≥. Clearly x and y and also y and z share a common unique image a∨ c up in J as well as a common unique image b∧ d down in M. £ Lemma 2.6.8. Given a symmetric, normal skew lattice, reflection is a congruence. Proof. If ≥ induces an isomorphism for classes A > B, then for all classes C both A∨C ≅ B∨C and A∧C ≅ B∧C under ≥. In the case of A∨C ≥ B∨C we have the diagram A∨C   A B∨C   M  B If M is the meet class of A with B∨C, then M lies somewhere between A and B. (Possibly M = B.) Since A ≅ B under ≥, so is A ≅ M. Since A∨C = A∨B∨C, A∨C ≅ B∨C by Theorem 2.5.4 (since if one side of the diagram is an isomorphism, so is the other). Let bʹ ∈ M be the unique element such that b’ ≥ b. Since B and M have identical cosets in B∨C, orthogonality yields b∨ c = bʹ∨ c. But Theorem 2.6.4 implies that a∨ c ≥ bʹ∨ c, and thus a∨ c ≥ b∨ c. Likewise, c∨ a ≥ c∨ b, a∧ c ≥ b∧ c and c∧ a ≥ c∧ b. (In fact, a∧ c ≥ b∧ c and c∧ a ≥ c∧ b follow from a ≥ b in any normal skew lattice without the added assumption that A ≅ B under ≥. This implication characterizes normal bands.) It is now clear that reflection must be a congruence. £ Reflection is the maximal congruence r on S inducing isomorphisms between D-classes in S and their image D-classes in the quotient skew lattice S/r. The latter is called the reduced skew lattice of S to be denoted Srd. Clearly (Srd)rd = Srd. We say that a symmetric, normal skew lattice S is reduced if Srd = S. The following result is of independent interest. 80 II: Skew Lattices Theorem 2.6.9 ( The Reduction Theorem) Let S be a symmetric, normal skew lattice with maximal lattice image T and reduced skew lattice Srd. If D is the maximal lattice image of Srd . then the canonical epimorphisms S → T and S → Srd induce an isomorphism of S with the fibered product of T and Srd over D, S ≅ T ×D Srd; both Sred and D, moreover, are distributive. S ⎯ → ⎯ S rd ↓ pullback ↓ T= S/D and D = S rd /D. T ⎯ → ⎯ D Proof. We need only show that when S is reduced, it is distributive. We do so by showing that none of the following types of subalgebras can arise in S where in what follows A, B. C, J and M represent distinct D-classes of S. J J      A A B C or  C B      M M Suppose first that M is trivial so that in both diagrams each element of C commutes with all elements in A and B. In the left diagram C must also commute with all elements of J so that C must trivial. Similarly in the left diagram A and B must also be trivial and thus the left diagram reduces to a lattice. Dropping the added assumption that M be trivial, we see that the natural rectangular functor K of S when restricted to the left diagram is a functor of isomorphisms between distinct classes. Thus K cannot be the rectangular functor of a reduced algebra. In the right subalgebra, each n ∈ J has commuting factorizations as a∨ c and b∨ c with a ∈ A, b ∈ B and c ∈ C where clearly a > b. As a consequence, A ≅ B under ≥. Even if M is not trivial, the skew lattice on the right must factor as M × the M-trivial case. Hence still A ≅ B under ≥, which again cannot happen in a reduced symmetric, normal skew lattice. Thus the right diagram also cannot occur in S. Hence lattice D is distributive and the normal skew lattice Srd must also be distributive by Theorems 2.3.2 and/or 2.3.4. £ The structure of symmetric, normal skew lattices has been reduced to that of lattices and symmetric, normal, distributive skew lattices that is, to that of lattices and strongly distributive skew lattices. We now examine the latter, first when S/D is finite where we show that S is decomposable, thus proving a significant special case of Theorem 2.5.4. So let S be strongly distributive with S/D finite and denote the latter by T. Being a finite distributive lattice, T has a finite set π of join-irreducible elements, including its minimal element 0. The class of prime filters of T is thus given by Pr(T) = { p∨T⎮ p ∈ π}. 81 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Recall that the center Z(S) of S is the union of all trivial D-classes of S. Since S is normal, Z(S) is a (possible empty) ideal of S that is isomorphic to its image in T which is also an ideal of T. Z(S) is empty precisely when the minimal D-class of S is nontrivial. In any case, all D-classes of S that are minimal in the complement of Z(S) correspond to join-irreducible elements in T. Lemma 2.6.10. Given S and T as above, let X be a minimal D- class in the complement of Z(S), let x be fixed in X and let P be the prime filter in T induced by the image of X in π. Set Sʹ = (S \ S∨ x∨S) ∪ x∨S∨ x and let T[X⎮P] be the P- primary algebra induced by X and P. Then i) Sʹ is a subalgbra of S that is also mapped onto T by the canonical epimorphism from S. ii) An isomorphism θ: S ≅ Sʹ ×T T[X⎮P] is given by the rule ( ⎧ x ∨ y ∨ x, y ∧ x ∧ y) for all y ∈ S ∨ x ∨ S θ( y) = ⎨ . y otherwise ⎩ iii) Upon comparison in T, Z(Sʹ) is properly larger than Z(S). Proof. Sʹ is a subalgebra since x commutes with all elements in the complement of S∨ x∨S. Thus θ is at least an isomorphism off of S∨x∨S and by Theorem 2.6.3, θ also yields an isomorphism of S∨ x∨S with ( x∨S∨ x) × X. Suppose that u ∈ S∨ x∨S and w ∈ S \ S∨ x∨S are given. Then θ( u∨ w) = θ( u)∨θ( w) is equivalent to x∨ u∨ w∨ x = x∨ u∨ x∨ w and ( u∨ w)∧ x∧( u∨ w) = ( u∧ x∧ u). Since x commutes with w, the first identity holds. Because x∧ w lies in Z(S), x∧ u ≥ x∧ w and x∧( u∨ w) = x∧ u. Similarly, ( u∨ w)∧ x = u∧ x and the second identity holds. Finally, θ( u∧ w) = θ( u)∧θ( w) is equivalent to u∧ w = ( x∨ u∨ x)∧ w. Distribution in the symmetric, normal case yields ( x∨ u∨ x)∧ w = ( x∧ w) ∨ ( u∧ w) ∨ ( x∧ w) = u∧ w since u∧ w ≻ x∧ w with x∧ w in Z(S) implies u∧ w ≥ x∧ w. £ As a consequence we have the following Primary Decomposition Theorem. Theorem 2.6.11. Let S be a strongly distributive skew lattice with finite maximal lattice image T and let Pr(T) be the set of prime filters of T including T . Then to each P in Pr(T) there corresponds to a rectangular algebra XP , that is unique to within isomorphism, such that S is isomorphic to the fibered product ∏T{T[XP⎮P]⎮P ∈Pr(T)}. S is reduced if and only if XP is nontrivial for each proper prime filter P . 82 II: Skew Lattices Proof. Repeated applications of Lemma 2.6.10 enable one to pass through the prime filters of T and successfully strip primary factors off of S to obtain the decomposition. To see uniqueness, let X be a D-class corresponding to a join-irreducible element p of T and let P = p∨T be the induced prime filter. If P = T, X was just the minimal D-class of S and XP = X. Otherwise there is a maximal D-class of S lying beneath X, call it Y. Applying the Lemma to the subalgebra X∪Y, X must factor as XP × Y where to within isomorphism XP is y∨X∨ y for any y in Y. Thus XP is indeed unique to within isomorphism. Finally, if any XP for some P ≠ T is trivial, then for Y < X as just given, K(X, Y) is an isomorphism and S is not reduced. But if no XP is trivial, except possibly for P = T, then no K(A, B) with A > B can be an isomorphism since then A as an element of T belongs to a prime filter P that excludes B and thus XP is not trivial so that B is to within isomorphism a proper factor of A. £ Theorem 2.6.5 follows immediately from Theorems 2.6.9 and 2.6.11. What happens when the maximal distributive lattice image D of S is not finite? S still factors as the fibered product of its maximal lattice image T and its maximal reduced image Sred: S ≅ T ×D Sred. Since Sred is necessarily distributive, the question reduces to what can be said about arbitrary distributive, symmetric, normal skew lattices. Being a variety of algebras, every such algebra decomposes into a subdirect product of subdirectly irreducible algebras. Which distributive, symmetric, normal skew lattices are the subdirectly irreducible? To answer this, let 2 denote the lattice 1 > 0, let R2 [L2] denote the right [left] rectangular skew lattice on {1, 2} and let 3R and 3L be the results of adjoining a zero element 0 to R2 and L2, respectively (Thus 1, 2 > 0 in both cases.) Finally, let 5 denote the fibered product algebra 3L ×2 3R of order 5. Equivalently, 5 = (L2 × R2)0. Theorem 2.6.12. The only nontrivial subdirectly irreducible strongly distributive skew lattices are copies of 2, R2, L2, 3R or 3L. Every strongly distributive skew lattice is thus a subdirect product of some of these algebras and in particular can be embedded in a power of 5. Proof. We show that a nontrivial subdirectly irreducible distributive, symmetric, normal skew lattice S must be a copy of one of the stated algebras. If S is such an algebra, a pair of elements a ≠ b in S must exist that are congruent under all nonidentity congruences on S. If a and b are not D-equivalent, then since D separates them, D is the identity Δ and S is a lattice that must be a copy of 2. So assume that a ≠ b in some D-class. We define two congruences on S as follows: a a x  y if a∧ x = a∧ y and x  * y if x∧ a = y∧ a Each is clearly an equivalence on S that, thanks to normality and Theorem 2.3.4, is indeed a a a congruence. Since a D b but a ≠ b, either  or * must separate a and b and thus equal Δ. In either case A must be the maximal D-class of S. (Otherwise some aʹ > a exists in a properly 83 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond a a higher D-class, but neither  nor  * separate aʹ from a and so neither is Δ.) If S = A, then S is rectangular and so, being subdirectly irreducible, is a copy of either R2 or L2. Otherwise lower c c D-classes exist and in particular there is an element c < a. Since both  and * identify a with c c c, both are the trivial congruence, the universal relation ∇. Since  and  * equal ∇ for all c < a, A must have a unique singleton lower class, {0} and thus S = A0. Since S is subdirectly irreducible, A itself is a copy of either R2 or L2 so that S is a copy of either 3R or 3L. £ Corollary 2.6.13. The variety of strongly distributive skew lattices is generated by 5. The variety of symmetric, normal, skew lattices is generated by 5 plus the variety of lattices. £ We saw in Section 2.3 normal skew lattices arise as maximal normal bands in rings. Before proceeding to the next section, we offer an example of somewhat different character, an example that both illustrates much that has occurred in this section and also sets the stage for developments in Chapter 4. Example 2.6.1. Given sets A and B, let PR(A, B) be the set of all partial function from A to B. PR(A, B) becomes a strongly distributive skew lattice upon setting f ∨ g = f ∪ g⎮G\F and f ∧ g = g⎮F∩G where F and G denote the actual functional domains in A of f and g respectively. PR(A, B) factors as ∏ a∈A PR({ a}, B) with each factor PR({ a}, B) being a copy of the primitive algebra B0, the right rectangular skew lattice on A with 0 adjoined. The various D-classes are indexed by the subset algebra 2A that forms the maximal lattice image of PR(A, B). For any subset F of A, its D-class is precisely BF. If G ⊆ F, K : BF → BG is just the natural coordinate-wise (BF , BG ) projection. If A is finite, PR(A, B) has the primary decomposition ∏ {2A[ B a P a ] a ∈ A } 2A where P a is the principal filter generated by a. £ Example 2.6.2. Given sets A and B, the left-handed variant of the above skew lattice on PL(A, B) is given by setting f ∨ g = g ∪ f⎮F\G and f ∧ g = f⎮F∩G. Remarks similar to those made above can be made here also. £ By a ring of partial functions is meant any subalgebra PR(A, B) for some pair of sets A and B. The following variation of Theorem 2.6.12 holds. By a full ring of partial functions is meant a subalgebra S of some PR(A, B) such that each a ∈ A correspond to a subset B a ⊆ B such that the primary decomposition of PR(A, B) given above restricts to the primary decomposition ∏ {2A[B a P a] a ∈ A } . 2A 84 II: Skew Lattices Theorem 2.6.14. Every right-handed, strongly distributive skew lattice can be embedded in a power of 3R. Equivalently, every such skew lattice can be embedded in some PR(A, {1, 2}). Thus, every right-handed strongly distributive skew lattice S is isomorphic to a ring of partial functions. If S/D is finite, then S is isomorphic to a full ring of partial functions. £ Example 2.6.3 (Example 2.3.4 revisited). This matrix example is isomorphic to the direct product of the primitive algebras (F × F)0 where F × F is the rectangular algebra, that is, it is isomorphic to the power [(F × F)0] n. £ As indicated above, a natural continuation of these ideas is given in Chapter 4, and in particular, in the first section. 85 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Historical remarks The material in Sections 2.1 and 2.2 appeared in Jonathan Leech’s 1989 paper in Algebra Universalis. The material in Sections 2.3 and 2.6 appeared in his 1992 paper in the Semigroup Forum. The material in Sections 2.4 and 2.5 originated in his 1993 paper in the Transactions of the American Mathematical Society. Highlights from these papers appeared later in his 1996 survey article in the Semigroup Forum. An important result from his 1993 Transactions paper, not presented here is the fact that the free symmetric skew lattice on two generators is infinite. As a consequence, the free skew lattice on two generators must be infinite. Thus (symmetric) skew lattices are not locally finite. This contrasts with the case for lattices where the free lattice on 2 generators has four elements, but the free lattice on three generators is infinite. In her 2011 paper in Algebra Universalis, Karin Cvetko-Vah showed that for symmetric skew lattices that are also categorical, the free algebra on two generators is finite of order 16. (But then thanks to the case for lattices, the free such algebra on three generators must be infinite.) Her result echoed the case for small skew lattices (≤ 2 generators) in rings that had been studied in Leech’s 2005 paper below. References Jonathan Leech, Skew lattices in rings, Algebra Universalis, 26 (1989), 48-72. Normal skew lattices, Semigroup Forum, 44 (1992), 1-8. The geometric structure of skew lattices, Transactions of the American Math. Society, 335 (1993), 823-845. Recent developments in the theory of skew lattices, Semigroup Forum, 52 (1996), 7-24. Small Skew Lattices in Rings, Semigroup Forum, 70 (2005), 307-311. Karin Cvetko-Vah, On strongly symmetric skew lattices, Algebra Universalis, 66 (2011), 99-113. Internal decompositions of skew lattices, Communications in Algebra 35 (2006), 243 -247. 86 III: QUASILATTICES, PARALATTICES & THEIR CONGRUENCES We take a closer look at quasilattices, paralattices, and especially refined quasilattices which by definition are simultaneously algebras of both types. Particular attention is given to their congruence lattices and to related topics such as Green’s equivalences and simple algebras. Since all skew lattices are refined quasilattices, our study has implications for skew lattices. In the first section we consider quasilattices where D(∨) = D(∧), with D denoting the common congruence. These are the noncommutative lattices for which the Clifford-McLean Theorem holds: the maximal lattice image of a quasilattice Q is Q/D and its maximal antilattice subalgebras are its D-classes. The congruence D also plays an major role in the congruence lattice Con( Q) of a quasilattice Q. For instance, given a family of congruences {θ i}, both D ∧ supi(θ i) = supi(D ∧ θ i) and its dual hold. (Theorem 3.1.1) As a consequence, Con( Q) is a subdirect product of the interval sublattices [D, ∇] and [Δ,D], where ∇ is the universal congruence where Δ is the identity congruence. (Theorem 3.1.2) In particular, Con( Q) is naturally a copy of the direct product [D, ∇] × [Δ,D] precisely when Q itself factors as the direct product T × A of a lattice T and an antilattice A. (Theorem 3.1.3) Due to these theorems, the only simple quasilattices (in that the congruence lattice reduces to {Δ, ∇}) are simple lattices and simple antilattices. Section 2 addresses the topic of simple antilattices. Its main result Theorem 3.2.3 states that finite simple antilattices exist for all composite orders greater than 5. Antilattices of odd prime order are trivially non-simple. In the remaining cases, algebras of orders 1 or 2 are always simple, while simple antilattices of order 4 are shown to be impossible. In the next section we look at noncommutative lattices that are regular in the strongest sense: L(∧), R(∧), L(∨) and R(∨) are congruences relative to both ∨ and ∧. Flat quasilattices (where D(∨) is either L(∨) or R(∨) and likewise D(∧) is either L(∧) or R(∧), with the two remaining Green’s relations being equality) are trivially regular since D is both unambiguous and a congruence on quasilattices (Theorem 3.3.2). Moreover, every regular quasilattice factors as the fibered product of its four possible maximal flat images (Theorem 3.2.4). In Section 4 we study paralattices, and especially refined quasilattices that are paralattices and quasilattices simultaneously. A paralattice S is also a quasilattice if and only if both D(∨) and D(∧) are congruences, in which case they meld into a single relation D. A regular paralattice is 87 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond thus a refined quasilattice. In fact it is the fibered product of its maximal skew lattice and skew* lattice images. (Corollary 3.4.6.) As a consequence we have the following sublattice of the lattice of quasilattice varieties: Refined quasilattices  Regular paralattices   skewlattices skew * lattices   lattices In general a refined quasilattice is partially regular in that L(∨) and R(∨) are congruences relative to ∨, while L(∧) and R(∧) are congruences relative to ∧. Even if not fully regular, refined paralattices behave very much like skew(*) lattices. This aspect is developed in some detail in the discussion from Theorem 3.4.7 through Theorem 3.4.14. The latter asserts that each refined quasilattice, if not isomorphic, is at least isotopic to some skew lattice. (The definition of “isotopic” is given in the section). Indeed every refined quasilattice can be viewed as the result of taking a left-handed skew lattice and mildly scrambling its ∨- and ∧-computational details. (See the remarks after Theorem 3.4.14.) Thus here is a sense in which refined quasilattices are roughly skew lattices. In Section 3.5 we study the effects of various distributive identities. Theorem 3.5.1 asserts that a double band (S; ∨, ∧) satisfying both a∧( b ∨ c)∧ a = ( a∧ b∧ a) ∨ ( a∧ c∧ a) and its dual a∨( b ∧ c)∨ a = ( a∨ b∨ a) ∧ ( a∨ c∨ a) is a quasilattice if and only if it a paralattice. Theorem 3.5.2 asserts that such a quasilattice [paralattice] satisfies both strengthened identities a∧( b ∨ c)∧ d = ( a∧ b∧ d) ∨ ( a∧ c∧ d) and its dual a∨( b ∧ c)∨ d = ( a∨ b∨ d) ∧ ( a∨ c∨ d) if and only if it factors as the product of a distributive lattice and an antilattice. Finally, any given quasilattice [paralattice] factors as the product of a distributive lattice and a regular antilattice if and only if the even stronger identities, a∧( b ∨ c) = ( a∧ b) ∨ ( a∧ c), ( b ∨ c)∧ d = ( b∧ d) ∨ ( c∧ d) and their duals, hold. The sixth and final section is rather “recreational” in nature. Extending Section 3.2, we consider ways that magic squares, finite planes and other rectangular designs can be used to create simple antilattices. Many classic magic squares, beginning with the classic Lo Shu, provide examples of simple antilattices. 8 1 6 ⎡8 1 6⎤ 1 ⎡ 2 3⎤ ⎢ ⎥ ⎢ ⎥ 3 5 7 ⎯ → ⎯ (∧) 3 5 7 ⎢ ⎥ (∨) 4 5 6 ⎢ ⎥ 4 9 2 ⎢4 9 2 ⎣ ⎦⎥ ⎢7 8 9 ⎣ ⎦⎥ Conditions for simplicity are given for various classes of antilattices derived from finite designs. (See, e.g., Theorems 4.6.10 and 4.6.11.) 88 III: Quasilattices, Paralattices and their Congruences 3.1 Congruences on quasilattices Recall that a quasilattice is a noncommutative lattice ( Q; ∨, ∧) for which the natural quasi-orders (∨)≺ and (∧)≻ dualize each other: x (∨)≺ y iff x (∧)≻ y, that is, y∨ x∨ y = y iff x∧ y∧ x = x. Corollary 1.3.5 asserts that a noncommutative lattice is a quasilattice if and only if D(∨) = D(∧) with the common relation D being a congruence, in which case Q/D is the maximal lattice image of Q and the D-classes are the maximal rectangular subalgebras of Q. Thus quasilattices are precisely the class of those noncommutative lattices that follow the Clifford-McClean Theorem. Indeed we might say that quasilattices are the most natural noncommutative generalizations of lattices in that they have the rough shape of a lattice. They also have some striking properties that have no precursors for regular bands. Theorem 3.1.1. For any quasilattice ( Q; ∨, ∧), the following hold: i) For all congruences θ on Q, D o θ = θ o D = θ ∨ D. ii) Given a family of congruences {θ i} on Q: D ∨ infi(θ i) = infi(D ∨ θ i) and D ∧ supi(θ i) = supi(D ∧ θ i). Proof. Let x, y ∈ Q be such that x Doθ y. Hence u ∈ Q exists such that x D u θ y. Set w = ( x∨ y∨ x) ∧ y ∧ ( x∨ y∨ x). Then x = ( x∨ u∨ x)∧ u∧( x∨ u∨ x) θ w D y so that D o θ ⊆ θ o D. But D = D–1 and –1 –1 –1 θ = θ , so that θ o D = θ o D –1 = (D o θ)–1 ⊆ (θ o D) –1 = D–1oθ = D o θ, and D o θ = θ o D follows with D o θ being precisely θ ∨ D. Thanks to (i), we next need to show D o infi(θ i) = infi(D o θ i). That we have D o infi(θ i) ⊆ infi(D o θ i) is clear. So let x infi(D o θ i) y in Q. Then for each i, some ui in Q exists such that x D ui θ i y. Set w = ( y∨ x∨ y)∧ x∧( y∨ x∨ y). Then y θ i ui = ( ui∧ x∧ ui)∨ x∨( ui∧ x∧ ui) θ w D x or just, x D w θ i y. Since w works for all i, x D w ( infiθ i) y and infi(D o θ i) ⊆ D o infi(θ i) is seen. Finally, clearly supi(D ∧ θ i) ⊆ D ∧ supi(θ i). So let x (D ∧ supi(θ)) y for some x, y ∈ Q. Thus x D y and u 1,… , un exist such that x = u 0 θ1 u 1 θ2 u 2 … un–1 θ n un = y. For j ≤ n set wj = ( uj∧ x∧ uj)∨ x∨( uj∧ x∧ uj). Then x D wj. In particular x = w 0, y = wn, and wj –1 θ j wj for j ≤ n. Thus x D∩θ1 w 1 D∩θ2 w 2 … wn–1 D∩θ n y so that x supi(D∩θ i) y and supi(D∧θ i) = D ∧ supi(θ i) follows. £ As a consequence we have: 89 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 3.1.2. Given a quasilattice ( Q; ∨, ∧), the map χ: Con( Q) → [D, ∇] × [Δ, D] defined by χ(θ) = (D∨θ, D∩θ) is an embedding of complete lattices yielding Con( Q) to be a subdirect product of [D, ∇] and [Δ, D]. Proof. That χ is a homomorphism of complete lattices follows from the above theorem. To see that χ is 1-1, observe that for any congruence θ on Q, xθ y iff xθ x∧ y∧ x, x∧ y∧ xθ y∧ x∧ y and y∧ x∧ yθ y. In general, x∧ y∧ x θ y∧ x∧ y iff x∧ y∧ x θ∩D y∧ x∧ y as x∧ y∧ x D y∧ x∧ y. Also x θ x∧ y∧ x iff x D∨θ x∧ y∧ x, and likewise y∧ x∧ y θ y iff y∧ x∧ y D∨θ y. The θ ⇒ θ∨D direction is clear. So suppose that x D∨θ x∧ y∧ x. Thus by Theorem 3.1.1(i), u ∈ Q exists such that x D u θ x∧ y∧ x. But then x = x∧ u∧ x θ x∧( x∧ y∧ x)∧ x = x∧ y∧ x. Similarly y∧ x∧ y D∨θ y implies y∧ x∧ y θ y. Consequently: x θ y if and only if x D∨θ x∧ y∧ x, x∧ y∧ x θ∩D y∧ x∧ y and y∧ x∧ y D∨θ y. But this means that θ is determined by D∩θ and D∨θ, so that χ is an embedding. Since [Δ, D] and [D, ∇] are sublattices of Con( Q) with D∩θ = θ for all θ ∈ [Δ, D] and D∨θ = θ for all θ ∈ [D, ∇], Con( Q) is embedded as a subdirect product. £ When does Con( Q) factor directly as [D, ∇] × [Δ, D]? To begin, a quasilattice Q splits if it is isomorphic with the product T × A of a lattice T and an antilattice A. In what follows ρ denotes the smallest congruence containing ≥. Theorem 3.1.3. Given a quasilattice Q, the following are equivalent: i) Q splits. ii) D ∩ρ = Δ. iii) A congruence θ exists such that D∩θ = Δ and Doθ = ∇. iv) For all x, y ∈ Q, x ρ y iff x∨ y = y∨ x [ alternatively, x∧ y = y∧ x]. v) Con( Q) ≅ [D, ∇] × [Δ,D] under the map θ → (θoD, θ∩D). Proof. Assume (i). Indeed, let Q = T × A with T and A as above. The D-classes of Q consist of pairs ( t, a) having common T-coordinates and the ρ-classes consist of pairs ( t, a) having common A-coordinates. From this, (ii) follows. Since Doρ = ∇ always holds, (ii) implies (iii). Suppose that (iii) holds for a congruence θ. Thus each θ-class meets each D-class at a unique element. Let χ: Q → Q/D × Q/θ be the homomorphism defined by χ( x) = ( x D, xθ), where x D and xθ are the respective congruence classes of x. Since D∩θ = Δ, χ is one-to-one. Since each D -class meets each θ-class, every possible pair ( x D, yθ) is in the image of χ, so that χ is an isomorphism. Since each θ-class meets each maximal rectangular subalgebra of Q, Q/θ is rectangular. Since Q/D is always a lattice, we have shown that Q splits. Hence (i) through (iii) are equivalent. 90 III: Quasilattices, Paralattices and their Congruences From the explicitly split case, Q = T × A, (iv) is easily seen. Conversely, since x∨ y = y∨ x implies x = y in each D-class, (iv) implies that each ρ-class uniquely meets each D-class, that is, (iv) implies (ii). But (ii) in turn implies (v). Indeed, let (θ1, θ2) ∈ [D, ∇] × [Δ, D]. Upon setting θ = (ρ∧θ1)∨θ2, the distributive properties of D yield (θoD, θ∩D) = (θ1, θ2). Hence the indicated injective homomorphism is also surjective and (v) follows. Finally, given (v), a congruence θ exists such that θoD = ∇ and θo D = Δ and (iii) holds. £ Every split quasilattice is a refined quasilattice. Within the variety of refined quasilattices, split quasilattices are characterized as follows. Theorem 3.1.4. Split quasilattices form the subvariety of refined quasilattices S for which ( S, ∨) and ( S, ∧) are normal, in that u∨ x∨ y∨ v = u∨ y∨ x∨ v and u∧ x∧ y 91∧ v = u∧ y∧ x∧ v hold, Proof. Both identities hold for lattices as well as for rectangular quasilattices, and hence also for split quasilattices. Conversely, suppose that both identities hold on a refined quasilattice. Then x∨ y = y∨ x iff x∧ y = y∧ x. For x∨ y = y∨ x implies x∨ y ≥ y, x so that x∧ y = ( x∨ y)∧ x∧ y∧( x∨ y) = ( x∨ y)∧ y∧ x∧( x∨ y) = y∧ x. The other direction is seen in similar fashion. Next, define a binary relation ρ0 by xρ0 y if x∨ y = y∨ x (or equivalently, x∧ y = y∧ x). Clearly ρ0 is both reflexive and symmetric. From x ρ0 y ρ0 z, x∨ y∨ z = y∨ x∨ z∨ y = y∨ z∨ x∨ y = z∨ y∨ x follows. Denoting this common value by u we get u ≥ x, z and x∧ z = u∧ x∧ z∧ u = u∧ z∧ x∧ u = z∧ x. Hence ρ0 is also transitive and thus an equivalence. By the identities in the theorem statement, ρ0 is seen to be a congruence. Since ≤ ⊆ ρ0, we get ρ ⊆ ρ0. Conversely, given x ρ0 y, from x ≤ x∨ y ≥ y first x ρ y and then ρ0 = ρ follows. Hence Theorem 3.1.3(iv) is satisfied and N splits. £ Corollary 3.1.5. A noncommutative lattice Q is a split quasilattice if and only if ρ∩δ = Δ. In general, Q/ρ∩δ is the maximal split quasilattice image of any quasilattice Q. Proof. Q/ρ∩δ is isomorphic to a subalgebra of the product Q/ρ × Q/δ where Q/ρ is an antilattice and Q/δ is a lattice. Thus Q/ρ∩δ splits also and is indeed isomorphic to Q/ρ × Q/δ under the map x(ρ∩δ) → ( xρ, xδ). £ We have seen that Con( Q) is a subdirect product of [Δ, D] and [D, ∇]. If { Di⎮ i ∈ I} is the set of all D-classes of Q, then an embedding D*: [Δ, D] → ∏ iCon( Di) of complete lattices is given by D*(θ) = 〈θ ∩ Di × Di)⎮ i ∈ I〉 Since ∏ iCon( Di) is in turn embedded in Con(∏ iDi), this suggests that [Δ, D] shares similarities with congruence lattices of rectangular quasilattices. On the other hand, the interval [ρ, ∇] is isomorphic to the congruence lattice Con( N/ρ) of the greatest rectangular image Q/ρ of Q. This leads us to inquire how [Δ, D] and [ρ,∇] are related. 91 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Consider the embedding χ: Con( Q) → [D, ∇] × [Δ, D] given by χ[θ] = (D ∨ θ, D ∩ θ). Since ρoD = ∇, restricting χ to [ρ, ∇] yields an embedding χ0: [ ρ, ∇] → [Δ, D] of complete lattices defined by χ0[θ] = D∩θ. When is χ0 an isomorphism? Precisely when ρ∩D = Δ. This condition is clearly necessary as it states that Δ lies in the image of χ0. Given this condition, then for all θ ∈ [Δ, D], D ∩(θ ∨ρ) = (D ∩θ)∨(D ∩ρ) = θ∨Δ = θ, which implies that χ0 is also surjective. But ρ∩D = Δ if and only if Q splits. Thus: Theorem 3.1.6. The interval [ρ, ∇] of rectangular congruences on a quasilattice Q is isomorphic to a complete sublattice of the interval [Δ, D] under the map θ → D∩θ. This embedding is an isomorphism if and only if Q splits. £ To complete our basic picture of quasilattice congruences we give a partial complement of the Clifford-McLean Theorem. In its proof, the terms ∨– morphism and ∧- morphism refer to homomorphisms with respect to the stated operations. Theorem 3.1.7. Given a lattice Λ with disjoint antilattices Dλ assigned to each λ ∈ Λ, a quasilattice structure exists on Q = ∪ D λ λ such that the maximal antilattices in Q are precisely the D and N/D Con( D ). λ ≅ Λ. In addition, ∨ and ∧ can be defined so that [Δ, D] ≅ ∏λ λ Proof. Let Φ = {ϕ( λ, µ): Dλ → Dµ⎮λ ≤ µ} be an ascending family of ∨–morphisms such that for all λ ∈ Λ, ϕ( and secondly, λ, λ) is the identity map on Dλ ϕ(µ, ν)ϕ(λ, µ) = ϕ(λ, ν) for all λ ≤ µ ≤ ν in Λ. Similarly let Ψ = {ψ(µ, λ): Dµ → Dλ⎮λ ≤ µ} be a descending family of ∧–morphisms satisfying dual properties. To define ∨, set x∨ y = ϕ( if x , y and λ, ∈ D ∈ D π)( x)∨ϕ(µ, π)( y) ∈ Dπ λ µ π = λ∨µ. Similarly, define ∧ on Q using Ψ in dual fashion. The operations ∨ and ∧ induced from Φ and Ψ yield a quasilattice ( Q; ∨, ∧) for which the D are the maximal antilattices and for λ which Q/D ≅ Λ as stated. To complete the first assertion we need only exhibit at least one pair (Φ, Ψ). This is done as follows. First, to each λ ∈ Λ pick some d in D . Next, for each λ λ λ ∈ Λ, let ϕ(λ,λ) = ψ(λ,λ) be the identity map on D as required. Finally for each strict comparison λ λ < µ in Λ, let ϕ( be the constant map sending all D to d in D . Similarly define λ, µ): Dλ → Dµ λ µ µ ψ( as the constant map sending all of D onto d . As so defined, ( µ, λ): Dµ → Dλ µ λ Φ, Ψ) satisfies all the required conditions. Moreover, for such a pair (Φ, Ψ), the embedding D*: [Δ, D] → ∏ Con( D ) defined by D*( λ λ θ) = 〈θ ∩ Dλ× Dλ⎮λ ∈Λ〉 is onto. For given 〈θ Con( D ) is at least an equivalence on Q. Thanks to the λ⎮λ ∈ Λ〉 ∈ ∏λ λ 〉, the union θ = ∪λθλ pointed character of (Φ, Ψ) as defined, θ is a congruence inducing 〈θλ⎮λ∈Λ〉. £ 92 III: Quasilattices, Paralattices and their Congruences Not all quasilattices Q = ∪ D λ λ for which Q/D ≅ Λ arise from some such pair (Φ, Ψ) of ascending and descending morphisms. Those that do are characterized by both (Q, ∨) and (Q, ∧) being normal. This is just the double assertion of Theorem 1.2.16. The above construction is relevant to the question: Given a band B with multiplication ∧, is ( B, ∧) the ∧- reduct of some quasilattice ( B, ∨, ∧)? If ( B, ∨, ∧) exists, it is called a quasilattice closure of B. It is not unique, unless B is a semilattice so that ( B, ∨, ∧) is a lattice. Clearly, it is necessary that the maximal semilattice image ( B/D, ∧) form a lattice in that given any pair of D-classes M and N in B, a D-class J exists which is the supremum of M and N in ( B/D, ∧). This condition is also sufficient. Indeed, assume that B has join classes. On each D-class M define a rectangular join ∨ M. (Perhaps, let x∨ M y = x.) Using an ascending family Φ of ∨-morphisms between the M, create an operation ∨ on all of B that yields a quasilattice, ( B, ∨, ∧). Proposition 3.1.8. A band B is a ∧- reduct of a quasilattice if and only if its maximal semilattice image B/ D is the ∧- reduct of a lattice. Returning to quasilattice congruences, note that while the interval [D, ∇] is distributive, in general Con( N) need not satisfy any identity beyond those common to all lattices. Indeed [Δ, D] need only satisfy identities common to congruence lattices of antilattices. But consider the antilattice defined on a given set N by x∧ y = x = x∨ y. In this case Con( N) is just the lattice Equ( N) of all equivalences on N. But such lattices collectively satisfy only lattice identities common to all lattices. We shall see in the next section that not all antilattices are so behaved. Consider the following questions. Given quasilattice Q, when is Con( Q) distributive? Even if Con( Q) is not distributive, under what conditions will instances of distribution occur? In response we present three assertions, the first two following immediately from Theorem 3.1.1 and the complete embedding D*: [Δ, D] in ∏ iCon( Di) where the Di are all maximal antilattices in Q. 1. Con( Q) is distributive precisely when [Δ, D] is distributive. The latter occurs when each D- class of Q has a distributive congruence lattice. (For example, this is the case if all D-classes are simple as algebras.) 2. Given particular quasilattice congruences η and θ i for i ∈ I, η ∧ s upi(θ i) = s upi(η∧θ i) holds in Con( Q) if either η or at least one of the θ i lies in [D, ∇]. We present our third assertion as a theorem where we consider the remaining Green’s equivalences L(∨), R(∨), L(∧) and R(∧) which are defined for any noncommutative lattice. 93 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 3.1.9. If Q is a quasilattice, then L(∨), R(∨), L(∧) and R(∧) permute with any congruence in [Δ, D], ∧- distribute over suprema in [Δ, D] and ∨- distribute over infima in [Δ, D], even if these four equivalences are not congruences themselves. If either L(∨), R(∨), L(∧) or R(∧) is also a congruence, then it possesses these distributive properties in Con( Q). Proof. To begin, L o θ = θ o L and R o θ = θ o R hold for any congruence θ on a rectangular band. Hence L(∨), R(∨), L(∧) and R(∧) permute with any congruence on an antilattice. Likewise, L and R distribute as stated over congruences on a rectangular band. Thus the first statement must hold at least for antilattices. The general statement involving [Δ, D] now follows since all calculations take place in D-classes. Since the join of all four equivalences with D is just D, the second statement follows from the first and Theorem 3.1.2. £ 3.2 Antilattices that are simple as algebras Recall that an algebra A is simple if Con(A) = {Δ, ∇}. Algebras of order 2 are simple. By Theorem 3.1.2 , a simple quasilattice is either a simple lattice or a simple antilattice. Our interest is in the latter case. We begin with some remarks about when simplicity cannot occur, starting with a definition. An antilattice N is flat if L(∨), R(∨), L(∧) and R(∧) are each either Δ or ∇. In the finite case, N is flat precisely when the arrays defining ∨ and ∧ are either single columns or rows. Lemma 3.2.1. All antilattices of prime order are flat; moreover every equivalence on a flat antilattice is a congruence. Thus an antilattice cannot be simple if its order is an odd prime. Proof. Given the assumption of prime order, L(∨) and R(∨) reduce to either Δ or ∇. Hence either x∨ y = x holds uniformly or else x∨ y = y holds uniformly. Similarly, either x∧ y = x holds uniformly or else x∧ y = y does. For such operations, every equivalence must be a congruence. £ By contrast, let A be the antilattice on { a, b, c, d} determined from the arrays: (∨) a b (∧) a d c d c b . Recall that x∨ y and x∧ y are the element lying in the row of x and the column of y of the relevant array, where the rows are the R-classes and the columns are the L-classes for the respective operations. Here L(∨) = L(∧) is the only proper nontrivial congruence. The lattice of congruences is thus: Δ < L < ∇. In general: 94 III: Quasilattices, Paralattices and their Congruences Proposition 3.2.2. No 4-element antiilattice is simple. Proof. Any such antilattice A falls into one of three cases. First case: A is flat. In this case Con( A) equals Equ( A) and has order greater than 2. Second case: either ( A, ∨) or ( A, ∧) is flat, but not both. Then both L and R for the nontrivial operation are congruences that are distinct from Δ and ∇. Third case: neither ( A, ∨) nor ( A, ∧) is flat. Then the elementary combinatorics of 2×2 squares forces one of L(∨) or R(∨) to equal one of L(∧) or R(∧), with the equated equivalence being a nontrivial congruence. £ Thus simple antilattices of finite order > 2 must have composite order greater than 5. We may now state: Theorem 3.2.3. Simple antilattices exist for all composite orders > 5. In all such cases, none of R(∨), L(∨), R(∧) or L(∧) are congruences. Proof. Given a set A of order mn ≥ 6 with n ≥ m ≥ 2, store its elements in each of the following m× n arrays, so that a∨ b is in the row of a and the column of b of the array on the left and a∧ b is similarly obtained from the array on the right. (First array indices are used in the second array to show how the elements have been rearranged.) a 11 a 12 a 13 ... a 1 n a 21 a 22 a 23 ... a 2 n (∨) a 31 a 32 a 33 ... a 3 n      am 1 am 2 am 3 ... amn a 11 a 12  a 1, n− m+1 a 1, n− m+2  a 1, n−2 a 1, n−1 a 21 a 22 a 23  a 2, n− m+2 a 2, n− m+3  a 2, n−1 a 31 a 32 (∧) a 33 a 34  a 3, n− m+3 a 3, n− m+4  a 41 a 42 a 43         am, m am, m+1  am, n am−1, n  a 3, n a 2, n a 1, n Let θ be a congruence on A and let distinct a, b in A exist such that aθ b. We show that θ = ∇. To begin, let a 11θ a 21 be given. Then ( a 11∨ a 1 i)θ( a 21∨ a 1 i) for i = 1, 2, …, n yields a sequence of congruent pairs a 1 iθ a 2 i in the first two rows of the ∨-array. Upon taking ∧-products of these θ-related pairs, all elements in the first two rows of the ∧-array are seen to be θ-related. In particular, a 21θ a 31. This leads first to a sequence of relations a 2 iθ a 3 i in the ∨-array and then to all elements in the first three rows of the ∧-array being θ-related. The process continues until the entire ∧-array, and hence A, is absorbed into a single θ-class, showing that θ = ∇. 95 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Suppose next that we are just given aijθ akl for some i < k. Then ai 1 = ( aij∨ a 11) θ akl∨ a 11 = ak 1. Applying f( x) =( x ∧ a 11)∨ a 11 to both sides of ai 1θ ak 1, as often as needed, eventually yields a 11θ a 12 and hence θ = ∇. Finally, suppose that ahiθ ahk with i ≠ k. If ahi and ahk lie in different rows in the (∧)-array, then ahi∧ ahk and ahk∧ ahi are θ–equivalent, but have differing first indices, thus returning us to the previous case to obtain θ = ∇. Otherwise, look at am m∧ ahi θ amm∧ ahk in the final row of the ∧-array. If both ∧-products have differing first indices we again return to the previous case. Otherwise, a 1 i = a 11∧( amm ∧ ahi) θ a 11∧( amm ∧ ahk) = a 1 k. with i < iʹ and k < kʹ. ʹ We repeat this cycle of calculations until a pair of θ-related elements with distinct first indices is eventually encountered, which must happen due to the design of the ∧-array. We are then returned to the previous case. £ By contrast, for any flat antilattice A, Con( A) is the lattice Equ( A) of all equivalences on the underlying set of A. Flat antilattices of all types together generate the variety of regular antilattices for which R(∨), L(∨), R(∧) and L(∧) are congruences. Such an algebra A factors as the direct product of flat antilattices of each type, A ≅ A( l, l) × A( l, r) × A( r, l) × A( r, r) with Con( A) correspondingly factoring as Con( A) ≅ Equ( A( l, l)) × Equ( A( l, r)) × Equ( A( r, l)) × Equ( A( r, r)). Flatness and regularity are studied in a broader context in the next section. 3.3 Regular quasilattices Recall that a noncommutative lattice is flat if one of the following pairs of identities is satisfied: ( r, l): a∨ b∨ a = b∨ a and a∧ b∧ a = a∧ b. ( l, r): a∨ b∨ a = a∨ b and a∧ b∧ a = b∧ a. ( l, l): a∨ b∨ a = a∨ b and a∧ b∧ a = a∧ b. ( r, r): a∨ b∨ a = b∨ a and a∧ b∧ a = b∧ a. Thus, being ( r, l)-flat means that D(∨) = R(∨) and D(∧) = L(∧), or equivalently, L(∨) = R(∧) = Δ. Modified remarks hold for the other three types of flatness. A noncommutative lattice is regular if R(∨) L(∨), R(∧) and L(∧) are all congruences, in which case D(∨) and D(∧) are also congruences. While flat quasilattices are regular, this is not the case for flat paralattices. In general: 96 III: Quasilattices, Paralattices and their Congruences Theorem 3.3.1. The condition that L(∨) be a congruence determines a subvariety for any variety of noncommutative lattices. It contains as a further subvariety those algebras for which L(∨) = Δ. When L(∨) is a congruence on a noncommutative lattice N, then L(∨) = Δ on N/θ for any congruence θ in [L(∨), ∇]. Similar remarks hold for D(∨), R(∨), D(∧), R(∧) and L(∧). Proof. A typical pair of L(∨)-related elements are b∨ a and a∨ b∨ a. L(∨) is compatible with ∨ and ∧ if and only if the equations, [ c∨( b∨ a)]∨[ c∨( a∨ b∨ a)] = c∨( b∨ a) and [ c∨( a∨ b∨ a)]∨[ c∨( b∨ a)] = c∨( a∨ b∨ a) together with their -∨ c, c∧- and -∧ c variants, hold. Clearly L(∨) is a congruence when L(∨) = Δ, a condition that is equivalent to the equation a∨ b∨ a = b∨ a being satisfied. The first assertion follows. If L(∨) is a congruence, then L(∨) = Δ on N/L(∨). Thus, if θ is as stated, N/θ, being a homomorphic image of N/L(∨), must also satisfy L(∨) = Δ. The verifications of the remaining cases are similar, depending on the descriptions of typical R(∨)-related elements ( a∨ b∨ a, and a∨ b), typical D(∨)-related elements ( a∨ b∨ a and b∨ a∨ b) and their ∧-variants. £ Theorem 3.3.2. Flat quasilattices are regular. Regular quasilattices, in turn, form the subvariety of quasilattices generated from all flat quasilattices. Proof. In a flat quasilattice, each of R(∨), L(∨), R(∧) and L(∧) is either D or Δ, and hence is a congruence. Conversely, given a regular quasilattice Q, define κ: Q → Q/L(∨) × Q/R(∨) by κ( x) = ( x L(∨), x R(∨)). Since L(∨)∩R(∨) = Δ, κ is an embedding of Q into a product of quasilattices for which either D = R(∨) or D = L(∨) holds on each factor. Repeating the process on each factor, but using R(∧) and L(∧) instead, we obtain a final embedding of Q into a product of four flat quasilattices, each representing a specific type of flatness. The theorem follows. £ Given a noncommutative lattice Q, let ϕ( l, l) denote the least congruence on θ for which Q/θ is ( l, l)-flat. Similarly, let ϕ( l, r), ϕ( r, l) and ϕ( r, r) denote least congruences for the other types of flatness. For regular quasilattices, Theorems 3.1.8 and 4.3.1 yield: Lemma 3.3.3. Given a regular quasilattice Q, the equivalences R(∨), L(∨), R(∧) and L(∧) all permute with each other. Moreover, ϕ( l, l) = R(∨)oR(∧), ϕ( l, r) = R(∨)oL(∧), ϕ( r, l) = L(∨)oR(∧) and ϕ( r, r) = L(∨)oL(∧). £ 97 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Given a regular quasilattice Q, let Q( l, l), Q( l, r), Q( r, l), and Q( r, r) denote the maximal flat images of each type. Define Φ: Q → Q( l, l) × Q( l, r) × Q( r, l) × Q( r, r) by: Φ( x) = ( xϕ( l, l), xϕ( l, r), xϕ( r, l), xϕ( r, r)). Φ is a monomorphism, since the involved congruences intersect to Δ. To describe its image, observe that Q shares with its four maximal flat images a common maximal lattice image, Q/D. If δ( u,v): Q( u,v) → Q/D, for ( u, v) ∈ {( l, l), ( l, r), ( r, l), ( r, r)}, are epimorphisms induced by the inclusions of the ϕ( u, v) into D, then Φ( Q) lies in {( u, v, w, y) ∈ Q( l, l) × Q( l, r) × Q( r, l) × Q( r, r)⎮δ( l, l)( u) = δ( l, r)( v) = δ( r, l)( w) = δ( r, r)( y)}, the fibered product over Q/D of the various maximal flat images of Q. In fact: Theorem 3.3.4. If Q is a regular quasilattice, then Φ is an isomorphism of Q with the fibered product over Q/D of the four maximal flat images of Q. Q ≅ Q( l, l) × Q/D Q( l, r) × Q/D Q( r, l) × Q/D Q( r, r) Proof. Restricting our attention to each D-class A of Q, the respective rectangular band structures yield first A ≅ A/R(∨) × A/L(∨) and then in turn A/R(∨) ≅ A/R(∨)/R(∧) × A/R(∨)/L(∧) ≅ A/R(∨)oR(∧) × A/R(∨)oL(∧) = A/ϕ( l, l) × A/ϕ( l, r) with A/L(∨) similarly factoring as A/ϕ( r, l) × A/ϕ( r, r). Thus class A factors as a product of its maximal flat images. In passing from D-classes to Q, the result follows. £ The above theorem is the quasilattice version of a result of Kimura [1958] about regular bands. For a general discussion of permuting congruences and fibered products, see Grätzer [1979]. 3.4 Paralattices and refined quasilattices Skew lattices are simultaneously quasilattices and paralattices and thus possess the features of both. While quasilattices possess a coherent Clifford-McLean structure, paralattices need not. Nor need (flat) paralattices be regular. Thus some of our basic intuitions about bands and quasilattices no longer always hold. We begin our look at paralattices with a special case. Unlike skew lattices and more generally quasilattices, it is possible for say ∨ to be commutative, but not ∧. (For quasilattices, if say ∨ is commutative, then D = ∨ Δ and hence D∧ = Δ, making ∧ commutative.) A band with joins, or a ∨- band, is a band B whose natural 98 III: Quasilattices, Paralattices and their Congruences partial order ≤ has natural joins. Thus, for any pair of elements x and y, their supremum x∨ y exists in ( B, ≤). Upon denoting the band multiplication by ∧, one obtains a paralattice ( B; ∨, ∧) where ( B, ∨) is a semilattice, or equivalently, where D(∨) = Δ. Thus, in this case D will denote D(∧). Viewed as paralattices, ∨ -bands form a subvariety characterized simply by x∨ y = y∨ x. Flat ∨ - bands are ∨ - bands for which ∧ is either left regular ( x∧ y∧ x = x∧ y) or right-regular ( x∧ y∧ x = y∧ x). Both types of flat ∨ - bands form subvarieties. In terms of B i and C j identities of Section 1.3, we have the following result and its left dual: Theorem 3.4.1. A flat ∨- band satisfying a∧ b∧ a = b∧ a (so that D = R ) is ∧ ∧ characterized by the associativity of ∨ and ∧ together with the following four identities, B1 and C2: a ∧ ( a ∨ b) = a = ( b ∧ a) ∨ a. B3 and C3: a ∧ ( b ∨ a) = a = a ∨ ( b ∧ a). Proof. These identities necessarily hold for such a ∨-band, as b∧ a ≤ a ≤ a∨ b = b∨ a in the coherent natural partial ordering. Conversely, any associative algebra ( N; ∨, ∧) satisfying B1, B3, C2 and C3 is at least a double band by Theorem 1.3.4. In this case B1 and C2 together assert that ≺ ≺ ≤ ∨ R dualizes ∧≺L while B3 and C3 jointly assert that ∨≺L dualizes ∧≺L making ∨ R = ∨≺L = ∨ and thus ≤. Thus N is a flat ∧≺L = ∧ ∨-band of the indicated type. £ Thanks to the dual (union) version of Theorem 1.3.13 we have: Theorem 3.4.2. The congruence lattice of a ∨- band is distributive. £ How prevalent are ∨-bands among bands? Several established classes of bands turn out to be ∨-bands. Example 3.4.1. If B is a rectangular band and set S = B1, the extension obtained by adjoining an identity element 1, then (B1, ≤) is a join semilattice characterized by 1 ≥ b for b in B. 1  • • • • • • • Examples 3.4.2. Let B be the free (left, right or 2-sided) normal band BX (where xyzw = xzyw) on alphabet X. Then B 1 is a ∨-band. Indeed elements of B (in the 2-sided case) look like aAc where A is any finite subset of X (thanks to middle commutativity). In ( B, ≤), aAc ≤ aʹ Aʹ cʹ if and only if a = aʹ, c = cʹ and A ⊆ Aʹ. Thus any finite subset of B with a common upper bound in ( B, ≤) has a least upper bound: 99 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond aA 1 c ∨ aA 2 c ∨ … ∨ aAnc = a( A 1 ∪ A 2 ∪ … ∪ An) c. ( B 1, ≤) is hence an upper semilattice. In similar fashion one handles the free left normal case (with expressions aA) and the right normal case (with expressions Ac). Indeed with a little more work one can show that if B is the free (left, right or 2-sided) regular band on X (where xyxzx = xyzx), then B 1 is a ∨-band. With even further work one can show that if B is a free band on X, then B 1 is a ∨-band. £ Dual ∨- bands are paralattices for which ∧ is commutative. Dual ∨-bands arise as reducts (S; ∨, ∩) of structurally enriched skew lattices ( S; ∨, ∧, ∩) for which the natural partial ordering ≤ has natural meets, with inf( x, y) denoted by x∩ y. In particular, they arise in the study of certain types of skew Boolean algebras. (See Section 4.2 below.) For ∨-bands and their duals (where D = D(∨)) we have: Theorem 3.4.3. D is a congruence on a [ dual] ∨- band B if and only if D = Δ and B is a lattice. In general a paralattice is also a quasilattice if and only if both D(∨) and D(∧) are congruences. Regular paralattices in particular are quasilattices. Proof. Given a D-class A of a ∨-band B, pick elements a, b in A. If D = D(∧) is a congruence, then a∨ b lies in A also with a ≤ a∨ b and b ≤ a∨ b. Being a common D-class, this forces a = a∨ b = b. Thus A is a singleton set. Hence D = Δ and B is a lattice. More generally, let P be a paralattice for which both D(∨) and D(∧) are congruences. Then P/D(∨) is a ∨-band. Since D(∧) is a congruence on P, D(∧) is also a congruence on P/D(∨) by Theorem 3.3.1. Hence P/D(∨) is a lattice by our first assertion and λ ⊆ D(∨). Similarly, λ ⊆ D(∧). Since the converse inclusions hold, D(∨) = D(∧) and P is a quasilattice by Corollary 1.3.5. Conversely, when P is also a quasilattice, D(∨) = D(∧) with the common equivalence being a congruence. £ Corollary 3.4.4. The class of all regular paralattices is just the class of regular refined quasilattices. In particular, every flat refined quasilattice is necessarily regular and is either a skew lattice or a skew* lattice. Proof. The first statement is clear. Next, take, e.g., a ( l,l)-flat refined quasilattice, P. Since P is a flat quasilattice, ≤ dualizes ∨≺L = ∨≺ dualizes ∧≺L = ∧≺. Since P is a flat paralattice, ∨≺R = ∨ ≤. Hence N is a flat skew* lattice and thus necessarily regular. ∧≺R = ∧ £ Theorem 3.4.5. The variety of regular paralattices is the join of the varieties of skew lattices and skew* lattices. In particular, every regular paralattice factors as the fibred product of its maximal skew lattice image with its maximal skew* lattice images over its maximal lattice image. 100 III: Quasilattices, Paralattices and their Congruences Proof. Clearly skew* lattices are regular paralattices. Thus the join of both varieties lies in the variety of regular paralattices. One the other hand, the larger variety is generated by its four subvarieties of flat algebras by Theorem 3.3.4. But these four subvarieties are either subvarieties of the varieties of skew lattices or the varieties of skew* lattices. The “join” assertion follows. The final remark follows again from the isomorphism of Theorem 3.3.4. £ More generally: Proposition 3.4.6. Given a refined quasilattice, both R(∨) and L(∨) are congruences with respect to ∨, while R(∧) and L(∧) are congruences with respect to ∧. Proof. First observe that ≤ is surjective between comparable D-classes: given classes, A ≤ B, then for all a ∈ A and b ∈ B, there exist bA ∈ A and aB ∈ B such that bA ≤ b and a ≤ aB. Indeed we may choose bA = b∧ a∧ b and aB = a∨ b∨ a. It follows from Theorem 1.2.19 that both ∨ and ∧ satisfy the identity xyxzx = xyzx. Since ≤ is surjective, u and v exist in the D-class of x such that x∧ y ≤ u and z∧ x ≤ v. Because u, x and v lie in a common D-class, u∧ 101x∧ 101v = u∧ 101v and thus x∧ y∧ x∧ z∧ x = ( x∧ y)∧ u∧ x∧ v∧( z∧ x) = ( x∧ y)∧ u∧ v∧( z∧ x) = x∧ y∧ z∧ x. From the theory of bands, the identity x∧ y∧ x∧ z∧ x = x∧ y∧ z∧ x implies R(∧) and L(∧) are ∧-congruences. Similarly one shows that R(∨) and L(∨) are ∨-congruences. £ From refined quasilattices to skew lattices and back again Many results established for skew lattices extend to refined quasilattices. In particular the following extensions of Theorem 2.2.1 and its immediate consequences hold, with the proofs being essentially the same. Theorem 3.4.7. Let ( S, ∨, ∧) be a refined quasilattice with D- classes A and B. If J and M are the respective join and meet classes of A and B in S/D, then J = { a∨ b⎮ a ∈ A, b ∈ B & a∨ b = b∨ a} and M = { a∧ b⎮ a ∈ A, b ∈ B & a∧ b = b∧ a}. £ Corollary 3.4.8. Given a refined quasilattice ( N, ∨, ∧) and an infinite cardinal number ℵ , then the union of all D- classes of order [ equal to or] less than ℵ is a subalgebra of N. In α α particular, the union of all finite D- classes is a subalgebra of N. £ 101 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Corollary 3.4.9. Let ( S, ∨, ∧) be a refined quasilattice with a ∈ S. The following are equivalent: 1. For all b ∈ S, a∨ b = b∨ a. 2. For all b ∈ S, a∧ b = b∧ a. 3. The D- class of a is the singleton, { a}. Thus the union of all singleton D- classes is the center of S. £ These results, in contrast with that of Theorem 3.1.7, show the effect of a coherent natural partial order on the structure of a quasilattice. It turns out that (even irregular) refined quasilattices and skew lattices are closely connected in a very precise way. To see this, we borrow a term arising in a number of other contexts both mathematical and scientific. An isotopy of quasi-lattices ( Q, ∨, ∧) and ( Qʹ, ∨ʹ, ∧ʹ) with respective natural pre-orders ≻ and ≻ʹ is a bijection f: Q → Qʹ such that x ≻ y in Q iff f( x) ≻ʹ f( y) in Qʹ. An isotopy of paralattices ( P, ∨, ∧) and ( Pʹ, ∨ʹ, ∧ʹ) with natural partial orders ≥ and ≥ʹ is a bijection f: P → Pʹ such that x ≥ y in P iff f( x) ≥ʹ f( y) in Pʹ. Finally, an isotopy of refined quasi-lattices ( S, ∨, ∧) and ( Sʹ, ∨ʹ, ∧ʹ) is any bijection from S to Sʹ that is simultaneously an isotopy of quasi-lattices and an isotopy of paralattices. In each case, all isomorphisms are isotopies. Also, in each of these cases, given an algebra ( S, ∨, ∧), upon defining ∨• by x∨• y = y∨ x and ∧• by x∧• y = y∧ x, all three derived algebras ( S, ∨•, ∧), ( S, ∨, ∧•) and ( S, ∨•, ∧•) are all isotopes of ( S, ∨, ∧) under the identity map on S. Indeed x∨• y∨• x = x∨ y∨ x and x∧• y∧• x = x∧ y∧ x on S, in the case of quasilattices, thus guaranteeing that ≤ is the same for all four algebras. Likewise x∨• y = y∨• x iff x∨ y = y∨ x on S with both outcomes agreeing, and x∧• y = y∧• x iff x∧ y = y∧ x on S with both outcomes agreeing in the case of paralattices, thus guaranteeing that ≤ is the same for all four algebras. Note that antilattices are isotopic precisely if the have the same size. In general we have the following assertions, the first of which is near-obvious: Proposition 3.4.10. Let quasilattices ( Q, ∨, ∧) and ( Qʹ, ∨ʹ, ∧ʹ) be given. Then any isotopy f: Q → Qʹ induces a unique isomorphism f*: Q/D → Qʹ/D making the following diagram commute, where δ: Q → Q/D and δʹ Qʹ → Qʹ/D denote the canonical epimorphisms. f Q ⎯ → ⎯ Q′ δ ↓ ↓ δ′ Q / D f * ⎯ → ⎯ Q′ / D 102 III: Quasilattices, Paralattices and their Congruences Moreover, bijections f : D → D α α αʹ between all corresponding D- classes of Q and Qʹ are induced upon restricting f to the various D -classes of S. Conversely, given quasilattices ( Q, ∨, ∧) and ( Qʹ, ∨ʹ, ∧ʹ), suppose that f*: Q/D → Qʹ/D is an isomorphism and that bijections f : D → D exist α α αʹ between all corresponding pairs of D -classes (relative to f* ). Then an isotopy f: Q → Qʹ is given by f = ∪ f . Any isotopy of (Q, ∨, ∧) with (Qʹ, ∨ʹ, ∧ʹ) thus arises in this manner. £ α The existence of an isotopy between quasilattices thus depends on having isomorphic maximal lattice images and corresponding D-classes of the same size. For refined quasilattices more constraints occur. Proposition 3.4.11. An isotopy f: ( S, ∨, ∧) → ( Sʹ, ∨ʹ, ∧ʹ) of refined quasilattices preserves both commuting joins and meets. Thus a∨ b = b∨ a in S implies f( a∨ b) = f( a) ∨ʹ f( b) = f( b) ∨ʹ f( a) in Sʹ. Dually, a∧ b = b∧ a in S implies f( a∧ b) = f( a) ∧ʹ f( b) = f( b) ∧ʹ f( a) in Sʹ. Proof. Suppose that a∨ b = b∨ a in S. If c denotes this common join, then c is the unique element in the join-class of the D-classes of a and b such that both c ≥ a and c ≥ b. Being an isotropy of paralattices, we have f( c) ≥ both f( a) and f( b) in S ʹ. Being an isotropy of quasilattices, we have f( c) lying in the join class of the D-classes of f( a) and f( b) in Sʹ. By uniqueness, f( a) ∨ʹ f( b) = f( c) = f( b) ∨ʹ f( a). Similarly, f preserves commuting meets. £ Michael Kinyon observed (in a personal comminication) that every refined quasilattice is isotopic to a skew lattice in a fairly simple way. To see this first recall a fact about regular bands. Lemma 3.4.12. If ( S, •) is a band and an operation •L is defined on S by e•L f = efe, then ( S, •L) is a band if and only if ( S, •) is a regular band, in which case ( S, •L) is a left regular band. Dually, defining •R on S by e•R f = fef, ( S, •R) is a band if and only if ( S, •) is a regular band, in which case the band ( S, •R) is right regular. Moreover, e ≤ f in ( S, •) iff e ≤ f in ( S, •L), and e ≺ f in ( S, •) iff e ≺ f in ( S, •L). Similar remarks relate both ≤ and ≺ for ( S, •) and ( S, •R). Proof. (Here x• y is denoted by xy.) Clearly •L is idempotent. Is it associative? Note that a•L( b•L c) = a•L( bcb) = abcba while ( a•L b)•L c = ( a•L b) c( a•L b) = abacaba. Thus •L is associative iff abcba = abacaba holds on S. Clearly this happens if (S, •) is regular. Conversely, if ( S, •L) is associative, then expanding a•L b•L c both ways gives abcba = abacaba for all a, b, c. Replace a by ba to get babcbba = babbacbabba, which simplifies to (1) babcba = bacba The mirror argument gives (2) abcbab = abcab. 103 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Multiply both sides of (1) on the right by bcb to get babcbabcb = bacbabcb. Simplifying on the left gives (3) babcb = bacbabcb. Using (2), the right side of (3) simplifies as follows: bacbabcb =(2) bacbacb = bacb. Thus (3) becomes babcb = bacb, which is regularity. So assume that ( S, •) is regular. Then a•L b•L a = ( aba)•L a = a( aba) a = aba = a•L b, making ( S, •L) left-regular. If e ≤ f in ( S, •), then efe = f = fef, that is e• Lf = f = f•L e, so that e ≤ f in ( S, •L). Conversely e• Lf = f = f•L e reduces to efe = f = fef so that ef = efefef = f = fefefe = fe in ( S, •). In similar fashion, e ≺ f in ( S, •) iff e ≺ f in (S, •L), since f•L e•L f = fefef which immediately reduces to fef. Thus fef = f iff f•L e•L f = f. £ Thus, if ( S, •) is regular, then it is isotopic as a band to both ( S, •L) and ( S, •R). Our next theorem rests on the lemma and a special case of Corollary 3.4.4. We give an alternative proof. Lemma 3.4.13. If ( S, ∧, ∨) is a refined quasilattice that is left-handed in that x∧ y∧ x = x∧ y and x∨ y∨ x = y∨ x, then it is a skew lattice. Proof. Being a quasilattice, x∧( x∨ y) = x∧( x∨ y)∧ x = x. Being a paralattice, ( y∨ x)∧ x = ( x∨ y∨ x)∧ x = x. The dual identities are similarly seen. £ This leads to Kinyon’s observation (via email) about refined quasi-lattices. Theorem 3.4.14. Given a quasilattice ( S, ∨, ∧), it is a refined quasilattice if and only if ( S, ∨R , ∧L) is a left-handed skew lattice, in which case ( S, ∨, ∧) and ( S, ∨R , ∨L) are isotopic under the identity map on S . In addition, they share all instances of commutation in which case the outcomes agree for the corresponding pairs of operations. Proof. As seen in the proof above, a∧L b∧L a expanded reduces to a∧ b∧ a. Likewise, b∨R a∨R b reduces to b∨ a∨ b. Thus whenever ( S, ∨R , ∧L) is a quasilattice, both algebras share a common natural quasi-order. Also in general, a∧ b = b∧ a iff a∧L b = b∧L a with both outcomes being equal, and a∨ b = b∨ a iff a∨R b = b∨R a, with both outcomes being equal. Thus ( S, ∨R , ∧L) has a coherent natural partial order iff ( S, ∨, ∧) does, in which case both algebras share the same natural partial order. It follows that if ( S, ∨, ∧) is a refined quasilattice if and only if ( S, ∨R , ∧L) is a left-handed skew lattice, thanks to the previous lemmas. £ Every refined quasilattice is thus just a “scrambled skew lattice” . Indeed given a left- handed skew lattice (S, ∨ , ∧) with S/D denoted by T, various refined quasi-lattices ( S, ∨*, ∧*) may be recovered from it by (1) doing a fibered product factorization ( S 1, ∨1) ×T ( S 2, ∨2) of the right regular band (S, ∨), replacing ∨2 by its left-handed dual operation and finally shifting the resulting operation on S 1 ×T S 2 back to S to get ∨*; and (2) likewise factoring the left-regular band (S, ∧) as say ( S 3, ∧3) ×( T, ∧) ( S 4, ∧4) and replacing ∧4 by its right-handed dual and then shifting the 104 III: Quasilattices, Paralattices and their Congruences resulting operation from S 3 × T S 4 back to S to get ∧*. Every refined quasilattice can in principle be recovered from its derived skew lattice in this manner. When S 1 ×T S 2 and S 3 × T S 4 are just reducts of a common fibered factorization of (S, ∨ , ∧) as a skew lattice, the algebra ( S, ∨*, ∧*) is also a skew lattice. Note that results 3.4.7 – 3.4.9 above, are trivial consequences of the above theorem and their skew lattice predecessors in Section 2.2. 3.5 The effects of the distributive identities Connections between refined quasilattices and distributive properties exist which particularly involve split quasilattices; however, distributive identities, much like absorption identities, proliferate in the absence of commutativity. To begin, a noncommutative lattice is fully distributive if it satisfies the identities: D1: a ∧ ( b ∨ c) = ( a ∧ b) ∨ ( a ∧ c). D2: ( a ∨ b) ∧ c = ( a ∧ c) ∨ ( b ∧ c). D3: a ∨ ( b ∧ c) = ( a ∨ b) ∧ ( a ∨ c). D4: ( a ∧ b) ∨ c = ( a ∧ c) ∨ ( b ∧ c). The four identities are mutually independent, unlike the case for lattices. Remark. If one replaces (∨, ∧) by (+, •), these identities describe a semiring that is also distributive in that addition distributes over multiplication. When both operations are idempotent, one has an idempotent, distributive semiring or an ID-semiring. That is, ID-semirings are just fully distributive double bands, but in ring notation. ID-semirings were introduced by Pastijn and Romanowska [1982] and studied in several subsequent papers. For instance, varieties of ID-rings where both operations are middle commutative (i.e., normal in band terminology) were classified by Pastijn in [1983]. Two separate lines of research thus meet at fully distributive quasilattices. We will consider connections between ID-semirings and fully distributive quasilattices after Corollary 4.5.3 below. A noncommutative lattice is bidistributive if it satisfies a slightly weaker pair of identities: D5: a ∧ ( b ∨ c) ∧ d = ( a ∧ b ∧ d) ∨ ( a ∧ c∧ d). D6: a ∨ ( b ∧ c) ∨ d = ( a ∨ b ∨ d) ∧ ( a ∨ c∨ d). Finally, a noncommutative lattice is distributive if it satisfies an even weaker pair of identities: D7: a ∧ ( b ∨ c) ∧ a = ( a ∧ b ∧ a) ∨ ( a ∧ c∧ a). D8: a ∨ ( b ∧ c) ∨ a = ( a ∨ b ∨ a) ∧ ( a ∨ c∨ a). Recall that all skew lattices in rings are distributive as are all skew Boolean algebras. 105 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond The power of these varying forms of distribution is seen in the following results. Theorem 3.5.1. A distributive, noncommutative lattice is a paralattice if and only if it is a quasilattice. Proof. Any paralattice satisfying D7 and D8 is a quasilattice. Just expand a∧( b∨ a∨ b)∧ a and a∨( b∧ a∧ b)∨ a using D7 and D8 and then simplify using B6, B7, C6 and C7 to obtain respectively B5 and C5. On the other hand, let N be a quasilattice satisfying both D7 and D8. To see that N is also a paralattice, suppose to the contrary that there exist a, b ∈ N such that a∧ b = b = b∧ a, but either a∨ b ≠ a or b∨ a ≠ a. Then we obtain either a∧( a∨ b)∧ a = a ≠ a∨ b = ( a∧ a∧ a)∨( a∧ b∧ a) or a∧( b∨ a)∧ a = a ≠ b∨ a = ( a∧ b∧ a)∨( a∧ a∧ a) and thus deny D7. Similarly, if a,b ∈ N exist such that a∨ b = b = b∨ a, but either a∧ b ≠ a or b∧ a ≠ a then we obtain denials of D8. Hence the two natural partial orderings indeed dualize each other and N is a paralattice. £ Passing to D5 and D6 we obtain a much stronger result. Theorem 3.5.2. A bidistributive quasilattice (paralattice) factors into the product of a distributive lattice and an antilattice. Conversely, every such product is a bidistributive quasilattice (paralattice). Proof. Let N be a (necessarily fine) bidistributive quasilattice. We show that if x ≥ y, z with y D z, then y = z. Since both u ≥ u∧ x∧ y∧ u, u∧ y∧ x∧ u and u∧ x∧ y∧ u D u∧ y∧ x∧ u in general, the implication yields the identity u∧ x∧ y∧ u = u∧ y∧ x∧ u. So let x ≥ y,z in N. Since y R(∧) y∧ z L(∧) z, with x ≥ y∧ z also, we show that y = z under the added assumption that either y L(∧) z or y R(∧) z. Let us assume that y L(∧) z. Consider y∨ z and z∨ y. D5 gives y∧( y∨ z) = y∧( y∨ z)∧ x = ( y∧ y∧ x)∨( y∧ z∧ x) = y∨ y = y and ( y∨ z)∧ y = x∧( y∨ z)∧ y = ( x∧ y∧ y)∨( x∧ z∧ y) = y∨ z. Hence y R(∨) y∨ z L(∨) z, in a common L(∧)-class of N. Thus, we may assume further that in addition to y L(∧) z either y L(∨) z or y R(∨) z. Supposing that y L(∨) z, then by D4, y = y∨ z = ( x∧ y∧ z)∨( x∧ x∧ z) = x∧( y∨ x)∧ z = x∧ x∧ z = z. If we take y R(∨) z instead, then D5 again yields 106 III: Quasilattices, Paralattices and their Congruences y = z∨ y = ( x∧ x∧ z)∨( x∧ y∧ z) = x∧( x∨ y)∧ z = x∧ x∧ z = z. Thus if x ≥ y,z with y L(∧) z, then y = z. If we had first supposed that y R(∧) z, then it would follow in similar fashion that y = z. Thus x ≥ y,z in N with y D z implies that y = z, from which the identity u∧ x∧ y∧ u = u∧ y∧ x∧ u follows. Similarly, if x, y ≥ z in N with x D y, then D6 can be used to show that x = y, from which follows the identity u∨ x∨ y∨ u = u∨ y∨ x∨ u. By Theorem 3.1.4, N factors into the product of its (necessarily) distributive maximal lattice image and an antilattice. For the converse, observe that antilattices are always bidistributive as, of course, are distributive lattices. Thus split quasilattices whose lattice factors are distributive are indeed bidistributive quasilattices. £ An even stronger result occurs when full distribution is assumed: Corollary 3.5.3. A quasilattice is fully distributive if and only if it is the product of a distributive lattice and a regular antilattice. In particular, every fully distributive quasilattice is regular. Proof. Let N be a fully distributive quasilattice. N factors as the product of a distributive lattice T and an antilattice A. Being a fine quasilattice, L(∨) is a ∨-congruence by Theorem 3.4.5. But L(∨) is a also a ∧-congruence: given b L(∨) c, D1 implies a∧ b L(∨) a∧ c and D2 implies b∧ d L(∨) c∧ d. Thus L(∨) is a full congruence. Similarly, R(∨) is a full congruence. Likewise, D3 and D4 imply that R(∧) and L(∧) are full congruences. Thus both N and its rectangular factor A are regular. Conversely, let N factor as the product of a distributive lattice T and a regular antilattice A. Being regular, A factors into a product of flat algebras, A ≅ A( l,l) × A( l,r) × A( r,l) × A( r,r), with each factor having operations ∨ and ∧ that each satisfy ab = a or ab = b. Conversely, identities D1 – D4 are easily seen to hold on such structures and, of course on T and hence on N. £ As mentioned above, two lines of research meet at fully distributive quasilattices. In particular, our Corollary 4.5.3 follows from Theorem 2.6 of Pastijn [1983]. By the latter theorem, the lattice of varieties of all ID-semirings satisfying the identities, x + y + z + w = x + z + y + w and xyzw = xzyw, is described. The sublattice of those varieties that also satisfy semiring versions of B5 and C5 is easily seen to be the join of the variety of distributive lattices and the variety of rectangular ID- semirings (satisfying x + y + z = x + z and xyz = xz). Thus every fully distributive quasilattice is the subdirect product of a distributive lattice and a rectangular ID-semiring (that is essentially a regular rectangular quasilattice). But such a subdirect product splits by Theorem 3.5.2 and the corollary follows. 107 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Pastijn’s theorem also shows that the lattice of all varieties of fully distributive quasilattices is Boolean of order 32. The larger lattice of varieties is also Boolean, but of order 210. (Pastijn works with identities, aba + a + aba = a and ( a+ b+ a) a( a+ b+ a) = a, the ID-semiring equivalents of B5 and C5.) Returning to refined quasilattices from the last section, one may ask about the effect of distributivity of ( S, ∨, ∧) on its isotopic left-handed skew lattice variant ( S, ∨*, ∧*) where x∨* y = y∨ x∨ y and x∧* y = x∧ y∧ x, Propositon 3.5.4. If a refined quasilattice ( S, ∨, ∧) is distributive, then its isotopic left-handed skew lattice ( S, ∨*, ∧*) is also distributive. Proof. Given that x∨* y = y∨ x∨ y = y∨* x∨* y and x∧ *y = x∧ y∧ x = x∧* y∧* x, suppose that ( S, ∨, ∧) is distributive. Then x ∧* ( y ∨* z) ∧ * x = x∧( y ∨* z)∧ x = x∧( z ∨ y ∨ z)∧ x = ( x∧ z∧ x) ∨ ( x∧ y∧ x) ∨ ( x∧ z∧ x) and ( x∧* y∧ *x) ∨* ( x∧* z∧ *x) = ( x∧ y∧ x) ∨* ( x∧ z∧ x) = ( x∧ z∧ x) ∨ ( x∧ y∧ x) ∨ ( x∧ z∧ x), so that ( S, ∨*, ∧*) satisfies D7 and likewise its dual D8. £ Conversely, if ( S, ∨*, ∧*) is distributive, then (S, ∨, ∧) is easily seen to satisfy the weaker identity x∧( z∨ y∨ z)∧ x = ( x∧ z∧ x)∨( x∧ y∧ x)∨( x∧ z∧ x) and its dual. If in addition ( S, ∨, ∧) is already known to be a skew lattice, then one can show it must be distributive. 3.6 Deriving simple antilattices from magic squares Recall that a magic square is a square array of distinct numbers where all rows, columns and the two diagonals have a common magic sum. A classic instance is the Lo-Shu with a magic sum of 15: 8 1 6 3 5 7 . 4 9 2 Given a magic square, its derived antilattice arises by letting the given square be the ∧-array and letting the ∨-array be the square array storing the same numbers entered in their natural ordering. Thus in the case of the Lo-Shu one gets: 108 III: Quasilattices, Paralattices and their Congruences ⎡8 1 6⎤ 1 ⎡ 2 3⎤ (∧) ⎢3 5 7 ⎥ ( ⎢ ⎥ . ⎢ ⎥ ∨) 4 5 6 ⎢ ⎥ 4 9 2 ⎣ ⎦ 7 8 9 ⎣ ⎦ Square arrays also come from finite planes, that is, vector spaces of dimension 2 over finite fields. For instance, given the field &5, the plane &5 × &5 can be represented as the 5×5 array of ordered pairs on the left below, but with parentheses deleted. Alternatively, one could view these pairs as base 5 representations of integers in base 10. Thus 3,2 represents 325 = 1710. The planar array could thus be encoded using the numbers in the right array. ⎡ ⎡ ⎤ 0, 0 1, 0 2, 0 3, 0 4, 0⎤ 0 1 2 3 4 ⎢ ⎢ ⎥ 0,1 1,1 2,1 3,1 4,1 ⎥ 5 6 7 8 9 ⎢ ⎥ ⎢ ⎥ 0, 2 1, 2 2, 2 3, 2 4, 2 ⎢ ⎥ 10 ⎢ 11 12 13 14 ⎥ ⎢ ⎢ ⎥ 0, 3 1, 3 2, 3 3, 3 4, 3 ⎥ ⎢ ⎥ 15 16 17 18 19 ⎢ ⎥ 0, 4 1, 4 2, 4 3, 4 4, 4 ⎣ ⎦ ⎢20 21 22 23 24 ⎣ ⎦⎥ In either case, this plane has 25 points and 30 lines, the latter arranged in six classes of five parallel lines each. The rows of the array consist of all lines of slope 0, the columns consist of all lines of undefined slope, the main diagonal plus the four broken descending diagonals yield all five lines of slope 1, and the counter-diagonal plus all four broken ascending counter-diagonals yield all five lines of slope 4. In all, between the rows, columns, diagonals and counter- diagonals, 20 out of 30 lines are accounted for, with only lines of slopes 2 or 3 left out. Alternatively, &5 × &5 can be represented by storing the five lines of slope 1 in the five rows and the five lines of slope 4 in the five columns in the left array below. ⎡0,0 1,1 2,2 3,3 4,4⎤ ⎡ 0 6 12 18 24⎤ ⎢2,3 3,4 4,0 0,1 1,2⎥ ⎢ ⎥ ⎢ ⎥ 13 19 20 1 7 ⎢ ⎥ 4,1 0, 2 1, 3 2, 4 3, 0 ⎢ ⎥ 21 2 8 14 15 ⎢ ⎥ 1 ⎢ ,4 2,0 3,1 4,2 0,3⎥ ⎢ 9 10 16 22 3⎥ ⎢ ⎥ ⎢ ⎥ 3, 2 4, 3 0, 4 1, 0 2,1 ⎣ ⎦ 17 23 4 5 11 ⎣ ⎦ In the right array, not only do all rows, columns, the main diagonal and the counter-diagonal sum to 60, but so do all broken diagonals and counter-diagonals. This makes the right array a pandiagonal magic square. Returning to the left array, the (broken) diagonals and counterdiagonals are precisely the lines of slope 3 and 2 respectively. Indeed, the line arrangement of the left array forces the right array to be pandiagonal. That finite planes can yield pandiagonal squares is well known. Together, the two representations of this plane in integer format describe an antilattice induced from a magic square. As will be shown below, both antilattices derived from the two magic squares thus far encountered are simple. Is this true in general? 109 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond To begin, given antilattice A and any pair a, b ∈A, recall that the principal congruence θ( a, b) is the smallest congruence on A relating a and b. Clearly: θ( a, b) = ∩{θ ∈ Con(A)⎮ a θ b}. In particular, θ( a, b) = Δ precisely when a = b. Clearly: Lemma 3.6.1. An algebra A of any type is simple if and only iff θ( a, b) = ∇ for all a ≠ b in its underlying set . For antilattices, this obvious criterion can be simplified. Consider an antilattice A deter- mined by a pair of rectangular arrays. Let R 0 and C 0 represent a row and a column of, say, the ∨- array of A. ( Which array is not important. But R 0 and C 0 must come from the same array.) Lemma 3.6.2. ( Simplicity Criterion for Antilattices) Given an antilattice A determined by a pair of rectangular arrays, let R 0 and C 0 denote respectively a row and a column of the ∨- array. Then A is a simple algebra iff θ( a, b) = ∇ for all a ≠ b in R 0 and all a ≠ b in C 0 . In particular, any given θ( a, b) must equal ∇ if A is generated from { a, b} using both ∨ and ∧. Proof. The condition is clearly necessary. To see sufficiency, suppose that the condition holds for row R 0 and column C 0 intersecting at element c in the ∨-array. Given a ≠ b in A, both c∨ a and c∨ b lie in R 0, while a∨ c and b∨ c lie in C 0. Since a ≠ b, either c∨ a ≠ c∨ b in R 0 or a∨ c ≠ b∨ c in C 0. Say c∨ a ≠ c∨ b, so that θ( c∨ a, c∨ b) = ∇. But since c∨ a θ( a, b) c∨ b, θ( c∨ a, c∨ b) refines θ( a, b) so that θ( a, b) = ∇ also. Thus θ( a, b) = ∇ for all a ≠ b ∈ A and A is simple. Since ∨ and ∧ are idempotent, the subalgebra 〈 a, b〉 generated from { a, b} lies in the θ( a, b)-class of a. The final statement follows. In the case of a square antilattice determined from a pair of n× n arrays, this theorem says that the number of principal congruences needing to be checked can be reduced from ( n 4 – n 2)/2 to just n 2 – n. Although the check to see that θ( a, b) = ∇ for a ≠ b in either R 0× R 0 or C 0× C 0 can be initially tedious, as the check continues some random recursion enters the process. Thus, if say θ( a, b) has been shown to equal ∇ and a θ( c, d) b is encountered in the check of θ( c, d), then one can immediately conclude that θ( c, d) = ∇ also holds. 1 ⎡ 2 3⎤ ⎡8 1 6⎤ Example 3.6.1. The Lo-Shu antilattice (∨) ⎢4 5 6⎥ ( ⎢ ⎥ is simple. ⎢ ⎥ ∧) 3 5 7 ⎢ ⎥ 7 8 9 ⎣ ⎦ 4 9 2 ⎣ ⎦ To begin, take {1, 2}. From the ∧-array, it is clear that 6, 9 ∈〈1, 2〉, the subalgebra generated from {1, 2}. But {1, 2, 6, 9} clearly generates the ∨-array and thus the algebra. Hence θ(1,2) = ∇. Similar remarks hold for any other pair a ≠ b in any row or column of either array. These remarks deserve a more precise analysis. Given distinct elements a and b in a common row (column) of a 3×3 array, the elements c and d lying in neither the row (column) or the two columns (rows) of a and b is called the dual pair. The relationship is symmetrical. Thus {1, 2} and {6, 9} form dual pairs in the ∨-array above, but not in the ∧-array. Any pair of dual pairs in a 3×3 array generates the entire array under the ambient idempotent operation. Given two distinct elements in a common row or column of one of the above Lo-Shu arrays, this pair 110 III: Quasilattices, Paralattices and their Congruences immediately generates its dual pair in the opposite array. In this sense, these two arrays are complementary 3×3 arrays, so that any pair of elements lying in a common row or column in either array generates the entire antilattice which thus is simple. What can be said in general about congruences on an antilattice? Given a rectangular array A, a cartesian partition of A is a partition P that is induced in cartesian fashion from a partition of the rows and a partition of the columns of A. For example, one cartesian partition of ⎡ a b c d e⎤ ⎢ f g h i j ⎥ ⎢ ⎥ k l m n o ⎣ ⎦ ⎡⎡ a b c⎤ ⎡ d e⎤⎤ ⎢ ⎥ is given by: ⎢⎣⎢ f g h ⎦⎥ ⎣⎢ i j ⎦⎥⎥ . [⎢ k l m ⎣ ][ n o] ⎦⎥ In all, 260 = 52 × 5 cartesian partitions of this array are given by the fifty-two partitions of { a, b, c, d, e} and the seven partitions of { a, f, k}. Given such a partition P, an equivalence P# on A is given by a P# b if a and b lie in the same P-class. Such an equivalence is called a cartesian equivalence on A. In the case of rectangular bands, the congruences on an array that are consistent with a single rectangular band operation (using just one of ∨ or ∧) are precisely its cartesian equivalences. Thus: Proposition 3.6.3. Given an antilattice A, its congruences arise from pairs of cartesian partitions of its two arrays sharing the same equivalence classes. The general 3×3 case. The Lo-Shu is one of infinitely many possible 3×3 magic squares that can arise if we agree to store integers besides 1 - 9. Others include the following two squares: ⎡71 89 17 ⎤ ⎡252 171 363⎤ ⎢5 59 113⎥ ⎢ ⎥ ⎢ ⎥ 373 262 151 ⎢ ⎥ 101 29 47 ⎣ ⎦ 161 353 272 ⎣ ⎦ The magic square on the left consists entirely of primes, with a magic sum of 177, the least possible such sum for any magic square of primes. The magic square on the right consists of 3-digit palindromes. Like the Lo Shu case, both examples induce simple antilattices. Is this true for all 3 × 3 magic squares? To answer this, we begin with the following result. 111 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Lemma 3.6.4. Given 3×3 arrays, A and Aʹ, each storing the same 9 distinct elements, the following assertions are equivalent: 1. A and Aʹ form a complimentary pair of 3×3 arrays. 2. If two distinct elements are either row-related or column-related in either array, they are unrelated in either sense in the other array. 3. The rows [columns] in A either all become (extended) diagonals in Aʹ or all become extended counterdiagonals in Aʹ; similar remarks hold in passing from Aʹ to A. Proof. Clearly (1) implies (2). For the converse, observe that the status of (2) is unchanged if either array undergoes row or columns interchanged! Thus, we assume (2) in the case where ⎡ a b c ⎤ elements a and b lie in a common row of A, as in ⎢ d e f ⎥ . Assertion (2) implies that A ⎢ ⎥ ʹ is of g h i ⎣ ⎦ ⎡ a f x⎤ ⎡ a i x⎤ ⎡ a f h ⎤ ⎡ a i e ⎤ form ⎢ i b y ⎥ or ⎢ ⎥ . Applying (2) further, A ⎢ ⎥ or ⎢ ⎥ its ⎢ ⎥ f b y ⎢ ⎥ ʹ must be either i b d ⎢ ⎥ f b g ⎢ ⎥ u v w ⎣ ⎦ u v w ⎣ ⎦ e g c ⎣ ⎦ h d c ⎣ ⎦ trnaspose . In either case we have a complementary pair of arrays. Similarly, assuming a and b lie in the same column of A, (2) forces Aʹ to be a complementary array. Likewise, if a and b are row-[column-] related in Aʹ, then (2) forces A to be a complementary to Aʹ. Thus, (1) and (2) are equivalent. Clearly (3) implies (1) and (2). Given the latter, every row/column in either array must be an (extended) [counter]diagonal in the other array. This can only happen if (3) holds. We are ready to state our main result about antilattices induced from 3×3 magic squares. ⎡ a b c ⎤ ⎢ ⎥ Theorem 3.6.5. Given a 3×3 array A = d e f ⎢ ⎥ consisting of distinct positive ⎢ g h i ⎣ ⎦⎥ integers in their natural ( increasing) order and a second 3 × 3 magic square Aʹ storing the same ⎡ b i c⎤ ⎢ ⎥ integers, then either A and Aʹ are complementary or Aʹ is a dihedral variation of f e d ⎢ ⎥ . ⎢ g a h ⎣ ⎦⎥ 1 ⎡ 3 4 ⎤ ⎡3 11 4⎤ (An instance of the latter is the pair A = ⎢5 6 7⎥ and A ⎢ ⎥ .) ⎢ ⎥ ʹ = 7 6 5 ⎢ ⎥ 8 9 11 ⎣ ⎦ 8 1 9 ⎣ ⎦ Proof. Using a dihedral replacement of Aʹ if need be, distinct β > γ > 0 and α > β + γ exist such that: 1 ⎡ 1 1⎤ ⎡−1 1 0 ⎤ ⎡ 0 1 − 1⎤ ⎡ α − β α + β + γ α − γ ⎤ Aʹ = α 1 ⎢ 1 1⎥ + ⎢ ⎥ + ⎢ − ⎥ = α ⎢ + β − γ α α − β + γ ⎥ . ⎢ ⎥ β 1 0 − 1 ⎢ ⎥ γ 1 0 1 ⎢ ⎥ ⎢ ⎥ 1 1 1 ⎣ ⎦ 0 ⎣ − 1 1⎦ 1 ⎣ − 1 0⎦ ⎣ α + γ α − β − γ α + β ⎦ 112 III: Quasilattices, Paralattices and their Congruences Clearly a = α – β – γ and b = α – β. What is the next smallest element? If it is α – γ, then the ascending sequence α – β – γ < α – β < α – γ < α – β + γ < α < α + β – γ < α + γ < α + β < α + β + γ must occur. In this case we have the displayed array. Otherwise, we must have: α – β – γ < α – β < α – β + γ < α – γ < α < α + γ < α + β – γ < α + β < α + β + γ yielding an array complementary to A. This theorem has the following consequence: Corollary 3.6.6 . Given the arrays A and Aʹ of the prior theorem, the induced antilattice A is simple if and only if A and Aʹ are complementary. Otherwise, Con(A) is a 3 -element chain. In general, all antilattices induced from 3 × 3 magic squares are congruence distributive. Proof. In the complementary case, any pair of distinct elements generates A, which thus is simple. Otherwise, a single nontrivial, proper congruence is given by [ a, b, c, g, h, i⎮ d, e, f]. The 4 × 4 case We next consider antilattices induced from 4 × 4 magic squares storing 1 - 16. While just one 3×3 magic square stores 1 - 9 (with eight dihedral variations), 880 essentially distinct magic squares store 1 to 16. A list of all 880 squares was given by Bernard Frénicle de Bessy in a posthumous 1693 publication. A mathematical analysis was given in the 1983 paper of Dame Kathleen Ollerenshaw and Sir Hermann Bondi [5]. Thanks to the following observation, these 880 cases decompose into 220 classes of 4. Lemma 3.6.7 . Given a 4 × 4 magic square A , let squares B, C and D be induced from A by simultaneous row and column permutations determined by (2 3), (1 2)(3 4) and (1 3 4 2) respectively. Then A – D are all magic squares, but none are dihedrally equivalent. Moreover all four squares induce the same antilattice. Thus one can get by checking the leading array in each row of four squares in the Ollerenshaw-Bondi list. Among these, the nonsimple cases are easily spotted, thanks to a theorem about semimagic squares (all rows and columns add up to 34). In its statement, the index of a congruence µ counts its number of congruence classes. Theorem 3.6.8. If a semi-magic square A storing 1 – 16 induces a nonsimple antilattice A, then A has a maximal congruence µ of index 2 whose corresponding congruence class partition is either πR = {1 – 4, 13 – 16⎮5 – 12} or πC = {1, 4, 5, 8, 9, 12, 13, 16⎮2, 3, 6, 7, 10, 11, 14, 15} 113 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond where πR and πC are outer/inner partitions splitting rows[columns] 1 & 4 against rows[columns] 2 & 3 in the standard array. Examples 3.6.2. Consider the following magic squares in the Ollerenshaw-Bondi listing: 1 ⎡ 7 12 14⎤ 1 ⎡ 4 15 14⎤ 1 ⎡ 16 6 11⎤ ⎢ ⎥ ⎢ ⎥ 13 ⎢ 4 10 7⎥ (1) 10 16 3 5 (9) 13 16 3 2 (25) . 15 ⎢ 9 6 4 ⎥ 12 ⎢ 9 6 7 ⎥ 12 ⎢ 5 15 2⎥ ⎣⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 8 2 13 11⎦ ⎣8 5 10 11⎦ ⎣8 9 3 14 ⎦ Square (1) induces a simple antilattice because 1 - 4 lie in distinct rows and columns (denying πR) and 1, 5, 9, 13 lie in distinct rows and columns (denying πC). By contrast both πR and πC work for (9), while πC works, but not πR, for square (25). Thus both (9) and (25) are nonsimple. ( Caveat. In the Ollerenshaw-Bondi list, the arrays actually store 0 – 15, instead of 1 – 16, and do so in base 4 notation.) A survey of the 220 leading squares in the Ollerenshaw-Bondi list yields, upon applying the test of Theorem 3.6.8, the following statistic: Corollary 3.6.9. Of the 880 magic squares storing 1 – 16, 416 cases yield simple antilattices and 464 yield nonsimple antilattices, giving a breakdown of 47.27% to 52.73%. Proof of Theorem 3.6.8. (All arrays in this proof are identified to within row and column permutations.) To begin, all possible cartesian partitions of a 4×4 square with distinct elements can only have indices among the following: 1, 2, 3, 4. 6, 8, 9, 12, 16. Thus if A is nonsimple, the index |µ| of its maximal proper congruence µ must lie among 2, 3, 4, 6, 8, 9, 12. If |µ| = 2, then any cartesian partition of the standard array is one of four cases: one row and three rows, or one column and three columns, or two rows and two rows, or two columns and two columns. The first two cases are impossible when A is included, as no row or column in the standard array has the magic sum of 34. In the final cases, the sum of each pair of rows or columns must be 2 × 34 = 68. This occurs only for {row 1 ∪ row 4⎮row 2 ∪ row 3} or {column 1 ∪ column 4⎮column 2 ∪ column 3}, just as stated. |µ| = 3 is impossible in the antilattice context since that would mean a row or column in the standard array would sum to 34 (because it would appear as a row or column in A), which is impossible. |µ| = 4 is possible. But in this case, the quotient algebra A/µ would have order 4 and thus be nonsimple by Proposition 3.2.2. Hence µ was not really maximal after all. |µ| = 6 is also possible with the cartesian partition of the standard array having either ⎡ 1 [ × 2] 1[× 2]⎤ ⎡ 1 [ ×1] 1[× 3]⎤ ⎢ ⎥ ⎢ ⎥ template 1 [ × 2] 1[× 2] ⎢ ⎥ or template 1 [ ×1] 1[× 3] ⎢ ⎥ or a transpose. In all these cases, the [⎢2 × 2 ⎣ ] [2 × 2]⎦⎥ [⎢2 ×1 ⎣ ] [2 × 3]⎦⎥ 114 III: Quasilattices, Paralattices and their Congruences two bottom cells would still be adjacent in the cartesian partition of the magic square, thus inducing a congruence of index two. Similarly for the remaining indices of 8, 9 and 12, congruences with these indices must be refined by a properly larger congruence of index 2, thus returning us to the |µ| = 2 case. Examples 3.6.3. Pandiagonal magic squares first appear in the 4×4 case. In considering pandiagonal magic squares in general, two such squares of the same dimension are equivalent if either is obtained from the other using a combination of dihedral operations, along with cyclic permutations of the rows and/or the columns. Any given pandiagonal square storing 1 through 16 is thus one of 8 × 4 × 4 = 128 equivalent pandiagonal magic squares. In the 4×4 case, 48 dihedrally distinct pandiagonal squares exist, all being equivalent to exactly one of the following three pandiagonal magic squares: ⎡ ⎤ 1 ⎡ 8 10 15⎤ 1 8 11 14 ⎢ ⎥ ⎡ 1 14 4 15⎤ ⎢ ⎥ ⎢ ⎥ 12 13 3 6 ⎢ ⎥ ⎢ ⎥ 12 13 2 7 ⎢ ⎥ 8 11 5 10 ⎢ ⎥ . 7 2 16 9 ⎢ ⎥ ⎢6 3 16 9 ⎥ 13 2 16 3 ⎢ ⎥ 14 ⎣⎢ 11 5 4 ⎦⎥ ⎢ ⎥ 12 ⎣⎢ 7 9 6 ⎦⎥ 15 ⎣ 10 5 4⎦ The first two pandiagonal magic arrays induce simple antilattices; but not the third array. Two classic constructions In 1693, Simon de la Loubère gave the following rule for constructing magic squares for any odd order n. Again we will work with base n ordered pairs, often in abbreviated notation. De la Loubère’s Rule. Place 00 in the middle of the first row. In ascending (broken) diagonal fashion place in order the remaining 01 through 0, n–1. Beneath 0, n–1 place 10 and again in ascending diagonal fashion place 11 through 1, n–1. Beneath 1, n–1 place 20, and repeat the process until an entir e n× n array is filled. The resulting array is a magic square of odd order n storing 0 through n 2–1 in base n. The array to the right is the n = 5 case in base 5 notation storing 0 – 24. ⎡00 01 02 03 04 ⎤ ⎡31 43 00 12 24⎤ 10 ⎢ 11 12 13 14 ⎥ ⎢ ⎥ ⎢ ⎥ 42 04 11 23 30 ⎢ ⎥ (∨) 20 21 22 23 24 ⎢ ⎥ (∧) 03 10 22 34 41 ⎢ ⎥ ⎢30 31 32 33 34 ⎥ 14 ⎢ 21 33 40 02⎥ ⎢ ⎥ ⎢ ⎥ 40 41 42 43 44 ⎣ ⎦ 20 32 44 01 13 ⎣ ⎦ Theorem 3.6.10. The magic squares of odd order n given by De la Loubère’s rule along with their corresponding standard array yield simple antilattices precisely when n is prime. 115 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. Suppose that x θ y where x ≠ y. By Lemma 3.6.2, we may assume that x and y either lie in a common row or in a common column of the ∨-array. If x and y are in the same column, then their meets yield θ-related elements u and v in distinct columns of the ∨-array. From { u, v, u∨ v, v∨ u} we gain a pair of distinct θ-related elements lying in a common row of the ∨-array. Thus at the outset we may assume that x θ y with x and y distinct elements in a common row of the ∨-array. If the order of the magic square is p = 2 n + 1 (and the order of the algebra is p 2), then n 0 ∨ x and n 0 ∨ y must be distinct θ-related elements in the middle row of the ∨-array, say ni and nj. But ni and nj are also lie the main ascending diagonal of the ∧-array and from them we can generate via f(X, Y) = ni ∨ (X ∧ Y), all n, i± mk where k = j – i. If p is prime, the main ascending diagonal in the ∧-array must lie in a common θ-class. Since this diagonal generates the entire algebra, θ = ∇. If p is composite, say p = ab with 1 < a, b < p, then define an equivalence α by ij α kl if both i ≡ k (mod a) and j ≡ l (mod a). That α is a ∨-congruence is clear. In the case of ∧, observe that in the ∧-array any horizontal or vertical displacement of a positions from any starting position yields an α-related element. Conversely, any pair of α-related elements are connected by a sequence of such displacements. Thus given x α y and u α v, the ∧-columns of x∧ u and y∧ v, being the ∧-columns of u and v, differ in their position by a multiple of a. Likewise the ∧-rows of x ∧ u and y ∧ v, being the ∧-rows of x and y, differ in their position by a multiple of a. It follows that x ∧ u α y ∧ v so that α is a ∧–congruence also. Clearly α is neither Δ or ∇. A variation of the above rule had been given previously by Claude Gaspar Bachet de Méziriac (who in 1621 published the famous edition of Diophantus’ Arithmetica). Bachet de Méziriac’s Rule. Place 00 directly above the middle position of an n× n array. In ascending ( broken) diagonal fashion place in order, the remaining 01 through 0, n–1. Next, place 10 two rows directly above 0, n–1. In ascending ( broken) diagonal fashion place, in order, 11 through 1, n–1. Next, place 20 two rows directly above 1, n–1. Repeat the process until an entire n× n array is filled to produce a magic square storing 0 through n 2–1 in base n. Theorem 3.6.11. The magic squares of odd order given by Bachet de Méziriac’s rule induce simple antilattices precisely when the order is prime. ⎡00 01 02 03 04 05 06⎤ ⎡63 20 54 11 45 02 36⎤ 10 ⎢ 11 12 13 14 15 16 ⎥ ⎢ ⎥ ⎢ ⎥ 26 53 10 44 01 35 62 ⎢ ⎥ 20 21 22 23 24 25 26 ⎢ ⎥ 52 16 43 00 34 61 25 ⎢ ⎥ ⎢30 31 32 33 34 35 36⎥ ⎢15 42 06 33 60 24 51⎥ ⎢ ⎥ ⎢ ⎥ 40 41 42 43 44 45 46 ⎢ ⎥ 41 05 32 66 23 50 14 ⎢ ⎥ 50 51 52 53 54 55 56 ⎢ ⎥ 04 31 65 22 56 13 40 ⎢ ⎥ ⎢60 61 62 63 64 65 66 ⎣ ⎦⎥ ⎢30 64 21 55 12 46 03 ⎣ ⎦⎥ (Standard array for 0 – 48, base 7) (de Méziriac array for 0 – 48. base 7) 116 III: Quasilattices, Paralattices and their Congruences Proof. Suppose first that p is prime with p = 2 n + 1 and let θ be a congruence with x θ y where x ≠ y. As with the previous theorem, things may be reduced to the case where x = ni and y = nj in the middle row of the ∨-array and the ascending diagonal of the ∧-array. If F(X, Y) = n 0 ∨ (X ∧ Y), then F( ni, nj) = n ( i + j)/2 in the same θ-class as ni = n 0 ∨ ni. (Here ( i + j)/2 is calculated in & p.) Since n 0, n 1, … , n p–1 generates the algebra, simplicity follows if we can show that from ni and nj one can F-generate the entire n th ∨-row. This is equivalent to showing that from any two i ≠ j in & p, all of & p is generated via the function f( x, y) = ( x + y)/2. Let S be the set of all numbers in & p thus generated. If 0 ∈ S, then S is closed under addition and thus is a nontrivial subgroup of & p which forces S = & p. Indeed f(0, ( x + y)/2) = ( x + y)/4, f(0, ( x + y)/4) = ( x + y)/8, etc. Hence all ( x + y)/2 n lie in S. Since some power of 2 equals 1 in & p, we get x + y ∈ S so that S is as claimed. Otherwise, suppose 0 ∉ S. From f( x + k, y + k) = f( x, y) + k, the general case can be shifted to the 0-case, so that no matter what pair i, j is given, the f-generated set is all of & p. If p is composite, say p = ab with 1 < a, b < p, then define an equivalence α by ij α kl if both i ≡ k (mod a) and j ≡ l (mod a). The argument that α is a congruence is identical to that in the case of de la Loubère’s rule. A number of further examples of magic squares that induce simple antilattices are given in Leech [2005b]. Historical remarks Nearly all results in the first five sections appeared in a 2002 paper by Gratiela Laslo and Jonathan Leech that studied congruences on noncommutative lattices. The paper was written while Laslo was working her dissertation at the University of Cluj-Napoca; several results are from that dissertation. The material on isotopy is of more recent vintage, and was developed from remarks in an email from Michael Kinyon. It appears here for the first time. The final section on recreational mathematics and antilattices appeared Leech’s 2005 paper. References G. Laslo and J. Leech, Green’s equivalences on noncommutative lattices, Acti Sci. Math (Szeged) 68 (2002), 501-533. J, Leech, Magic squares, finite planes and simple quasilattices, Ars Combinatoria 71 (2005), 75 - 96 117 IV: SKEW BOOLEAN ALGEBRAS In this chapter such fundamental concepts as normal bands, skew lattices and generalized Boolean algebras are integrated. We have already seen in Theorem 2.3.7 that maximal normal bands in rings form strongly distributive skew lattices that are characterized by a∧( b ∨ c) = ( a∧ b) ∨ ( a∧ c) and ( a ∨ b)∧ c = ( a∧ c) ∨ ( b∧ c). More is true: any such maximal normal band S forms a noncommutative variant of a generalized Boolean algebra called a skew Boolean algebra. In particular S possesses a zero element 0 for which 0∨ a = a = a∨0 and 0∧ a = 0 = a∧0 hold; S is also closed under a difference operation a \ b given by a – aba. Such a system (S; ∨, ∧, \, 0) satisfies many identities, a subset of which provides a defining set of identities for a skew Boolean algebra. The strongly distributive skew lattices of partial functions encountered Section 2.6 give archetypal examples: to ∨ and ∧ we adjoin the empty partial function ∅ as the zero and the difference f \ g given by the restriction of f to F \ F∩G, where F and G are the respective supports of f and g (their sets of actual inputs). In this chapter we will work with the right-handed case PR(A, B). The skew Boolean version (with the expanded signature) is denoted by P(A, B). In addition to the above classes of examples, every primitive skew lattice with a singleton lower class A > {0} is strongly distributive and has a zero element 0; moreover a difference operation is given by the simple rule: x \ y = x, if y = 0, and 0 otherwise. These primitive algebras play a basic role in the theory, doing for skew Boolean algebras what the Boolean algebra 2 and its isomorphic copes do for (generalized) Boolean algebras. In the Section 4.1, skew Boolean algebras (S; ∨, ∧, \, 0) are formally defined as structural enhancements of strongly distributive skew lattices. Variants of some of the familiar results about generalized Boolean algebras are then proved. In particular, skew Boolean algebras are shown to be subdirect products of primitive skew Boolean algebras; moreover every skew Boolean algebra can be embedded into a power of 5, a 5-element primitive algebra. (See Corollaries 4.1.6 and 4.1.7.) Not surprisingly, every right-handed skew Boolean algebra can be embedded in some partial function algebra P(A, B). In Section 4.2, special attention is given to classifying finite algebras, and in particular, to classifying finitely generated (and thus finite) free skew Boolean algebras. In the process, not only do we look at an important class of examples, we also engage in some of the basic algebraic procedures of skew Boolean algebras. A fundamental concept in this section is that of an orthosum, both an orthosum of elements and an orthosum of subalgebras. The main results are Theorems 4.2.2 and 4.2.6, with the latter describing the structure of finitely generated free algebras. A skew Boolean algebra is, of course, just a strongly distributive skew lattice with added operations and axiomatic constraints. While section 4.3 focuses mostly on the relation between 119 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond skew Boolean algebras and strongly distributive skew lattices, this is also a convenient context to consider alternative characterizations of skew Boolean algebras. Theorem 4.3.5 gives an independent set of five identities that characterizes all right-handed skew Boolean algebras. In the Section 4.4 we look at skew Boolean algebras with finite intersections ∩, that is, algebras for which the natural partial order ≥ has meets that are called intersections and denoted by x∩ y. For a partial function algebra P(A, B), f∩ g is the usual intersection of partial functions f and g viewed as subsets of A × B. All skew Boolean algebras S for which S/D is finite have intersections as do, more generally, all complete skew Boolean algebras. Indeed, having at least finite intersections is often the rule. Here, similarities with Boolean algebras are much tighter: if ∩ is included in the signature, then all congruences are determined by their kernel ideals – the congruence classes of 0 – and thus their congruence lattices are distributive (Theorem 4.4.8). We also show that free skew Boolean algebras have finite intersections (Theorem 4.4.18). The lattice of all subvarieties of these algebras is described in the section’s concluding Theorem 4.4.24. Sections 4.1 – 4.4 give a structural hierarchy lying at the core of this chapter’s subject. skew Boolean algebras with finite intersections  skew Boolean algebras  strongly distributive skew lattices In Section 4.5 we study a functor ω that object-wise takes a generalized Boolean algebra B and constructs a skew Boolean cover of B, that is, a skew Boolean algebra SB such that (1) SB/D ≅ B and (2) SB has a trivial center {0}. Its underlying set ω(B) consists of all naturally ordered pairs in B, {( b, bʹ)⎪ b ≥ bʹ}. Initially, ω(B) has obvious join and meet operations: ( b, bʹ) ∨ ( c, cʹ) = ( b∨ c, bʹ∨ cʹ) and ( b, bʹ) ∧ ( c, cʹ) = ( b∧ c, bʹ∧ cʹ). We “twist” these outcomes to produce noncommutative variants of ∨ and ∧. This provides another class of skew Boolean algebras that have finite intersections as well as other properties that we investigate. The process also gives us a “workout” of much that was discussed in previous sections. In Chapter 6, where we return to skew lattices in rings, special attention is given to skew Boolean algebras that occur when the set of idempotents in a ring is closed under multiplication, in which case the idempotents form such an algebra. In particular we will examine how this occurrence affects the structure of the ring. All this occurs in Sections 6.4 to 6.7. 120 IV: Skew Boolean Algebras In the seventh and final chapter, we examine further the role of skew Boolean algebras in universal algebra, in particular in the study of what might be termed “generalized Boolean phenomena,” a topic of continuing interest in universal algebra. In particular we will present a number of results by Robert Bignall and his student, Matthew Spinks. Both gentlemen have made significant contributions in this area. This chapter concludes with a discussion of historical aspects of this topic and references. 4.1 Skew Boolean algebras We have seen that any maximal normal band S in a ring R is a skew lattice in R under multiplication and the cubic join ∇. Indeed, every such band is the full set of idempotents in the subring it generates in R. There is more. To begin, if E(R) is commutative, then E(R) is a generalized Boolean lattice since for each e in E(R), the principal lattice ideal ⎡ e⎤ = { f⎮ f ≤ e} is a Boolean lattice. Indeed, if e ≥ f in E(R), then e – f ∈ E(R) also; moreover e ≥ e – f with f ∧ ( e – f) = 0 and f ∨ ( e – f) = e. All this is true even when E(R) is not commutative, provided it forms a band under multiplication. Since E(R) is then a normal skew lattice by Theorem 2.3.7, each ⎡ e⎤ is a sublattice; what is more, for all f ∈ ⎡ e⎤ one has e – f ∈ ⎡ e⎤ with f ∨ ( e – f) = e and f ∧ ( e – f) = 0. This leads us to a definition: A skew Boolean algebra is an algebra (S; ∨, ∧, \, 0) such that (i) (S; ∨, ∧, 0) is a strongly distributive skew lattice with 0 and (ii) \ is a binary operation satisfying ( e∧ f∧ e) ∨ ( e\ f) = e and ( e∧ f∧ e) ∧ ( e\ f) = 0. By Theorem 2.3.4, (i) is equivalent to (S; ∨, ∧) being distributive, symmetric and normal and having the 0-identities hold. (i) and (ii) together imply e∧ f∧ e and e\ f commute. They also imply that each ⎡ e⎤ is a Boolean sublattice of S with e\ f being the unique complement of e∧ f∧ e in ⎡ e⎤. Conversely given (i), if each [e] forms a Boolean lattice, then (ii) follows. These observations give: Theorem 4.1.1. Skew Boolean algebras form a variety; moreover, every congruence on the skew lattice reduct of a skew Boolean algebra is also a skew Boolean algebra congruence. In particular, the Green’s equivalences are all skew Boolean algebra congruences. £ This variety will be denoted by SBA. The broader perspective of skew lattices yields: Theorem 4.1.2. A normal, symmetric skew lattice with 0 forms a skew Boolean algebra if and only if its maximal lattice image S/D is a generalized Boolean lattice, in which case \ is implicitly determined by (ii) above. 121 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. Let S be a normal, symmetric skew lattice with 0 such that S/D is a generalized Boolean lattice. Then S is distributive by Theorem 2.3.2. Since each ⎡ e⎤ is a Boolean lattice, \ is implicitly determined by (ii). The converse is clear. £ Examples 4.1.1. The following skew lattices are examples of skew Boolean algebras: (a) Maximal normal bands in rings form skew Boolean algebras upon setting e∧ f = ef , e∨ f = e∇ f and e\ f = e – efe. (See Cvetko-Vah and Leech, [2011] and [2012].) (b) Any partial function set P(A, B) with ∨ and ∧ defined as before and with 0 = ∅ and f \ g = f⎮F \ G where F and G are the supports in A for f and g respectively (c) More generally, any ring of partial functions provided the underlying lattice of subsets of A is a generalized Boolean lattice and contains the empty set. (d) Given a rectangular skew lattice D, a primitive skew Boolean algebra is formed by D0 upon setting x \ y = x if y = 0, but 0 otherwise. £ Theorem 4.1.3. Skew Boolean algebras satisfy: (iii) e \ f = e \ ( e∧ f∧ e). (iv) e \ ( f ∨ g) = ( e \ f) ∧ ( e \ g) and e \ ( f ∧ g) = ( e \ f) ∨ ( e \ g). (v) e \ ( e \ f) = e ∧ f ∧ e. Proof. These identities are immediate consequences of each inner ideal ⎡ e⎤ being a Boolean lattice. £ Given a skew lattice S, an ideal of S is a subset I of S such that given x, y in I, and z in S x∨ y, z∧ x and x∧ z are in I. Given any element a in a skew lattice S, the principal ideal of a is the set ( a) = { x ∈ S⎮ x ≺ a}. Clearly, x ∈( a) if and only if x ≺ b for all b ∈D a. If S has a zero element 0, the annihilator of a is the set ann( a) = { x ∈ S⎮ x ∧ a = 0}. Due to Theorem 2.1.2, x ∈ ann( a) if and only if x ∧ b = 0 = b ∧ x for all b ∈D a. If the skew lattice is distributive, then ann( a) is easily seen to be an ideal. Clearly ( a) and ann( a) can also be parameterized by the relevant D-class A = D a as ( A) and ann( A) respectively since due to Theorem 2.1.2, any b in D a induces the same pair of sets,. When S is a skew Boolean algebra, the situation can be sharpened to give a decomposition of primary importance in understanding these algebras. Theorem 4.1.4. Given a D- class A of a skew Boolean algebra S , then: (i) Both ( A) and ann( A) are ideals of S. (ii) All elements of ( A) commute with all elements of ann( A). (iii) In particular, for all u ∈( A) and all v ∈ ann( A), u∧ v = 0 = v∧ u. (iv) The map µ: ( A) × ann( A) → S defined by µ( e 1, e 2) = e 1 ∨ e 2 is an isomorphism. 122 IV: Skew Boolean Algebras Proof. As stated above, (i) holds for all distributive skew lattices with zero elements. From v∧ b = 0 = b∧ v for all v ∈ ann( A) and all b ∈D a, (iii) follows by Theorem 2.12. Next, (ii) follows from (iii) and symmetry. Given ( e 1, e 2), ( f 1, f 2) ∈( A) × ann( A), ( e 1∨ e 2) ∨ ( f 1∨ f 2) = ( e 1∨ f 1) ∨ ( e 2∨ f 2) follows from (ii). By Theorem 2.3.4, ( e 1 ∨ e 2) ∧ ( f 1 ∨ f 2) expands to ( e 1 ∧ f 1) ∨ ( e 1 ∧ f 2) ∨ ( e 2 ∧ f 1) ∨ ( e 2 ∧ f 2) = ( e 1 ∧ f 1) ∨ ( e 2 ∧ f 2). Hence µ is at least a homomorphism. For any s ∈S, the decomposition s = ( s∧ a∧ s) ∨ ( s \ a) represents s as the join of an element in S1 with an element in S2. Hence µ is also “onto”. Finally, suppose that ( e 1 ∨ e 2) = ( f 1 ∨ f 2) for e 1, f 1 in S1 and e 2, f 2 in S2. Letting u be this common join we have u ≥ e 1, f 1, e 2 and f 2. Thus e 1 = e 1∧ u = e 1 ∧ ( f 1 ∨ f 2) = ( e 1∧ f 1) ∨ ( e 1∧ f 2) = ( e 1∧ f 1) ∨ 0 = e 1∧ f 1 and in similar fashion f 1 = e 1∧ f 1. Likewise e 2 = f 2 so that µ is indeed an isomorphism. £ Corollary 4.1.5. Every skew Boolean algebra S with a finite maximal lattice image is isomorphic to a product of primitive skew Boolean algebras that are determined to within isomorphism by its minimal non- 0 D -classes. £ Recall that 2 is the Boolean lattice {1 > 0}, 3L is the left-handed primitive skew Boolean algebra {1 L 2 > 0}, 3R is its right-handed variant and 5 is the fibered product, 3L ×2 3R. Corollary 4.1.6. The nontrivial directly irreducible skew Boolean algebras are the primitive algebras. The nontrivial subdirectly irreducible skew Boolean algebras are 2, 3L and 3R . Thus every skew Boolean algebra is a subdirect product of copies of 2, 3L and 3R. £ Proof. The first statement is clear by Theorem 4.1.4. A nontrivial, subdirectly irreducible algebra is thus primitive, and must be either left-or right handed, thanks to the factorization of Theorem 2.1.5. If both, it is a copy of 2. Otherwise, it is a copy of 3L or 3R by Theorem 2.6.12. £ Corollary 4.1.7. Every skew Boolean algebra can be embedded in a power of 5. Every left-handed (right-handed) skew Boolean algebra can be embedded in a power of 3L ( of 3R). Alternatively, every right-handed skew Boolean algebra can be embedded in some algebra of partial functions with codomain {1, 2}. £ Corollary 4.1.8. A (quasi-)identity in ∨ , ∧ and \ holds for all skew Boolean algebras if and only if it holds on 5. It holds for all left-handed (right-handed) skew Boolean algebras if and only if it holds on 3L (or 3R ). The question of when a (quasi-)identity holds in any of these varieties is thus decidable. £ The above results reveal the following simple lattice. Here 〈A〉 denotes the subvariety generated by the algebra A. 123 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 4.1.9. The lattice of all subvarieties of skew Boolean algebras is given by: 〈5〉   〈3L〉 〈3R〉   〈2〉 ↑ 〈1〉 where 1 is the subvariety of trivial algebras, 〈5〉 is the variety of all skew Boolean algebras, and 〈3L 〉, 〈3R〉 and 〈2〉 are the respective subvarieties of all left- and right-handed skew Boolean algebras and generalized Boolean algebras. £ Recall that an algebra is locally finite if every finite subset generates a finite subalgebra. A variety of algebras is called locally finite if each of its algebras are locally finite. Theorem 4.1.10. Skew Boolean algebras are locally finite. Proof. Let a skew Boolean algebra S be generated from a finite set X. Clearly only finitely many distinct functions from X to 5 exist. Let ϕ = ∏ i∈Iϕ i : S → 5I be an embedding of S into a power of 5. Since each ϕ i is determined by its behavior on X, only finitely many of the ϕ i’s can be distinct, in which case only finitely many are needed for an embedding, since in a family of functions that collectively separate points in their common domain, repetition is superfluous. Put otherwise, we may assume that |I| is finite, in which case the embedding gives |S| ≤ 5|I|. £ Complete algebras We now turn to issues of completeness. A symmetric skew lattice is join [ meet] complete if all commuting subsets have suprema [minima] in the natural partial ordering. It is complete if it is both join and meet complete. By a maximal element in a skew lattice is meant any element in the maximal D-class of S, should the latter exist. The unique maximal element, if it exists, is essentially the constant 1. Lemma 4.1.11. Join complete, symmetric skew lattices have maximal D -classes. In particular, every join complete symmetric, normal skew lattice S has lattice sections, all given as m∧S∧ m for some maximal element m of S. Proof. For then every maximal totally ≤-ordered subset must contain a maximal element m that in turn is ≤-maximal in the skew lattice. The rest is clear. £ A nontrivial skew Boolean algebra is completely reducible if it is isomorphic to a product of primitive skew Boolean algebras. Since primitive algebras are trivially complete, so is any completely reducible algebra. A primitive class in a skew Boolean algebra is any D-class lying 124 IV: Skew Boolean Algebras directly above 0. An atom or primitive element is any element in a primitive class. A skew Boolean algebra is atomic if for every nonzero x ∈S an atom a ∈S exists such that a ≤ x. Theorem 4.1.12. A skew Boolean algebra is completely reducible if and only if it is complete and atomic. Proof. Completely reducible algebras are clearly complete and atomic. For the converse, let S be a complete atomic skew Boolean algebra. For any x ∈S, the set x∧S∧ x is a complete atomic Boolean algebra with (unique) maximal element x. Thus each x > 0 in S is the supremum of its underlying atoms, all of which commute, with distinct sets of commuting atoms giving distinct suprema. Next, let {X 0 j} be an indexed collection of all primitive classes of S and let P j = X j be the corresponding primitive skew Boolean algebras. Define σ: ∏ j P j → S by setting σ[〈 ej〉] = sup〈 ej〉. By our remarks, σ is at least a bijection. Let 〈 ej〉, 〈 fj〉 ∈ ∏ j P j be given with sup j〈 ej〉 = e and sup j〈 fj〉 = f. Then e∨ f ≥ ej∨ fj for each j. Indeed, ( e∨ f) ∧ ( ej∨ fj) = ( e∧ ej) ∨ ( f∧ ej) ∨ ( e∧ fj) ∨ ( f∧ fj) = ej ∨ ( f∧ ej) ∨ ( e∧ fj) ∨ fj = ej ∨ ( f∧ ej) ∨ fj. Case 1. Both ej, fj ≠ 0. Here ej∨( f∧ ej)∨ fj = ej∨ fj since all ∨-factors are D-equivalent. Case 2. fj = 0. Then f∧ ej = 0 since its image in S/D is 0. Hence fj∨( f∧ ej)∨ fj = ej = ej∨ fj. Case 3. ej = 0. Here ej∨( f∧ ej)∨ fj = fj = ej∨ fj again. Similarly, ( ej∨ fj)∧( e∨ f) = ( ej∨ fj) so that e∨ f ≥ ej∨ fj is verified. Each primitive join ej∨ fj is either an atom of S or 0. Claim e∨ f = sup( ej∨ fj). Clearly e∨ f ≥ sup( ej∨ fj). If the inequality is strict, this means that e∨ f ≥ an atom gk that does not lie below sup( ej∨ fj). Thus for this index k, ek = 0 = fk. But this yields a contradiction in the Boolean lattice S/D where D is an atom and neither D g e ≥ k D nor D D , yet D D . Thus e g f ≥ g e ∨ D f ≥ g ∨ y = sup( ej∨ fj) as claimed. Next, by mid- k k k commutativity, ( e∧ f) ∧ ( ej∧ fj) = ( e∧ ej) ∧ ( f∧ fj) = ej∧ fj, and similarly ( ej∧ fj) ∧ ( e∧ f) = ej∧ fj. Thus ( e∧ f) ≥ ( ej∧ fj) for each j and hence ( e∧ f) ≥ sup( ej∧ fj). An argument similar to that for the join guarantees that ( e∧ f) = sup( ej∧ fj). Thus σ: ∏ j P j → S is an isomorphism of skew lattices. Since \ is implicitly determined by the skew lattice structure, σ is an isomorphism and S is seen to be completely reducible. £ Theorem 4.1.13. Complete skew Boolean algebras satisfy the identities: 1) e ∧ sup( fi) = sup( e ∧ fi). 2) e ∨ inf( fi) = inf( e ∨ fi). 3) e \ sup( fi) = inf( e \ fi). 4) e \ inf( fi) = sup( e \ fi). Proof. In each case the obvious inequality (≤ or ≥) becomes equality in its complete maximal Boolean algebra image. Thus one already has equality in the skew Boolean algebra. £ 125 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Given complete skew Boolean algebras S and T, any homomorphism f: S → T sends commutative subsets to commutative subsets. If f also preserves suprema and infima, then f is a complete homomorphism of complete skew Boolean algebras. We state the following fuller description of the P(A, B) process. Theorem 4.1.14. Given maps α: Aʹ → A and β: B → Bʹ, P(α, β): P(A, B) → P(Aʹ, Bʹ) defined by P(α, β) f = β fα, for all f ∈P(A, B), is a homomorphism of complete Boolean skew algebras. Moreover, if Set and CSBA denote the respective categories of sets and complete skew Boolean algebras, then P: Set × Set → CSBA is a bifunctor from the category of sets to the category of complete skew Boolean algebras. 4.2 Finiteness, orthosums and free algebras We recast the assertions of Theorem 4.1.4 and Corollary 4.1.5 in more detail for the case where S/D is finite and especially when the algebra S itself is finite. It will be helpful to explore things in a slightly more general context, proceeding as follows. Two element a and b in a skew Boolean algebra are orthogonal if a ∧ b = 0 (and thus b∧ a = 0 and also a∨ b = b∨ a). A finite set of elements { a 1, … , an} is an orthogonal set if the ai are pairwise orthogonal, so that a 1 ∨ … ∨ an = aσ1 ∨ … ∨ aσ n for all permutations σ on {1, 2, … , n}. In this situation, a 1 ∨ … ∨ an is denoted by a 1 + … + an n or ∑ a . (Indeed, such notation will assume orthogonality.) Such a sum is referred to as an 1 i orthogonal sum, or an orthosum for short. A family of D-classes { D 1, … , Dr} is orthogonal when elements from distinct classes are orthogonal. For this it is sufficient that some transversal set { d 1, … , dr} is orthogonal. When this occurs, the Di are the primitive D- classes in the subalgebra they generate. In general: Proposition 4.2.1. Given an orthogonal family of nonzero D- classes { D 1, … , Dr} and two orthosums a 1 + … + ar and b 1 + … + br where ai, bi ∈ Di: i) ( a 1 + … + ar) ∨ ( b 1 + … + br) = ( a 1∨ b 1) + ( a 2∨ b 2) + … + ( ar∨ br); ii) ( a 1 + … + ar) ∧ ( b 1 + … + br) = ( a 1∧ b 1) + ( a 2∧ b 2) + … + ( ar∧ br); iii) ( a 1 + … + ar) \ ( b 1 + … + br) = ( a 1\ b 1) + ( a 2\ b 2) + … + ( ar\ br); iv) a 1 + … + ar = b 1 + … + br iff a 1 = b 1, a 2 = b 2, … , and ar = br. Indeed (i) – (iv) extend to the subalgebra r ∑ D 0 1 = { x D 0 i 1 + … + xr⎪ xi ∈ } generated from the i union D 1 ∪ … ∪ Dr where elements from distinct D 0 are also orthogonal. £ i 126 IV: Skew Boolean Algebras r ∑ D 0 1 is an internal direct product of the primitive subalgebras D 0 , … , i 1 D 0 r that we call the orthosum of the D 0 . Of course, ai \ bi = 0 whenever ai i D bi. A special case occurs when the Di are atomic D-classes lying directly over the class {0}. Here ai ∧ bj = 0 for elements ai and bi from distinct atomic D-classes, making them orthogonal. In general, nonzero orthogonal D-classes D r 0 1, … , Dr are the atomic D-classes of the generated subalgebra ∑ D 1 . In any case we i have a basic result for skew Boolean algebras with only finitely many D-classes: Theorem 4.2.2. If S is a nontrivial skew Boolean algebra with finitely many D- classes and having atomic D- classes { D r 0 1, … , Dr}, then S is the orthosum ∑ D 1 of primitive i subalgebras { D 0 , … , D 0 1 r } . £ The above decomposition is an internal form of the atomic decomposition of a skew Boolean algebra S, which must occur when S/D is finite. This internal form is unique. The external form, given as a direct product, is unique to within isomorphism. In the left-handed case for finite S, the standard atomic decomposition is S ≅ n × × … × ≤ ≤ 1 n n with 2 ≤ n n … ≤ n L 2 L rL 1 2 r , with nL being the unique left-handed primitive algebra on {0, 1, 2, … , n –1} with 0 being the 0-element. Standard decompositions are also unique. Consider 2 × 2 × 4 L × 5 L × 5 L or more briefly 22 × 4 2 L × 5 L . In this instance 22 provides the center of the algebra where “L” is superfluous. Similar remarks hold in the right-handed case. In the two-sided general case one uses notation such as 3 L•5 R to represent the primitive algebra 3 L×25 R given by the fibered product, as in: S ≅ 23 × 3 L•5 R × 5 L•4 R × 7 L•7 R. In this case a standard decomposition could be given by lexicographically ordering the factors. In any case, a finite skew Boolean algebra is classified when its standard atomic decomposition is given. Example 4.2.1. Partial function algebras serve as primary examples of SBAs. Note that P n L({1, … , n},{1, … , m}) ≅ Π PL({ i}, {1, 2, … , m}) i=1 n ≅ ( ) n ≅ PL({1}, {1, 2, ... , m}) ( m + 1) . L In particular, PL({1, … , n},{1}) ≅ 2 n. In this case each partial function f is determined by choosing a subset of {1, 2, … , n} to be f–1(1), resulting in a bijection between PL({1, … , n},{1}) and the power set of {1, 2, … , n} that preserves the generalized Boolean operations. We characterize congruences on and homomorphisms between skew Boolean algebras with finitely many D-classes. 127 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proposition 4.2.3. Let θ be a congruence on a skew Boolean algebra A viewed as an orthosum r ∑ D 0 1 of primitive subalgebras D 0 , the being the atomic i D- classes. Then: i Di 1) If d θ 0 for d ∈ Di, then Di ⊆ [0] , the congruence class of 0. θ 2) If d 1 θ d 2 with d 1 ∈ Di, d 2 ∈ Dj, but Di ≠ Dj, then Di ∪ Dj ⊆ [0] . θ 3) Thus if some Di ⊆ [0] , then upon re-indexing, D , with the remaining θ 1∪ … ∪ Dk ⊆ [0]θ Di refined by θ- classes and r (∑ r D 0 ) θ ≅ ∑ D 0 θ 1 i k+1 i i where 0 0 0 θ i = θ | D × D and D θ is primitive for each i ≥ k + 1. i i i i r Thus, given S = ∑ D 0 and a homomorphism of skew Boolean algebras f : S → T: 1 i 4) f[ S] is an orthosum with summands f D 0 ⎡ , each of which is either primitive or else ⎣ ⎤ i ⎦ just {0 T}. In the former case, f[ D ] is at least atomic in f[ S]. i 5) f D 0 ⎡ 0 0 ∩ 0 ⎡ ⎤ ≠ {0 T} implies i = j so that = D . ⎣ ⎤ i ⎦ f D ⎣ j ⎦ Di j Finally, in the purely primitive case: 6) Given left-[right]handed primitive algebras D 0 and , a non-0 homomorphism 1 D 02 from D 0 to is any map sending 0 to 0, and elements in D 1 to elements in D 2. 1 D 02 7) In general, all non-0 homomorphisms f : D 0 → are obtained as follows: 1 D 02 (a) Send 0 to 0. (b) Pick a ∈ D 1 and b ∈ D 2 and any map λ: L a → L b and any map ρ: R a → R b. (c) Finally set f(0) = 0 and for all x ∈L a and y ∈R a, set f( x∧ y) = λ( x) ∧ ρ( y). Proof. (1) should be clear. (2) For d 1 = d 1∧ d 1 is θ-related to d 1∧ d 2 = 0 and likewise d 2 θ 0. The conclusion now follows from (1). (3) – (5) should also be clear. (6) and (7) recall some of our remarks in the final part of Section 1.3 . £ As a consequence, we have the following particularly crisp result: 128 IV: Skew Boolean Algebras Theorem 4.2.4. Given a skew Boolean algebra S = r ∑ D 0 1 with finitely many D- classes i and with the D being its atomic D- classes, then its congruence lattice decomposes as follows: i Con( S) 0 0 ≅ Con( D ) ) . £ 1 × … × Con( Dr Recall that for a primitive algebra D 0, Con( D 0 r ) is essentially the lattice of rectangular partitions of D, augmented by a bottom element corresponding to the universal congruence on D 0. Free algebras: the finite case Given a non-empty set X: SBA X is the free skew Boolean algebra on X. RSBA X is the free right-handed skew Boolean algebra on X. LSBA X is the free left-handed skew Boolean algebra on X. GBA X is the free generalized Boolean algebra on X. Free algebras are, of course, unique to within isomorphism. Thus if we say “the free” we have in mind a particular concrete instance, from which we are free (in an alternative sense) to find other isomorphic variants. In this paper, the default free algebra F X on an alphabet X is the algebra of all terms (or polynomials) in X. In the current context, the terms are defined inductively as follows. 1) Each x in X is a term, as is the constant 0. 2) If u and v are terms, so are ( u ∨ v), ( u ∧ v) and ( u \ v). Two terms, u and v, are equivalent in F X iff u = v is an identity in the given variety of algebras. Clearly these criteria for equivalence differ among the four varieties of interest. Given an SBA equation of terms in X, u = v, one can check if it is a left-handed identity (or right-handed identity) by seeing if it holds for all evaluations on 3 L (or on 3 R). It is an SBA identity precisely when it holds for all evaluations on both 3 L and 3 R. Finally, it is a GBA identity if and only if it holds for all evaluations on 2. In our considerations, we are free to relax aspects of the syntax for parentheses if all ways of reinserting them lead to equivalent expressions. E.g., that would happen with x∨ y∨ z, but not with x∧ y∨ z. Given the universal character of the homomorphisms involved in the Clifford-McLean and the Kimura Factorization theorems for skew Boolean algebras: GBA X ≅ SBA X /D ≅ RSBA X /D ≅ LSBA X /D. RSBA X ≅ SBA X /L and LSBA X ≅ SBA X /R. SBA X ≅ LSBA X ×GBA RSBA X. X (Indeed, let V be any variety of algebras with W a subvariety of V. For each algebra A in V, let θA be the congruence on A such that A/θA is in W and the induced map ϕA : A → A/θA is a 129 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond universal homomorphism from A to W. Then if A is a free V-algebra on generating set X, then A/θ is a free W-algebra on generating set ϕA[ X]. In the above context, θA = D, L or R as appropriate, with X and ϕA[X] being equipotent under ϕA.) In what follows we first consider SBA n, LSBA n, etc. which denote SBA X, LSBA X etc. on alphabet X = { x 1, x 2, … , xn}. Their standard atomic decomposition is given in Theorem 4.2.6 below. But to obtain the latter we need to understand their atomic structure. The case for LSBA n and for RSBA n is described in Theorem 4.2.5, the content of which is our immediate goal. We focus on LSBA n. Since LSBA n has finitely many generators it will be finite and thus is determined by its atomic D-classes, all lying just above the class {0}. We first describe these classes. Each class consists of atoms all sharing a common form. The justification that they are indeed the atomic D-classes will follow. They are the 2 n–1 classes of one of the forms below where { y 1, y 2, … , yn} in the table represents an arbitrary permutation of { x 1, x 2, … , xn}. A typical class arises from a partition {L|M} of { x 1, x 2, … , xn} with k ≥ 1 elements in L and n– k elements in M used to form the term ( y 1∧ … ∧ yk) \ ( yk+1∨…∨ yn). This partition is ordered in that {L|M} is distinct from {M|L}. Thus, e.g., {1, 2|3, 4} ≠ {3, 4|1, 2}. FormType Number of Classes of this Form Class Sizes ⎛ ⎞ y ∨ ∨ 1 \ ( y 2 y 3 ∨ yn ) n = n ⎝⎜1⎠⎟ 1 ⎛ n⎞ ( y ∧ ∨ ∨ 1 y 2 ) \ ( y 3 y 4 ∨ yn ) ⎝⎜ 2⎠⎟ 2 ⎛ n⎞ ( y ∧ ∧ ∨ ∨ 1 y 2 y 3) \ ( y 4 y  ∨ 5 yn ) ⎝⎜ 3⎠⎟ 3    ⎛ ⎞ y ∧ ∧ 1 y 2 ∧ yn 1 = n ⎝⎜ n⎠⎟ n Given the left-handed identity x∧ y∧ z = x∧ z∧ y and the 2-sided identities x \ ( y ∨ z) = x \ ( z ∨ y) = ( x \ y) \ z = ( x \ z) \ y, (easily checked on 3L or on both 3L and 3R respectively), ( y 1∧ …∧ yk)\( yk+1∨…∨ yn) is invariant in outcome under any permutation of y 2, …, yk or of yk+1, …, yn. What does distinguish the elements in each class is the left-most element or variable, y 1. In all, a total of n 2 n–1 essentially distinct atoms exist to produce n 2 n–1 n-variable functions on 3L (or on 3R). This is verified in the remarks below. But first, an example: 130 IV: Skew Boolean Algebras Example 4.2.2. For X = { x, y, z, w}, the 15 atomic classes and the 4(23) = 32 atoms are: { x \ ( y∨ z∨ w)} { y \ ( x∨ z∨ w)} { z \ ( x∨ y∨ w)} { w \ ( x∨ y∨ z)} {( x∧ y)\( z∨ w), ( y∧ x)\( z∨ w)} {( x∧ z)\( y∨ w), ( z∧ x)\( y∨ w)} {( x∧ w)\( y∨ z), ( w∧ x)\( y∨ z)} {( y∧ z)\( x∨ w), ( z∧ y)\( x∨ w)} {( y∧ w)\( x∨ z), ( w∧ y)\( x∨ z)} {( z∧ w)\( x∨ y), ( w∧ z)\( x∨ y)} {( x∧ y∧ z)\ w, ( y∧ z∧ x)\ w, ( z∧ x∧ y)\ w} {( x∧ y∧ w)\ z, ( y∧ w∧ x)\ z, ( w∧ x∧ y)\ z} {( x∧ z∧ w)\ y, ( z∧ w∧ x)\ y, ( w∧ x∧ z)\ y} {( y∧ z∧ w)\ x, ( z∧ w∧ y)\ x, ( w∧ y∧ z)\ x} { x∧ y∧ z∧ w, y∧ z∧ w∧ x, z∧ w∧ x∧ y, w∧ x∧ y∧ z} . £ Theorem 4.2.5. Given the free left-handed skew Boolean algebra LSBA n on { x 1, … , xn}: i) LSBA n is a finite algebra whose atoms are the terms ( y 1∧…∧ yk) \ ( yk+1∨…∨ yn) for k ≥ 1 and where ( y 1, … , yn) is a permutation of { x 1, … , xn}. ii) Atoms ( y 1∧…∧ yk) \ ( yk+1∨…∨ yn) and ( z 1∧…∧ zl) \ ( zl+1∨…∨ zn) lie in the same atomic class if and only if k = l, ( z 1, … , zk) is a permutation of { y 1, … , yk} and thus ( zl+1, … , zn) is a permutation of { yk+1. … , yn}. iiiL) ( y 1∧…∧ yk) \ ( yk+1∨…∨ yn) = ( z 1∧…∧ zl) \ ( zl+1∨…∨ zn) if besides (ii), y 1 = z 1. For the free right-handed dual algebra RSBA n, (i) and (ii) again hold along with: iiiR) ( y 1∧…∧ yk) \ ( yk+1∨…∨ yn) = ( z 1∧…∧ zl) \ ( zl+1∨…∨ zn) if in addition to (ii), yk = zk. Proof. We consider the left-handed case. The right-handed assertion is similar. To begin, given a permutation ( z 1, … , zk) of { y 1, … , yk}, ( z 1∧…∧ zk)\( yk+1∨…∨ yn) and ( y 1∧ …∧ yk)\( yk+1∨…∨ yn) are L-related; they are not equal if z 1 ≠ y 1. Indeed y 1∧…∧ yk L z 1∧…∧ zk plus ( a\ c)∧( b\ c) = ( a∧ b)\ c implies they are L-related; they are not equal if z 1 ≠ y 1 since they are not equal when operating as functions on 3L. Give y 1 and z 1 values 1 and 2 respectively, the remaining front variables 1, and all n–k back variables 0. The outcome for ( y 1∧ …∧ yk)\ ( yk+1∨…∨ yn) is 1 and for ( z 1∧…∧ zk)\( yk+1∨…∨ yn) is 2. In general, given distinct partitions {L|M} and {Lʹ|Mʹ} of { x 1, x 2, …, xn} with L and Lʹ nonempty, some element m lies in L∩Mʹ or in Lʹ∩M, say the former. Viewing m as a generator, given any {L|M}-term u and any {Lʹ|Mʹ}-term v, we have u∧ m = u but v∧ m = 0 = m∧ v. Thus u∧ v = u∧ m∧ v = u∧0 = 0 = v∧ u. Thus all {L|M}-terms are orthogonal to all {Lʹ|Mʹ}-terms. Since ( x 1∧ …∧ xk) \ ( xk+1∨…∨ xn) = 0 is not an identity in 3L for k ≥ 1, all {L|M}-classes are non-0 classes and distinct {L|M}-classes are orthogonal. Returning to the example above, {( x∧ y)\( z∨ w), ( y∧ x)\( z∨ w)} is disjoint from {( x∧ w)\( y∨ z), ( w∧ x)\( y∨ z)} with pairs from distinct classes being orthogonal. To see that they are full D-classes of LSBA n, and that they (all the) atomic D-classes, observe first that they are the atomic D- classes in the subalgebra of LSBA n that they generate. 131 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond We need to show that this subalgebra is in fact all of LSBA n. We do so by showing that each generator xk of LSBA n is in the generated subalgebra. The identities above give us: x 1 = ( x 1∧ x 2) + ( x 1\ x 2) = ( x 1∧ x 2∧ x 3) + (( x 1∧ x 2)\ x 3) + (( x 1\ x 2)∧ x 3) + ( x 1\ x 2)\ x 3) = ( x 1∧ x 2∧ x 3) + (( x 1∧ x 2)\ x 3) + (( x 1∧ x 3)\ x 2) + ( x 1\( x 2∨ x 3)) = ( x 1∧ x 2∧ x 3∧ x 4) + (( x 1∧ x 2∧ x 3)\ x 4) + … The process keeps repeating on each new term until generator x 1 is resolved into an orthosum of 2 n–1 {L|M}-type terms – indeed into all the {L|M}-type terms with left-most entry x 1. Similar calculations work for the remaining generators. Thus the 2 n–1 distinct {L|M}-classes are all the atomic D-classes of LSBA n. £ In the generalized Boolean case, all atomic terms resulting from the same {L|M} decomposition are equated. Thus the particular left-most generator/variable no longer differentiates among outcomes. In the 2-sided case, in the Kimura fibered product construction each left-handed atomic class is matched off with the right-handed atomic class with the same {L|M} partition. In this case the data of (( y 1∧…∧ yk), ( yʹ1∧…∧ yʹ k)) can be combined as y 1∧…∧ yk∧ yʹ1∧…∧ yʹ k and then reduced via 2-sided normality. Returning to the previous example: Example 4.2.2 continued. These previous terms describe the atomic classes of both the left- and right-handed free algebras on { x, y, z, w}. E.g. {( x∧ y)\( z∨ w), ( y∧ x)\( z∨ w)} works in the left-hand case, while {( x∧ y)\( z∨ w), ( y∧ x)\( z∨ w)} works in the right-hand case. In both finite cases it is possible to describe the “atomic” terms using cyclic permutations in a way that the terms do double duty. But that won’t “stretch” to the 2-sided case. Here we adjoin both ( x∧ y∧ x)\( z∨ w) and ( y∧ x∧ y)\( z∨ w) to the class to get: {( x∧ y)\( z∨ w), ( y∧ x)\( z∨ w), ( x∧ y∧ x)\( z∨ w), ( y∧ x∧ y)\( z∨ w)}. For two terms to be equal in value, both end variables in the left part would have to agree. In general, the corresponding atomic classes would be squared in size . £ We thus obtain precise structural descriptions of all four relevant free algebras. In what follows D{L|M} is the {L|M}-induced D = L-class, PL{L|M} is the left-handed primitive algebra D 0 {L|M} and PR{L|M} is its right-handed counterpart. Also, given primitive algebras P and Q, P• Q denotes their fibered product over 2, P×2 Q. In the next theorem, the trivial algebra 1 on {0} is included to allow the full distribution of binomial coefficients. This factor corresponds to the front-empty partition {∅| X}. Hence: 132 IV: Skew Boolean Algebras [2.2] Theorem 4.2.6. The free left-handed skew Boolean algebra LSBA n on { x 1, …, xn} is a direct sum of the primitive algebras PL{L|M} where {L|M} ranges over partitions {L|M} of { x 1, …, xn} where L ≠ ∅. Thus: ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎝ ⎠⎟ ⎜1 ⎝ ⎠⎟ ⎜ 2 ⎝ ⎠⎟ ⎜ 3 ⎝ ⎠⎟ ⎜ n ⎝ ⎠⎟ LSBA n ≅ × 2 × 3 × 4 × ... × ( n +1) . 1 L L L Dually, the free right-handed skew Boolean algebra RSBA n on { x 1, …, xn} is a direct sum of the primitive algebras PR{L|M} where {L|M} shares the same range. Thus: ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎜1 ⎜ 2 ⎜ 3 ⎜ n ⎝ ⎠⎟ ⎝ ⎠⎟ ⎝ ⎠⎟ ⎝ ⎠⎟ ⎝ ⎠⎟ RSBA n ≅ 1 × 2 × 3 ×4 × ... × ( n +1) . R R R Finally, the free skew Boolean algebra SBA n on { x 1, …, xn} is a direct sum of the primitive algebras PL{L|M}•PR{L|M} where {L|M} again shares the same range. Thus: ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0⎟ ⎝⎜1 ⎟ ⎝⎜ 2⎟ ⎜ 3⎟ ⎜ n⎟ SBA ⎝ ⎠ ⎠ ⎠ ⎝ ⎠ ⎝ ⎠ n ≅ 1 × 2 ×(3 •3 ) ×(4 •4 ) × ... × (( n +1) •( n +1) ) . £ L R L R L R Corollary 4.2.7. For all n ≥ 1: ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 ⎟ ⎜ 2⎟ ⎜ 3⎟ ⎜ n⎟ i) | ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ LSBA n| = 2 3 4  ( n +1) = |RSBA n|. ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 ⎟ ⎜ 2⎟ ⎜ 3⎟ ⎜ n⎟ ii) |SBA ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ n| = 5 10  ( n 2 +1) . 2 Moreover, if αL( n), αR( n) and α( n) denote the number of atoms in LSBA n, RSBA n and SBA n, respectively, then: ⎛ n⎞ ⎛ n⎞ ⎛ n ⎞ ⎛ n⎞ iii) αL( n) = ⎝⎜1⎠⎟1+⎝⎜2⎠⎟ 2 +... +⎝⎜ n −1⎠⎟ ( n −1)+⎝⎜ n⎠⎟ n = αR( n) . ⎛ n⎞ ⎛ n⎞ ⎛ n ⎞ ⎛ n⎞ iv) α( n) = . £ ⎝⎜1⎠⎟ 1+ ⎝⎜ 2⎠⎟ 4 +  + ⎝⎜ n −1⎠⎟ ( n −1)2 + ⎝⎜ n⎠⎟ n 2 Standard combinatorial arguments give the following simplifications: Corollary 4.2.8. Given αL( n), αR( n) and α( n) as above: n+1 αL( n) = αR( n) = n 2 n–1 and α( n) = n( n+1)2 n–2 , so that α( n) = αL( n). 2 133 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ Proof. To see = n 2 n–1, differentiate the binomial expansion of (1 + x) n ⎝⎜1⎠⎟1+⎝⎜ 2⎠⎟ 2+...+⎝⎜ n⎠⎟ n and set x = 1. Setting x = 1 again in the second derivative of the binomial expansion of (1 + x) n ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ gives: = n( n–1)2 n–2. ⎝⎜ 2⎠⎟ 2 i1 + ⎝⎜ 3⎠⎟ 3i 2 + + ⎝⎜ n⎠⎟ n( n −1) Adding the equality of the previous expansion to this and simplifying gives ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ = n 2 n–1 + n( n–1)2 n–2 = n( n+1)2 n–2. £ ⎝⎜1⎠⎟1+ ⎝⎜ 2⎠⎟ 4+  + ⎝⎜ n⎠⎟ n 2 A short table of values follows with the sizes for n = 5 given to 4-digit accuracy. n | L SBA n | α L ( n) | SBA n | α( n) 2 12 4 20 6 3 864 12 10, 000 24 8 4 14, 929, 920 32 425 × 10 80 16 25 5 3.715 × 10 80 3.017 × 10 240 ⎛ n⎞ ⎛ ⎞ ⎛ ⎞ Since = 2 n – 1, we have: ⎝⎜1⎠⎟ + n ⎝⎜ 2⎠⎟ + ...+ n ⎝⎜ n⎠⎟ Corollary 4.2.9. A free (left-handed, right-handed or two-sided) skew Boolean algebra on n generators has 2 n – 1 primitive factors in its atomic decomposition. Thus any skew Boolean algebra on n generators has at most that many. Any generalized Boolean algebra on n generators thus has ≤ 2 n – 1 atoms, and is free if and only if it has exactly that many. £ Theorems 4.2.2 and 4.2.6 also lead to: Corollary 4.1.20. Every finite skew Boolean algebra can be embedded in a finite free skew Boolean algebra. Every finite left-handed [right-handed] skew Boolean algebra is isomorphic to a direct factor of a finite free left-handed [right-handed] skew Boolean algebra. 4.3 Connections with strongly distributive skew lattices A skew lattice is embedded in a skew Boolean algebra (S; ∨, ∧, \, 0) if it is embedded in its skew lattice reduct (S; ∨, ∧). Since such reducts are strongly distributive, theorems from the previous section and Section 2.6 lead to the following results. 134 IV: Skew Boolean Algebras Theorem 4.3.1. A skew lattice can be embedded in a skew Boolean algebra if and only if it is strongly distributive. Proof. (Only if) Skew lattice reducts of skew Boolean algebras are strongly distributive and strongly distributive skew lattices form a skew lattice subvariety. (If) By Theorem 2.6.12 every strongly distributive skew lattice can be embedded in a power of the skew lattice 5, which however, is the reduct of that power of the skew Boolean algebra 5. £ Theorem 4.3.2. For skew lattice S the following are equivalent: i) S can be embedded in a right-handed skew Boolean algebra. ii) S is right-handed and strongly distributive. iii) S can be embedded in a slew lattice of partial functions PR(A, B). Proof. The equivalence of (i) and (ii) is seen as in the previous proof, but with 5 replaced by 3. If S can be embedded in some partial function skew lattice PR(A, B), then since the latter is right- handed and strongly distributive so it S. The converse follows from Theorem 2.6.14. £ Theorem 4.3.3. Given an identity in ∨ and ∧ the following are equivalent: i) The identity holds in all skew Boolean algebras. ii) The identity holds in all strongly distributive skew lattices. iii) The identity holds in the skew lattice 5. Proof. Both (i) and (ii) are equivalent by Theorem 4.2.1, and each clearly implies (iii). Conversely, any identity holding on 5 must also hold on any sub-skew lattice of any power of 5, and extending via isomorphism, it must hold on any strongly distributive skew lattice. £ Of course, one has the right-handed specialization of the above. Theorem 4.3.3R. Given an identity in ∨ and ∧ the following are equivalent: i) The identity holds in all right-handed skew Boolean algebras. ii) The identity holds all right-handed strongly distributive skew lattices. iii) The identity holds in the skew lattice 3R. iv) The identity holds in all partial function algebras P(A, B). £ Proof. The equivalence of (i) – (iii) is seen similarly as in the general case. Their equivalence with (iv) comes from Theorem 2.6.14. £ 135 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond The override and the update operations J. Berendsen, D. Jansen, J. Schmaltz and F. Vaandrager in their 2010 paper, The axiomatization of overriding and update (Journal of Applied Logic, 8 (2010), 141-150.), gave an approach to studying partial function algebras that is similar in many ways to ours, but with some differences. To begin, given a pair of partial functions f and g in the partial function set P(A, B) with subsets F and G of A being their respective support, the authors considered the following operations: Override: f ▹ g = f ∪ g⎪(G \ F). Update: f[ g] = g⎪(F∩G) ∪ f⎪(F \ G). Minus: f – g = f ⎪(F \ G). The override f ▹ g is clearly the join f ∨ g, while the minus f – g is just f \ g. Stated in our notation, the authors defined the update by f[ g] = ( f∧ g) ∨ ( f \ g) where f∧ g is g \ (g \ f), the latter holding for all right-handed skew Boolean algebras. For right-handed, strongly distributive skew lattices in general, f[ g] is given also as either ( f∧ g) ∨ f or f ∧ ( g∨ f). Indeed, using any of these three ways, on 3R one has: x[ y] 0 1 2 0 0 0 0 . 1 1 1 2 2 2 1 2 Thus all three evaluations of x[ y] in any skew Boolean algebra agree. (The equation ( x∧ y)∨ x = x∧( y∨ x) and its ∨-∧ dual are examined in Section 5.1.) One can consider algebras with signature ([], ∨) as was suggested by Berendsen et al in their paper. The problem of interest for those authors was that of determining identities in [ ] and ∨ that hold for all partial function algebras (and thus more generally, for all right-handed, strongly distributive skew lattices). Clearly all identities in [ ] and ∨ that hold in all partial function algebras follow from the defining identities for right-handed skew Boolean algebras; more generally, they follow from the defining identities for right-handed, strongly distributive skew lattices. In their paper, the authors were interested particularly in finding a set of identities in just [ ] and ∨ that (1) held for all partial function algebras and (2) was powerful enough so that all identities in ∨ and [ ] that hold in all partial function algebras were consequences of the given identities. The search for such a set of identities is apparently still open. However, if 3* denotes the algebra ({0, 1, 2}; [ ], ∨) defined by the Cayley tables x ∨ y 0 1 2 x[ y] 0 1 2 0 0 1 2 and 0 0 0 0 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 2 136 IV: Skew Boolean Algebras that reflect what occurs on 3R, either as a skew lattice or as a skew Boolean algebra, then along with Theorem 4.3.3R, we are led to the following result of Cvetko-Vah, Leech and Spinks: Theorem 4.3.4. An identity in [] and ∨ holds in all right-handed skew Boolean algebras, or more broadly in all right-handed strongly distributive skew lattices, if and only if it holds in 3*. Thus, the question of whether an identity in [] and ∨ holds in these classes of algebras, or in particular in all partial function algebras of the form P(A, B) is decidable. Proof. The condition is clearly necessary. Since every right-handed SBA [strongly distributive skew lattice] is a subalgebra of a power of 3R [under the appropriate signature] the condition is also sufficient. The final statement is now clear. £ Our interest in such identities and especially in [ ] is due to the role of the latter in right- handed strongly distributive skew lattices. Returning to Section 2.7, the free categorical and symmetric right-handed skew lattice on generators x and y can be presented as follows: x ∨ y − R y ∨ x     x[ y] = x ∧ ( y ∨ x) = ( x ∧ y) ∨ x. x − R x[ y] y[ x] − R y where     y[ x] = y ∧ ( x ∨ y) = ( y ∧ x) ∨ y. y ∧ x − R x ∧ y In particular, this skew diamond and its subalgebras describe the 2-generator subalgebras that can occur in right-handed skew Boolean algebras or in right-handed strongly distributive skew lattices in general or, for that matter, in right-handed skew lattices in rings. Thus, just as skew joins/overrides can be seen as biased unions, and meets as biased intersections, the two other terms in this diagram besides x and y can be viewed as the outcomes of updating x and y relative to each other. More will be said about the update operation in Section 5.1. Axioms for (right-handed) skew Boolean algebras In their paper, Berendsen et al introduced five identities that describe the behavior of ▹, – and ∅ in combination. Using automated reasoning software they showed their equations to be independent and obtained a number of derived equations. They also raised questions about the algebras that satisfy these equations. In a responding paper by Cvetko-Vah, Leech and Spinks ( Skew lattices and binary operations on functions, Journal of Applied Logic, 11 (2013), pp. 253-265), the variety of algebras satisfying those five identities was shown to be term equivalent to the variety of right-handed skew Boolean algebras. Thus the five identities given in the first paper lead to the following equational basis (or set or characterizing identities) for right-handed skew Boolean algebras. 137 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 4.3.5. An independent set of identities that characterize the variety of all right-handed skew Boolean algebras ( S; ∧, ∨, \, 0) is given by: i) x ∨ x ≈ x. ii) x \ x ≈ 0. iii) x ∨ y ≈ ( y \ x) ∨ x. iv) x \ ( y \ z) ≈ ( x \ y) ∨ ( x \ ( x \ z)). v) ( x ∨ y) \ z ≈ ( x \ z) ∨ ( y \ z). Here the meet ∧ is defined by x∧ y : = y \ ( y \ x). £ An independent set of six identities characterizing the variety of skew Boolean algebras was given in Spinks’ dissertation: (See also Spinks [1998] Prop. LBSL-RC-7.) i) x∧( x∨ y) ≈ x; ii) ( y∧ x)∨ x ≈ x; iii) x ∧ ( y∨ z) ≈ ( x∧ y) ∨ ( x∧ z); iv) ( x∨ y) ∧ z ≈ ( x∧ z) ∨ ( y∧ z); v) (( x∧ y)∧ x) ∨ ( x\ y) ≈ x; vi) ( x\ y) ∧ y ≈ 0. These contrast with the 12 identities implicit in the definition of a skew Boolean algebra in Section 4.1. As of this writing, neither of these two sets of identities have been bettered with respect to size. Of interest in the first set is the fact that along with 0 which is just used once, the binary operations \ and ∨ suffice in their statements. Could ∧ be eliminated from the second set of 2-sided identities? Actually yes, thanks to the identity: x∧ y = ( y∨ x) \ {[( y∨ x) \ x] ∨ [( y∨ x) \ y]}. This identity is easily seen to hold on primitive algebras, and hence on all skew Boolean algebras. Returning to all six identities above, it can be used to first define ∧ in terms of ∨ and \, and then to eliminate all occurrences of ∧ in the six identities. A reduced signature (∨, \, 0) is thus possible for skew Boolean algebras, but not convenient. As with generalized Boolean algebras, ∨ cannot be eliminated in favor of ∧, \ and 0. A closer look at the role of \ occurs in the final chapter. 4.4 Skew Boolean algebras with intersections Recall that a skew lattice (S; ∨, ∧) has finite intersections if every pair e, f ∈S possesses a natural meet with respect to the natural partial order ≥ on S. When it occurs, the natural meet of any e and f in S is denoted by e∩ f. We say that (S; ∨, ∧) has intersections if the natural partial order ≥ on S has infima for arbitrary subsets. In general the following must hold: 138 IV: Skew Boolean Algebras Lemma 4.4.1. For any pair e, f in a skew lattice S the following are equivalent: i) e∧ f = f∧ e. ii) e∧ f = e∩ f, where e∩ f exists. iii) f∧ e = e∩ f, where e∩ f exists. Proof. That (i) implies (ii) and (iii) is clear. Conversely, given say (ii) one has f ∧ e = f ∧ e ∧ f ∧ e = f ∧ ( e∩ f) ∧ e = e∩ f = e ∧ f and (i) follows. The equivalence of (i) and (iii) is similar. £ Lemma 4.4.2. A skew lattice having finite intersections is an algebra S = ( S; ∨, ∧, ∩) such that ( S; ∩) is a meet semilattice, ( S; ∨, ∧) is a skew lattice and the following identities hold: e ∩ ( e∧ f∧ e) = e∧ f∧ e and e ∧ ( e∩ f) = e∩ f = ( e∩ f) ∧ e. Skew lattices with finite intersections thus form a variety of algebras. Proof. The identities state in essence that the two partial orders on S induced by ∧ and ∩ must contain each other and thus coincide. £ Theorem 4.4.3. (Theorem 1.3.11 restated) The variety of all skew lattices having finite intersections is congruence distributive. £ A skew lattice (S; ∨, ∧) is initially finite if for all e ∈ S, ⎡ e⎤ = { f ⎪ f ≤ e} is finite. Clearly: Theorem 4.4.4. If (S; ∨, ∧, 0) is a normal, symmetric skew lattice with a zero, then: i) If S is join complete, then S is complete. ii) If S is complete, then S has intersections. iii) If S is initially finite, then S has intersections. £ A skew Boolean algebra with finite intersections is called a skew Boolean ∩- algebra. (Read “skew Boolean intersection algebra”.) These algebras form a variety of algebras denoted by SBA∩. We next scan examples of skew Boolean algebras to see which are ∩-algebras. Example 4.4.1. Finite intersections trivially exist for generalized Boolean algebras. Example 4.4.2. A 0-primitive skew Boolean algebra S = D0 has arbitrary intersections. Given A ⊆ S, inf A is the sole element of A when A is a singleton set, and 0 otherwise. Example 4.4.3. Every completely reducible skew Boolean algebra, being the product of primitive algebras, has arbitrary intersections. 139 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond In particular, a partial function algebra P(A, B) is completely reducible and thus must have arbitrary intersections. Indeed, upon viewing partial functions from A to B as subsets of A × B, intersections in our sense are just intersections of subsets: f∩ g = f ∩ g. In particular, a skew Boolean algebra S for which S/D is finite has arbitrary intersections. Example 4.4.4. Let R be a C-ring: an associative ring such that every for each x ∈R a central idempotent C( x) exists such that x C( x) = x and C( x) is the least central idempotent with respect to this property. Cornish [1980] showed that a left-handed skew Boolean algebra can be defined from any C-ring upon setting: x∧ y = x C( y), x∨ y = x + y – x C( y) and x \ y = x – x C( y). The intersection of a pair of elements is given by x∩ y = [1 – C( x – y)] x. Example 4.4.5. Kudryavtseva and Leech [2016] have shown that free skew Boolean algebras have finite intersections. This is discussed later in this section. Example 4.4.6. Given any maximal normal band S in a ring R, we have seen that S forms a skew Boolean algebra upon setting e∧ f = ef , e∨ f = e∇ f and e f = e – efe. They have intersections if the ring is semisimple and Artinian, or in particular, is a matrix ring over a field. Example 4.4.7. (Counterexample) Let * be the set of natural numbers and let S denote the subset of P(*, {0, 1}) consisting of all partial functions having domains that are either finite or cofinite (in that the complement in * is finite). It is easily verified that S is closed under the skew Boolean operations, ∨ and ∧ and \. Thus S is a skew Boolean algebra. Clearly f∩ g does not exist for the partial functions f, g ∈ S, both with full domain *, but with ⎧0 if n is even f( n) = 0 and g( n) = ⎨ . 1 if n is odd ⎩ Recall that an ideal is a nonempty subset I of a skew lattice S such that for all x, y ∈ I and all z ∈S, x∨ y, x∧ z and z∧ x all lie in I. While arbitrary sub-skew Boolean algebras of skew Boolean ∩-algebras need not have finite intersections, ideals of skew Boolean ∩-algebras do. Extending Theorem 4.1.4, we have: Theorem 4.4.5. Given a principal ideal 〈 a〉 in a skew Boolean algebra S and its associated annihilator ideal ann( a), S has finite intersections if and only if both 〈 a〉 and ann(a) have finite intersections, in which case the map µ: 〈 a〉 × ann( a) → S defined by µ( x, y) = x∨ y is an isomorphism of skew Boolean ∩- algebras. In particular, given x, y in 〈 a〉 and u, v in ann( a), ( x ∨ u) ∩ ( y ∨ v) = ( x ∩ y) ∨ ( u ∩ v). £ As a consequence we have: 140 IV: Skew Boolean Algebras Theorem 4.4.6. In the variety of skew Boolean ∩- algebras the following assertions hold. i) The primitive algebras are precisely the nontrivial simple algebras. ii) The primitive algebras are the nontrivial directly irreducible algebras. iii) The primitive algebras are the nontrivial subdirectly irreducible algebras. Proof. By Theorem 4.4.5, a nontrivial simple algebra must be primitive. Conversely, let S be a primitive algebra and let θ be a congruence on S. Suppose that e θ f with e ≠ f in S. Then e and f are also θ-congruent to e∩ f = 0. Since either e ≠ 0 or f ≠ 0, this forces every element of the primitive algebra to be congruent to 0. Hence θ is the universal congruence. The only other congruence possible is thus the identity congruence Δ. Thus all primitive algebras are simple and (i) holds. (ii) is clear. Given (i) and (ii), (iii) easily follows. £ Given a primitive skew Boolean algebra P and an ∩-preserving homomorphism ϕ from P to a skew Boolean ∩-algebra S, it follows that ϕ is either the 0-homomorpism, or an embedding. The ideals of any skew lattice S form a complete lattice. The canonical map π: S → S/D induces an isomorphism of this lattice with the lattice of ideals of S/D, via the map I → I/D = S/D. If S/D is finite, then all ideals of both S and S/D are principal with both lattices of ideals being isomorphic to S/D. In the current context, ideals serve as the kernels of homomorphism. Lemma 4.4.7. Let θ be a congruence on a skew Boolean ∩- algebra S and let θ[0] be the congruence class of 0. Then the class θ[0] is an ideal of S and for all e, f ∈ S, e θ f if and only if ( e \ e∩ f) ∨ ( f \ e∩ f) ∈ θ[0]. Conversely, if I is an ideal of S, then the relation θ = θ I defined as above is a congruence on S. Moreover the maps θ → θ[0] and I → θI form an inverse pair of bijections. Proof. Observe that on any skew Boolean ∩-algebra, e = f if and only if ( e \ e∩ f) ∨ ( f \ e∩ f) = 0. This is certainly true in the primitive case. Thus it is true for all such algebras by Theorem 4.3.6 and the first assertion of the lemma follows. For the converse, first let I be a principal ideal 〈 a〉 with complement ann( a). By Theorem 4.4.5, an ∩-algebra homomorphism ϕ of S onto ann( a) exists for which –1 〈 a〉 = ϕ (0). If θ is the congruence associated with ϕ, then e θ f in S iff ϕ( e) = ϕ( f) in ann( a) iff (ϕ( e) \ ϕ( e)∩ϕ( f)) ∨ (ϕ( f) \ ϕ( e)∩ϕ( f)) = 0 in ann( a) iff ( e \ e∩ f) ∨ ( f \ e∩ f) ∈ 〈 a〉. In general, any ideal I is the directed union of principal ideals: I = ∪↑{〈 a〉⎪ a ∈ I}. The θI as defined above is the corresponding directed union ∪↑{θ ⎪ a ∈ I} of “principal congruences” and 〈 a〉 thus is also a congruence. That both processes are reciprocal is clear. £ 141 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 4.4.8. Given a skew Boolean ∩- algebra S , its congruence lattice Con (S) on is ∩ isomorphic to the lattice of ideals of S (or of its maximal generalized Boolean image S/D.). Finitely generated congruences correspond to the principal ideals of S/D and thus form a generalized Boolean sublattice of the congruence lattice. Proof. The general correspondence is clear by the previous lemma. By the lemma again, finitely generated congruences have finitely generated ideal kernels that must be principal. Conversely, a principal ideal 〈 a〉 corresponds to the congruence generated from ( a, 0). £ Theorem 4.4.9. The lattice Con(S) of all skew lattice congruences on a skew Boolean ∩- algebra S is the subdirect product of the interval [Δ, D] and the sublattice Con (S) of skew ∩ Boolean ∩- algebra congruences on S under the map Con(S) → [Δ, D] × Con (S) given by the ∩ rule θ → (θ∩D, θθ[0]). Proof. By Theorem 3.1.2, Con(S) is the subdirect product of [Δ, D] and the interval [D, ∇] under the map θ → (θ∩D, θ∨D). But [D, ∇] ≅ Con(S/D) which is isomorphic to the lattice of ideals of S/D and in turn to the lattice of ideals of S, and thus to Con (S). The ∩-respecting ∩ congruence corresponding to θ∨D has kernel ideal (θ∨D)(0) = θ(0) so that θ(θ∨D)(0) = θθ(0). £ Intersections and the canonical images of a skew lattice Given a skew lattice S, the canonical skew lattice maps, S → S/L, S → S/R and S → S/D, are all homomorphisms. How do intersections fit into this picture? We begin with: Theorem 4.4.10. Given a skew lattice S with finite intersections: i) The canonical map S → S/L preserves intersections iff S is right-handed, so that S/L ≅ S. ii) The canonical map S → S/R preserves intersections iff S is left-handed, so that S/R ≅ S. iii) Both maps preserve finite intersections if and only if S is a lattice, in which case ∩ is ∧. iv) The canonical map S → S/D preserves intersections iff S is a lattice, so that S/D ≅ S . Proof. Suppose the map S → S/L preserves finite intersections and let x and y be distinct in S, but L-related. Then x∩ y exists in a properly lower D-class in S, while their images merge in S/L, giving them a trivial intersection in the D-class of S/L corresponding to that of both x and y in S. Thus (i) is seen. That cases for (ii) and (iv) are similar and (iii) follows immediately. £ Since neither S/L nor S/R need be isomorphic to S in general, they need not, in general, inherit intersections. This raises questions: Under what added conditions must a skew lattice S 142 IV: Skew Boolean Algebras having finite intersections imply that both S/L and S/R do also? Do examples exist where S has finite intersections but not S/L or S/R? Conversely, if S/L and S/R do also: Theorem 4.4.11. If a skew lattice S with finite intersections has a lattice section, then both S/L and S/R also have finite intersections. Proof. If T ⊆ S is a lattice section in S, then a copy of S/R in S is given by T[L] = ∪ t ∈TL t. For all x ∈ S let tx be the unique element in D x ∩ T and set x L = x∧ tx in T[L]. Then x L R x and T[L] = { x L⎪ x ∈ S}. Given a, b ∈ T[L], we claim that its intersection a∩ b in S is already in T[L]. We use the fact that x ≥ y in S iff both x L ≥ y L in T[L] and dually, x R ≥ y R in the dual subalgebra T[L]. (In general, if x and y correspond to ( xʹ, xʺ) and ( yʹ, yʺ) in S/R ×S/D S/L, then x ≥ y iff both xʹ ≥ yʹ and xʺ ≥ yʺ.) Since a = a L and b = b L, it follows that a, b ≥ ( a∩ b)L in T[L]. But since both a∩ b D ( a∩ b)L and a∩ b ≥ ( a∩ b)L by definition of a∩ b, a∩ b is ( a∩ b)L in T[L]. Being closed under ∩, T[L] and also T[R] have finite intersections and so do their copies, S/R and S/L. £ Thus for a symmetric skew lattice S with finite intersections, if S/D is countable (so that S has a lattice section), then both S/R and S/L also have finite intersections. We can do better. Theorem 4.4.12. If a symmetric skew lattice S has finite intersections, so do S/R and S/L. Proof. So let S be a symmetric skew lattice and let xʹ and yʹ in S/R be given with pre-images x and y in S. Let S1 be the sub-algebra generated from x and y, let T be a lattice section of S1 and let T[L] be the maximal left-handed subalgebra T[L] = ∪ t ∈TL t of S1. If x L = x∧ tx and y L = y∧ ty in S1 as in the previous proof, then x L and y L are also pre-images of xʹ and yʹ in S. Let S2 be the generated result of adjoining x L∩ y L in S to S1 and let T2 be a lattice section of S2 extending T1. Then x L∩ y L must lie in T2[L], as above. Suppose next that xʹ, yʹ ≥ zʹ in S/R. If z is a pre-image of zʹ in S, let S3 be the extension of S2 generated from S2 and z, and let T3 be a lattice section of S3 extending T2. By what was seen in the above proof, z L in T3[L] is also a pre-image of zʹ such that both x L ≥ z L and y L ≥ z L hence x L∩ y L ≥ z L. Clearly, if θ is the image of x L∩ y L in S/R, then xʹ, yʹ ≥ θ ≥ zʹ. Given that xʹ and yʹ are fixed and zʹ is arbitrary but subject to the constraint xʹ, yʹ ≥ zʹ, θ must be xʹ∩ yʹ in S/R. Since this must be true for all xʹ, yʹ in S/R, the latter has intersections. Likewise, so does S/L. £ Indeed, the above arguments essentially show: if a skew lattice S with finite intersections is such that for each countable sublattice T of S/D, the inverse image S|T of T in S has a lattice section, then both S/R and S/L also have finite intersections. 143 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 4.4.13. If a normal skew lattice S has finite intersections, then both S/L and S/R have finite intersections. Proof. A skew lattice S has finite intersections if and only if each principal ideal S∧ x∧S has finite intersections. For a normal skew lattice S, each ideal S∧ x∧S has a lattice section, namely x∧S∧ x, with x∧S = ( x∧S∧ x)[R] ≅ S/L and S∧ x = ( x∧S∧ x)[L] ≅ S/R. Thus S/L and S/R must have finite intersections if S does. £ Conversely, one may ask: if both S/L and S/R have finite intersections, must S also? Or do counterexamples exist? Here is a case of the latter. Example 4.4.8. Let S be the skew lattice defined on , with the  n∈{ an, bn} { a being a totally ordered family of n, bn} D-classes. For x, y in a common class { a , n, bn} ⎧ x, if n is even. x∨ y = y∧ x = ⎨ y, if n is odd. ⎩ Thus even classes are right-handed and odd classes are left-handed. If say x ∈ { a but m, bm} y ∈ { a where m < n, then x∨ y = y = y∨ x and x∧ y = x = y∧ x. Thus, n, bn} both a > both > both for all n n+1, bn+1 an, bn a ∈ &. n−1, bn−1 Consequently neither a nor exist, for all n ∈ &. On the other hand, both S/ n ∪ bn an ∩ bn L and S/R, have a D-class structure . . . > { a } > { } > { } > { } > { } > … n+2, bn+2 cn+1 an, bn cn−1 an−2, bn−2 that is right-handed when n is even and left-handed when n is odd. Both a ∩ b = and k k ck−1 a ∪ b = . Cases of pairs of elements from distinct D-classes are trivial, as they involve k k ck+1 pairs x, y where x > y. Observe that this example is trivially symmetric (since totally quasi-ordered), has many lattice sections, is distributive and thus is categorical. But for normal skew lattices, and in particular for skew Boolean algebras, we have the equivalence: Theorem 4.4.14. A normal skew lattice S has finite intersections if and only if both S/L and S/R have finite intersections. In particular, a skew Boolean algebra S has finite intersections iff both S/L and S/R do. Proof. We have already seen (⇒). As for (⇐), without loss in generality we represent S as the fibred product S/L ×S/D S/R. So let both ( xʹ, xʺ) and ( yʹ, yʺ) in S be given where xʹ, yʹ ∈S/L and 144 IV: Skew Boolean Algebras xʺ, yʺ ∈S/R. Since S/L and S/R have finite intersections, xʹ∩ yʹ and xʺ∩ yʺ exist in S/L and S/R, respectively. If xʹ∩ yʹ and xʺ∩ yʺ share a common image in S/D, then ( xʹ, xʺ) ∩ ( yʹ, yʺ) is just ( xʹ∩ yʹ, xʺ∩ yʺ). In general, let u 0∧ v 0 be the meet in S/D of the respective images u 0 of xʹ∩ yʹ and v 0 of xʺ∩ yʺ in S/D. In the respective D classes of S/L and S/R indexed by u 0∧ v 0, unique elements wʹ and wʺ exist (by normality) such that both xʹ∩ yʹ ≥ wʹ in S/L and xʺ∩ yʺ ≥ wʺ in S/R. The intersection ( xʹ, xʺ) ∩ ( yʹ, yʺ) is then precisely ( wʹ, wʺ). £ The practical consequence of all this as follows: just as all skew Boolean algebras can be constructed in principle from pairs of left- and right-handed skew Boolean algebras (SL, SR) having a common maximal commutative Boolean image B using the fibered product, SL ×B SR, so also all skew Boolean ∩-algebras can be constructed from pairs of left- and right-handed skew Boolean ∩-algebras (SL, SR) with a common maximal generalized Boolean algebra image B (with SL and SR viewed as skew Boolean algebras) by exactly the same process, SL ×B SR, even when their corresponding ∩-outcomes may have distinct locations relative to B. It follows that the study of skew Boolean ∩-algebras can, in principle, be reduced to studying right-handed skew Boolean ∩-algebras, or their term-equivalent left-handed duals, since pairs sharing common maximal lattice images can be spliced together at will. All this is illustrated in the following case of infinite free skew Boolean algebras. Atom splitting and the case of infinite free skew Boolean algebras Consider the inclusion LSBA n ⊂ LSBA n+1 induced by { x 1, … , xn} ⊂ { x 1, … , xn, xn+1}. The atoms of LSBA n are no longer atomic in LSBA n+1. The left-handed identity x = ( x∧ y) + ( x\ y) gives the following “subatomic” decomposition of the original atoms. ( x 1∧ x 2∧…∧ xk) \ ( xk+1∨…∨ xn) = ( x 1∧ x 2∧…∧ xk∧ xn+1) \ ( xk+1∨…∨ xn) + ( x 1∧ x 2∧…∧ xk)\ ( xk+1∨…∨ xn+1) Both components of the new decomposition are atoms in LSBA n+1. If say ( x 1∧ x 2∧ x 3)\( x 4∨…∨ xn) = ( x 1∧ x 3∧ x 2)\( x 4∨…∨ xn) in LSBA n, then their corresponding pairs of atomic components in LSBA n+1 remain equal. But if say ( x 1∧ x 2)\( x 3∨…∨ xn) ≠ ( x 2∧ x 1)\( x 3∨ x 4∨…∨ xn) in LSBA n, then both corresponding pairs of components are likewise unequal in LSBA n+1. One thus has extended the decomposition where, while a given element remains the same, its atomic decomposition doubles in length as each new generator is added. Thus given u ∈ LSBA n with atomic decomposition u = a 1 +…+ ar in LSBA n, each atom ak splits as bk + ck in LSBA n+1, where bi = ai∧ xn+1 and ci = ai\ xn+1, to give a revised atomic decomposition u = b 1 + c 1 + … + br + cr in LSBA n+1. Given the uniqueness of atomic decompositions (to within commutativity) of elements in LSBA n or in LSBA n+1, we have: 145 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Lemma 4.4.15 Given u be an atom of LSBA n and let a = b + c be the atomic decomposition of a in LSBA n+1 where b = a∧ xn+1 and c = a\ xn+1. Then the following are equivalent: i) u ≥ a in LSBA n ( and thus in LSBA n+1). ii) u ≥ b in LSBA n+1. iii) u ≥ c in LSBA n+1. ! Comments. (1) Thus a is in the atomic decomposition of u in LSBA n iff b [or c and hence both] is in the atomic decomposition of u in LSBA n+1. (2) LSBA n+1 has n+1 “natural” copies of LSBA n in it, each generated by one of n+1 ⎛ ⎞ subsets of { x 1, … , xn+1} of size n. Likewise n + 1 ⎝⎜ n −1⎠⎟ natural copies of LSBA n–1 lie in LSBA n+1, etc. This leads us to infinite free algebras with necessarily infinite generating sets. If X is infinite, then LSBA X is the upward directed union of its finite free subalgebras: LSBA X = ∪{LSBA Y⎮∅ ≠ Y ⊆ X & | Y| < ∞}. Given u and v of LSBA X, each occurs in some finite free subalgebra, say u in LSBA Y and v in LSBA Z for finite subsets Y and Z of X. Thus u∧ v, u∨ v and u\ v are calculated in the larger finite subalgebra LSBA Y∪ Z or in any finite LSBA W where Y∪ Z ⊆ W. Of course calculations of u∧ v, u∨ v and u\ v do not change in passing from LSBA Y∪ Z to any properly larger LSBA W. What changes is their atomic decompositions; such changes, however, are derived from the original decompositions in LSBA Y∪ Z by (possibly repeated) atomic splitting. Ultimately in LSBA X for X infinite, no atoms exist. (If a is an atom, then it appears as such in LSBA Y for some finite Y; but it immediately looses its atomic status in a properly larger free subalgebra.) Atoms are only relevant in its finite subalgebras. This is a fundamental difference between finite and infinite free algebras. Another fundamental difference is as follows. Recall that the center of a skew lattice, consisting of elements that both ∧-commute and ∨-commute with all elements, is the union of all singleton D-classes. In LSBA n (or RSBA n or SBA n) it is the set of all n atoms of the form x 1\( x 2∨…∨ xn) and the subalgebra they generate consisting of all orthosums of such atoms. But, except for 0, none of these orthosums remain central in LSBA n+1. For each atom, xn+1∧( x 1\( x 2∨…∨ xn)) = (( xn+1∧ x 1)\( x 2∨…∨ xn)) ≠ (( x 1∧ xn+1)\( x 2∨…∨ xn)) = ( x 1\( x 2∨…∨ xn))∧ xn+1 146 IV: Skew Boolean Algebras with the two new, unequal atoms being D-related in LSBA n+1. Thus, given any non-0 central element c = a 1 + … + ak in LSBA n with atoms ai of the given form, xn+1∧ c = ( xn+1∧ a 1) + … + ( xn+1∧ ak) ≠ ( a 1∧ xn+1) + … + ( ak∧ xn+1) = c∧ xn+1. The case for RSBA n and RSBA n+1, or SBA n and SBA n+1, is similar. We thus have: Theorem 4.4.16. Given a finite free skew Boolean algebra on n generators, whether left-handed, right-handed or two sided, its center forms a Boolean algebra of order 2 n. In the case of an infinite free algebra, the center is just {0}. ! We turn to a common property of all free algebras. Trivially, finitely generated free algebras have intersections since they are finite. It turns out that infinitely generated free algebras also have finite intersections. Thanks to our observations on adjoining free generators and their effects on atoms, intersections are stable under the inclusion LSBA n ⊂ LSBA n+1. Thus ∩ for pairs of elements in LSBA3 remains the same for these elements in the bigger, say LSBA7. What changes is the decomposition of all outcomes into atoms. The pool of atoms that two elements share in LSBA n, doubles by splitting to give rise to the new pool of atoms in LSBA n+1 that both share. As a result: Theorem 4.4.17 Given any set X, the free left-handed [right-handed] skew Boolean algebra LSBA X [RSBA X] on X has intersections. Given elements x and y in LSBA X, x∩ y can be calculated in any subalgebra LSBA Y, where Y is any finite subset of X such that LSBA Y contains both x and y. Similar remarks hold for RSBA X. Proof. Suppose that x and y are encountered in LSBA Y where Y is a finite subset of X and that u in LSBA X is such that u ≤ both x and y. Then x∩ y relative to LSBA Y exists. By our remarks, this x∩ y remains the intersection in any LSBA Z where Y ⊂ Z if Z is finite. Now u must be encountered in some finite subalgebra LSBA U where U ⊆ X. Then both the current x∩ y and u must lie in the larger subalgebra LSBA Y∪ U. Since Y∪ U is finite, x∩ y is the intersection here also, and u ≤ x∩ y follows. Thus x∩ y remains the intersection of x and y throughout all LSBA X. The case for RSBA X is similar. ! Since RSBA X ≅ SBA X/L and LSBA X ≅ SBA X/R, by Theorem 4.4.14 we have: Theorem 4.4.18. Free skew Boolean algebras have finite intersections. (Indeed, they have arbitrary intersections since a properly infinite subset must have 0-intersection.) ! 147 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond The lattice of subvarieties of skew Boolean ∩ -algebras In what follows we observe the following notation. For all n ≤ ℵ , nL [respectively nR] 0 denotes the left-[right-]handed primitive skew Boolean ∩-algebra on n = {0, 1, 2, ... , n – 1} and ℵ = {0, 1, 2, . . . } with 0 the zero element. Given two primitive algebras A and B, we denote 0 their fibred product A ×2 B by A•B. Finally, given any skew Boolean ∩-algebra A, 〈A〉∩ denotes the principal subvariety of all skew Boolean ∩-algebras generated by A in that they satisfy all identities satisfied by A. Consider the following lattice of primitive subalgebras of the primitive algebra ℵ ℵ 0L • viewed as a skew Boolean 0R ∩-algebra ℵ ℵ 0L • 0R + , ↗ ↖ ℵ ℵ 0L • 3R 3L• 0R ↗ ↖ ℵ ℵ 0L 0R , + , + , + ↖ ↗ ↖ ↗ ↖ ↗ 4L 3L•3R 4R ↖ ↗ ↖ ↗ 3L 3R ↖ ↗ 2 ↑ 1 In this diagram, each n L is identified with the trivial fibered product, n L•2, and each nR is identified with the trivial fibered product, 2• n R. The embeddings → are induced from the standard chain of inclusions: {0} ⊂ {0, 1} ⊂ {0, 1, 2} ⊂ {0, 1, 2, 3} ⊂ … . Proposition 4.4.19 The map A → 〈A〉∩ applied to the above diagram of inclusions induces a corresponding diagram of strict inclusions of the respective varieties, with i) 〈ℵ ℵ ℵ 0L • nR〉∩ = ∪{〈 mL• nR〉∩ ⎪ m < } for all n < . 0 0 ii) 〈 mL •ℵ ℵ ℵ 0R 〉∩ = ∪{〈 mL• nR〉∩ ⎪ n < } for all m < . 0 0 iii) 〈ℵ 0L•ℵ ℵ 0R 〉∩ = ∪{〈 mL• nR〉∩ ⎪ m, n < }. 0 148 IV: Skew Boolean Algebras Proof. Clearly we have a diagram of inclusions. Since any equation in the operations of the signature contains only finitely many variables, the final three assertions are also clear. Thus we need only show the inclusions of the induced subvarieties to be proper in the case of the finite primitive algebras. We first show 〈 nL〉∩ ⊂ 〈 n+1L〉∩ to be proper for all finite n. To begin, x ≈ y holds on 〈1〉∩ but not on 〈2〉∩. Next, for n ≥ 2 we set Φ n( x 1, x 2, . . . , xn) = ( x 1 \ ( x 1∩ x 2)) ∧ ( x 2 \ ( x 2∩ x 3)) ∧ . . . ∧ ( xn \ ( xn∩ x 1)) and Ψ n( x 1, x 2, . . . , xn) = x 1 \ [( x 1∩ x 2) ∨ ... ∨ ( x 1∩ xn) ∨ ( x 2∩ x 3) ∨ ... ∨ ( x 2∩ xn) ∨ … ∨ ( xn–1∩ xn)]. Since Φ n( a 1, a 2, . . . , an) ≠ 0 only if none of a 1, a 2, . . . , an is 0. Φ n( x 1, x 2, . . . , xn) ∧ Ψ n( x 1, x 2, . . . , xn) ≈ 0 holds on nL but not on n+1L (respectively, on nR but not on n+1R) for all n ≥ 2. Thus all inclusions at least along the two lower sides of the above diagram are proper. But this forces all links in the above diagram to be proper. For instance, ∩ ∩ 〈 mL• nR〉 ⊂ 〈 m+1L• nR〉 for m < ℵ and 0 n ≤ ℵ is indeed proper since Φ m( x 1∧ y, x 2∧ y, . . . , xm∧ y) 0 ∧ Ψ m( x 1∧ y, x 2∧ y, . . . , xm∧ y) ≈ 0 must hold in 〈 mL• nR〉∩ but not in 〈 m+1L• nR〉∩. £ This ascending array of principal varieties leads us to several results, beginning with: Theorem 4.4.20. Skew Boolean ∩ -algebras are locally finite. Proof. We revise the argument from Theorem 4.1.10. Given a skew Boolean ∩ - algebra S generated from a finite set X of size n, if ϕ: S → P is a nontrivial homomorphism from S to a primitive algebra, then ϕ[S] is a primitive subalgebra Pʹ of P that is isomorphic to a subalgebra of n+1L• n+1R. It follows that a homomorphism of ϕʹ: S → n+1L• n+1R exists inducing the same congruence on S that ϕ has. Moreover, only finitely many distinct homomorphisms from S to n+1L• n+1R are possible since S is generated from X. By the argument of Theorem 4.1.10, S can be embedded in a finite power of n+1L• n+1R making S itself finite. £ Theorem 4.4.21. A (quasi-)identity of signature (∨, ∧, \, ∩, 0) in n variables holds for all skew Boolean ∩ -algebras if and only if it holds in n+1L• n+1R. Likewise the (quasi-)identity holds for all left-handed (right-handed) skew Boolean ∩ -algebras if and only if it holds in n+1L (or in n+1R ). The question of when a given (quasi-)identity holds for all (left-handed or right handed) skew Boolean ∩ -algebras is thus decidable. £ Proof. If a (quasi-)identity in variables x 1, … , xn holds for all skew Boolean ∩ - algebras, in particular it holds for n+1L• n+1R. Conversely if it holds for n+1L• n+1R, it holds for all powers of n+1L• n+1. Thus, given skew Boolean ∩ - algebra S with a 1, a 2, …, an ∈ S, the latter collectively generate a subalgebra Sʹ of S that (by previous arguments) can be embedded in some 149 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond finite power of n+1L• n+1R. Since the given (quasi-)identity holds for all powers of n+1L• n+1R, it holds for Sʹ and in particular for the assignments, x 1 to a 1, x 2 to a 2, … , and xn to an. Since S and the ai are arbitrary, the theorem follows in the two-sided case. The left-handed and right-handed cases are similar. £ This above ascending array of principal varieties is only part of the full lattice of all varieties of skew Boolean ∩-algebras. Given a partially ordered set P = ( P, ≥), an order ideal of P is any nonempty subset I of P satisfying the following implication: x ∈ I and x ≥ y in P implies y ∈ I also. The following result of Brian Davey is relevant. (See Theorems 3.3 and 3.5 in [Davey 79].) Theorem 4.4.22. [Davey 79] Let V be a locally finite, congruence distributive variety. Then its lattice of subvarieties of V is completely distributive and is isomorphic to the lattice of order ideals of its partially ordered set of principal subvarieties generated by finite, subdirectly irreducible algebras and ordered by subvariety inclusion. £ Theorem 4.4.23. The lattice of subvarieties of skew Boolean ∩- algebras is isomorphic to the set of all order ideals of the following lattice of principal subvarieties generated by the finite primitive algebras. , + , + , 〈4L•3R〉∩ 〈3L•4R〉∩ + ↖ ↗ ↖ ↗ ↖ ↗ 〈4L〉∩ 〈3L•3R〉∩ 〈4R〉∩ ↖ ↗ ↖ ↗ 〈3L〉∩ 〈3R〉∩ ↖ ↗ 〈2〉∩ ↑ 〈1〉∩ The varieties of skew Boolean ∩- algebras, left- and right-handed skew Boolean ∩- algebras, and generalized Boolean algebras correspond to the order ideals given respectively by the entire lattice, the infinite ideal on the lower left {〈1〉∩ , 〈2〉∩ , 〈3L〉∩ , 〈4L〉∩ , …}, the corresponding infinite ideal on the lower right, and the ideal {〈1〉∩ , 〈2〉∩}. Proof. This follows from Proposition 4.4.19 and Theorems 4.4.20 and 4.4.22. £ The bottom of the lattice of subvarieties is as follows. We will look closely at a class of algebras in 〈3L〉∩ in the next section. 150 IV: Skew Boolean Algebras 〈3L•3R〉∩ ↑ 〈3L, 3R〉∩ ↖ ↗ ↖ ↗ 〈3L〉∩ 〈3R〉∩ ↖ ↗ 〈2〉∩ ↑ 〈1〉∩ The sublattice of nontrivial varieties containing at least 〈2〉, is described in Cartesian fashion with abbreviated notation in the following figure to the left. Here m, n stands for ( m+1)L•( n+1)R with m and n counting the size of both non-0 D-classes. One has m,n ≥ p,q when both m ≥ p and n ≥ q, i.e., p,q lies non-strictly to the lower left of m,n.        5 5,1 5, 2 5, 3 5, 4 5, 5  • • • • 4 4,1 4, 2 4, 3 4, 4 4, 5  • • • • • 3 3,1 3, 2 3, 3 3, 4 3, 5  • • • • • 2 2,1 2, 2 2, 3 2, 4 2, 4  • • • • • • • 1 1,1 1, 2 1, 3 1, 4 1, 5  • • • • • • • • 1 2 3 4 5  • • • • • • • • • Within the quadrant, an ideal corresponds to a non-increasing array of the terraced form to the right and as such is described by a non-strictly decreasing function f from {1, 2, 3, …} to {0, 1, 2, 3, … , ℵ0}. The function f for the above array is thus n 1 2 3 4 5 6 7 8 9 10  f = f ( n) 7 7 5 5 5 3 3 2 1 0  corresponding to the variety 〈8L •2R〉∩ ∪ 〈6L•4R〉∩ ∪ 〈4L•7R〉∩ ∪ 〈3L•9R〉∩ ∪ 〈2L•10R〉∩. Since only finitely many strict decreases in the output are possible, the number of such functions [ideals] is countably infinite. Even the trivial ideal 〈1〉∩ can be represented as the zero function: z( n) = 0 for all n. The lattice operations are evaluated in point-wise fashion: ( f∨ g)( n) = max{ f( n), g( n)} and ( f∧ g)( n) = min{ f( n), g( n)}. We thus have: Theorem 4.4.24 The lattice of varieties of skew Boolean algebras with intersections is isomorphic to the lattice of non-strictly decreasing functions from the set {1, 2, 3,…} to the set {0, 1, 2, 3, … ℵ0} with the join and meet operations given pointwise. In either the right or left-handed cases, the lattice of varieties is isomorphic to the usual ordering on {1, 2, …, ℵ0}. £ 151 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond 4.5 Omega algebras and skew Boolean covers Given a lattice L = ( L; ∨, ∧), upon setting ω(L) = {( a, b) ∈ L × L⎪ a ≥ b} we obtain a sublattice of L × L that clearly is distributive when L is. But what if L is a (possibly generalized) Boolean lattice with zero 0, so that for each a ∈ L, the principal poset ideal ⎡ a⎤ = { b ∈ L⎮ a ≥ b} is a Boolean lattice? Proposition 4.5.1. Given a lattice L with minimal element 0, ω(L) forms a generalized Boolean lattice if and only if L , and hence ω(L), is trivial. Proof. If a > 0 for some a ∈ L, then ( a, 0) has no complement in ⎡( a, a)⎤. £ Given a generalized Boolean algebra B, alternative twisted meet and join operations can be defined on the underlying set of ω(B) to give a Boolean skew lattice as follows: ( a, aʹ) ∧ ( b, bʹ) = ( a∧ b, aʹ∧ b) and ( a, aʹ) ∨ ( b, bʹ) = ( a∨ b, ( aʹ \ b)∨ bʹ). Here all component operations are in B with a \ b being the relative complement of a∧ b in ⎡ a⎤. Lemma 4.5.2. Under the operations above, ω(B) is left-handed skew lattice with a zero. Proof. That ∧ is idempotent and associative is easily checked, as is the fact that ∨ is idempotent. Associativity of ∨ reduces to comparing the second coordinates of [( a, aʹ) ∨ ( b, bʹ)] ∨ ( c, cʹ) and ( a, a′) ∨ [( b, bʹ) ∨ ( c, cʹ)] to see if {[( aʹ \ b) ∨ bʹ] \ c} ∨ cʹ = [ aʹ \ ( b ∨ c)] ∨ ( bʹ \ c) ∨ cʹ. But [( aʹ \ b) ∨ bʹ] \ c = [ a ʹ \ ( b∨ c)] ∨ ( bʹ \ c) holds for generalized Boolean algebras. Next, expanding gives: ( a, aʹ) ∧ [( a, aʹ)∨( b, bʹ)] = ( a, aʹ) ∧ ( a∨ b, ( aʹ \ b) ∨ bʹ) = ( a∧( a ∨ b), a ʹ∧ ( a∨ b)) = ( a, aʹ) and ( a, aʹ) ∨ [( a, aʹ)∧( b, bʹ)] = ( a, aʹ) ∨ ( a∧ b, aʹ∧ b) = ( a ∨ ( a∧ b), [ aʹ \ ( a∧ b)] ∨ ( aʹ∧ b)) = ( a, aʹ). In similar fashion, the remaining pair of absorption identities are verified. Clearly (0, 0) is the zero element . Finally, ( a, aʹ) ∧ ( b, bʹ) ∧ ( a, aʹ) = ( a, aʹ) ∧ ( b, bʹ) since both reduce to ( a∧ b, aʹ∧ b). Thus the identity x∧ y∧ x = x∧ y holds, as well as its equivalent dual, x∨ y∨ x = y∨ x. o Lemma 4.5.3. (ω(B); ∨, ∧) is a symmetric skew lattice. If ( a, aʹ) and ( b, bʹ) commute, then ( a, aʹ) ∧ ( b, bʹ) = ( a∧ b, aʹ∧ bʹ) and ( a, aʹ) ∨ ( b, bʹ) = ( a∨ b, aʹ∨ bʹ). Proof. Given ( aʹ, a), ( bʹ, b) ∈ ω(B), ( a, aʹ) ∧ ( b, bʹ) = ( b, bʹ) ∧ ( a, aʹ) if and only if aʹ∧ b = a∧ bʹ in the second coordinate of the outcomes, with both equaling aʹ∧ bʹ. Likewise, ( a, aʹ) ∨ ( b, bʹ) = ( b, bʹ) ∨ ( a, aʹ) if and only if ( aʹ \ b) ∨ bʹ = ( bʹ \ a) ∨ aʹ in the second coordinate of the outcomes with both equaling aʹ∨ bʹ. Assuming aʹ∧ b = a∧ bʹ we get 152 IV: Skew Boolean Algebras ( aʹ \ b) ∨ bʹ = ( aʹ \ ( aʹ∧ b)) ∨ bʹ = ( aʹ \ a) ∨ ( aʹ \ bʹ)∨ bʹ = ( aʹ \ bʹ)∨ bʹ = aʹ∨ bʹ. Similarly ( bʹ \ a) ∨ aʹ = aʹ∨ bʹ so that ( aʹ \ b) ∨ bʹ = ( bʹ \ a) ∨ aʹ follows. Conversely given the latter, one has aʹ∧[( aʹ \ b) ∨ bʹ]∧ b = aʹ∧[( bʹ \ a) ∨ aʹ]∧ b which yields aʹ∧ bʹ = aʹ∧ b. Similarly, a∧[( aʹ \ b) ∨ bʹ]∧ bʹ = a∧[( bʹ \ a) ∨ aʹ]∧ bʹ yields a∧ bʹ = aʹ∧ b. £ Lemma 4.5.4. In ω(B): i) ( a, aʹ) ≻ ( b, bʹ) if and only if a ≥ b in B. ii) ( a, aʹ) D ( b, bʹ) if and only if a = b in B. iii) ( a, aʹ) ≥ ( b, bʹ) if and only if a ≥ b in B with bʹ = aʹ ∧ b. Proof. Since ( b, bʹ) ∧ ( a, aʹ) ∧ ( b, bʹ) = ( a∧ b, bʹ∧ a∧ b), we have ( b, bʹ) ∧ ( a, aʹ) ∧ ( b, bʹ) = ( b, bʹ) if and only if a∧ b = b in B. Thus (i) holds and (ii) follows immediately. As for (iii), ( a, aʹ) ∧ ( b, bʹ) = ( b, bʹ) = ( b, bʹ) ∧ ( a, aʹ) becomes aʹ∧ b = bʹ = bʹ∧ a in the second coordinate, and b = a∧ b in the first coordinate. But these conditions, plus b ≥ bʹ are equivalent to both a ≥ b and bʹ = aʹ∧ b holding in B. £ Theorem 4.5.5. Upon letting ( a, aʹ) \ ( b, bʹ) be the complement of ( a, aʹ) ∧ ( b, bʹ) in the Boolean sublattice ⎡( a, aʹ)⎤, (ω(B); ∨, ∧, \, (0, 0)) becomes a skew Boolean algebra for which ω(B)/D ≅ B . Moreover, ω(B) has finite intersections. Thus, as a skew Boolean ∩ -algebra ω(B) has a distributive congruence lattice. Proof. ω(B) is symmetric by Lemma 4.5.3. Since ⎡( a, aʹ)⎤ = {( b, aʹ∧ b)⎮ a ≥ b} ≅ ⎡ a⎤ by Lemma 4.5.4, we see that ⎡( a, aʹ)⎤ is a Boolean lattice, making ω(B) also normal and hence a skew Boolean lattice. The first assertion is now clear. Next, ( a, aʹ) ∩ ( b, bʹ) is given by ([ aʹ∧ bʹ] + [( a∧ b) \ ( aʹ∨ bʹ)], aʹ∧ bʹ) where + is the symmetric difference in B: x + y = ( x\ y) ∨ ( y\ x). That ( a, aʹ), ( b, bʹ) ≥ ([ aʹ∧ bʹ] + [( a∧ b) \ ( aʹ∨ bʹ)], aʹ∧ bʹ) follows from Lemma 4.5.4 and basic Boolean algebra. So let ( a, aʹ), ( b, bʹ) ≥ ( c, cʹ). Then a∧ b ≥ c and aʹ∧ bʹ ≥ cʹ with aʹ∧ c = cʹ and bʹ∧ c = cʹ imply c ∧ {[ aʹ∧ bʹ] + [ a∧ b \ ( aʹ∨ bʹ)]} = cʹ + ( c \ cʹ) = c and c ∧ [ aʹ∧ bʹ] = cʹ∧ cʹ = cʹ so that ([ aʹ∧ bʹ] + [ a∧ b \ ( aʹ∨ bʹ)], aʹ∧ bʹ) ≥ ( c, cʹ). Thus ( a, aʹ) ∩ ( b, bʹ) is as stated . The final assertion follows from Theorem 4.4.3 . £ 153 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Algebras of the form ω(B) with operations as defined above are called omega algebras. These algebras lie in the varieties of left-handed skew Boolean algebras and left-handed skew Boolean ∩-algebras, with the particular emphasis indicated by the context. Characterizing omega algebras Given a skew lattice S, recall that a lattice section of S is any sublattice S0 that intersects every D-class of S. Any such lattice must be isomorphic to the maximal lattice image S/D of S. Lemma 4.5.6. Given an omega algebra ω(B), S0 = {( a, a)⎪ a ∈ B} is a lattice section. £ Lemma 4.5.7. Given ( a, aʹ) ∈ ω(B), the map ( a, bʹ) → ( a, aʹ) ∩ ( a, bʹ) is a bijection of the D -class of ( a, aʹ) with the principal ideal ⎡( a, aʹ)⎤ of (ω(B), ≥). Proof. In ω(B) the ideal ⎡( a, aʹ)⎤ is just the set {( b, aʹ∧ b)⎪ a ≥ b}. Given aʹ ≤ a, then for all x ≤ a, ( a, aʹ) ∩ ( a, x) = ([ aʹ∧ x] + [ a \ ( aʹ∨ x)], aʹ∧ x). Given any b ≤ a, we seek some x ≤ a such that ([ aʹ∧ x] + [ a \ ( aʹ∨ x)], aʹ∧ x) = ( b, aʹ∧ b) in ⎡( a, aʹ)⎤. Setting x = aʹ∧ b ∨ [ a \ ( aʹ∨ b)] we get, aʹ∧ x = ( aʹ∧ b) ∨ 0 = aʹ∧ b and thus [ aʹ∧ x] + [ a \ ( aʹ∨ x)] = aʹ∧ b + [ a \ ( aʹ ∨ [ a \ ( aʹ∨ b)])] = aʹ∧ b + [ a \ ( aʹ ∨ [( a \ aʹ) ∧ ( a \ b)])] = aʹ∧ b + [( a \ aʹ ) ∧ ( aʹ ∨ b)] = b. Is x ≤ a unique? Suppose that y ≤ a also satisfies the desired conditions. Then aʹ∧ x = aʹ∧ b = aʹ∧ y and [ aʹ∧ x] + [ a \ ( aʹ∨ x)] = b = [ aʹ∧ y] + [ a \ ( aʹ∨ y)], from which also follows first a \ ( aʹ∨ x) = a \ ( aʹ∨ y) and then aʹ∨ x = aʹ∨ y. Cancelling aʹ∧ x = aʹ∧ y and aʹ∨ x = aʹ∨ y in B gives x = y. £ We thus have the following characterization of omega algebras: Theorem 4.5.8. A left-handed skew Boolean ∩ -algebra S is isomorphic to ω(B) for some generalized Boolean algebra B if and only if S has a lattice section S0 and for all e ∈ S the map a → e ∩ a is a bijection of the D -class D e with the principal poset ideal ⎡ e⎤ . Under these conditions S is isomorphic to both ω(S0) and ω(S/D). Proof. The conditions are clearly necessary. Conversely, given these conditions, for each e ∈ S0 let β e: D e → ω(S0) be given by β e( f) = ( e, e∩ f). Next we define β: S → ω(S0) to be the union ∪ eβ e. β is at least a bijection from S to ω(S0). It also preserves ∧. Given e 1 and e 2 in S0, then for all f 1 D e 1 and f 2 D e 2 in S we have β[ f 1∧ f 2] = ( e 1∧ e 2, e 1∧ e 2 ∩ f 1∧ f 2) = ( e 1∧ e 2, e 1∧ e 2 ∩ f 1∧ f 2∧ e 2) = ( e 1∧ e 2, e 1∧ e 2 ∩ f 1∧ e 2) and β[ f 1]∧β[ f 2] = ( e 1, e 1 ∩ f 1) ∧ ( e 2, e 2 ∩ f 2) = ( e 1∧ e 2, [ e 1 ∩ f 1] ∧ e 2) = ( e 1∧ e 2, e 1 ∩ f 1 ∩ e 2). 154 IV: Skew Boolean Algebras Since ∧ is a left normal operation, e 1∧ e 2 ∩ ( f 1∧ e 2) = e 1 ∩ e 2 ∩ ( f 1∧ e 2) = e 1 ∩ f 1 ∩ e 2. Indeed we have e 1 ∩ e 2 ∩ ( f 1∧ e 2) ≤ e 1 ∩ e 2 ∩ f 1 = e 1 ∩ f 1 ∩ e 2. But on the other hand, f 1 ∩ e 2 ≤ f 1∧ e 2 so that e 1 ∩ f 1 ∩ e 2 ≤ e 1 ∩ ( f 1∧ e 2) ∩ e 2 = e 1 ∩ e 2 ∩ ( f 1∧ e 2). β also preserves ∨. Given e 1 and e 2 in S0, then β[ f 1∨ f 2] = ( e 1∨ e 2, ( e 1∨ e 2) ∩ ( f 1∨ f 2)) and β[ f 1]∨β[ f 2] = ( e 1, e 1 ∩ f 1) ∨ ( e 2, e 2 ∩ f 2) = ( e 1∨ e 2, [( e 1 ∩ f 1) \ e 2] ∨ [ e 2 ∩ f 2]) for all f 1 D e 1 and f 2 D e 2 in S. It remains to show that ( e 1∨ e 2) ∩ ( f 1∨ f 2) = (( e 1 ∩ f 1) \ e 2) ∨ ( e 2 ∩ f 2) is an identity for left-handed skew Boolean ∩-algebras, subject to the conditions f 1 D e 1 and f 2 D e 2. This is indeed the case for left-handed primitive skew Boolean ∩-algebras. To see this, first assume e 1 = f 1 = 0. Here the equation reduces to the identity e 2 ∩ f 2 = 0 ∨ ( e 2 ∩ f 2). Likewise, in the case e 2 = f 2 = 0 the equation reduces to the identity e 1 ∩ f 1 = (( e 1 ∩ f 1) \ 0) ∨ 0. Otherwise all four elements lie in the unique nonzero class and we get e 2 ∩ f 2 = 0 ∨ ( e 2 ∩ f 2). Thus the conditional equality holds on primitive algebras. But this explicit D-condition can be removed upon replacing e 1, f 1, e 2 and f 2 by e 1∧ f 1, f 1∧ e 1, e 2∧ f 2 and f 2∧ e 2 respectively. Hence the above conditional identity holds for all left-handed skew Boolean ∩-algebras. We have shown that β also preserves joins and hence β: S ≅ ω(S0). £ Example 4.5.1. In particular, a primitive skew Boolean algebra S is isomorphic to an omega algebra if and only if it is left-handed of order 3 . In this case we are looking at a copy of 3L, which is what ω(2) is for the primitive Boolean algebra 2: 1 > 0. 1 – 2 (1,1) – (1, 0) 3L: . ω(2): . Theorem 4.5.9. ω(B) is complete if and only if B is complete. It is both complete and atomic if and only if B is thus, in which case ω(B) ≅ 3L | A(B)| where A(B) is the set of atoms of B . Thus a complete, atomic skew Boolean algebra is isomorphic to some omega algebra if and only if it is left-handed and each of its atomic D -classes has exactly two elements. Proof. If B is complete and A = {( ai, aʹ i) ⎢ i ∈ I} is a pairwise commuting subset of ω(B), we claim that sup(A) = ( supaj, supaʹ j) with both suprema taken in B. By Lemma 4.5.4, if ( b, bʹ) ≥ ( ai, aʹ i), then b ≥ supaj and bʹ ≥ supaʹ j. Thus, we need only show ( supaj, supaʹ j) ≥ each ( ai, aʹ i) in ω(B). Lemma 4.5.3 and the definition of ∧ in ω(B) give ai ∧ supj aʹ j = supj ( ai ∧ aʹ j) = supj ( aʹ i ∧ aʹ j) = aʹ i which is what Lemma 4.5.4 requires for ( supaj, supaʹ j) ≥ ( ai, aʹ i). Since ω(B) is normal with 0 and has commuting suprema, all nonempty subsets of ω(B) also have infima. 155 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond In general, for any B the atoms of ω(B) have either the form ( x, 0) or the form ( x, x) where x is an atom of B. When B is both complete and atomic, then for all ( a, aʹ) ∈ω(B), ( a, aʹ) = sup[{( x, x) ⎢ x ∈ A(B), x ≤ aʹ} ∪ {( x, 0) ⎢ x ∈ A(B), x ≤ a \ aʹ}] . Thus ω(B) is also atomic in which case ω(B) is isomorphic to a product of primitive algebras, all of which are copies of 3L since each atomic D-class of ω(B) has exactly two elements . The converse holds in general: if a skew Boolean algebra S is both complete and atomic, then so is its maximal (generalized) Boolean algebra image, S/D. £ Corollary 4.5.10. Every left-handed skew Boolean algebra S can be embedded as a skew Boolean algebra into some omega algebra. Proof. By Corollary 4.1.7, every left-handed skew Boolean algebra can be embedded in a power of 3L which in turn is isomorphic to the omega algebra of the same power of 2. o The ∩-version of Corollary 4.5.10 fails since, as seen in Section 4.4, as an ∩-algebra 3L does not generate the variety 〈ℵ0L 〉∩ of all left-handed skew Boolean ∩-algebras. But we have: Theorem 4.5.11. As algebras with ∩, all omega algebras lie in 〈3L〉∩. A skew Boolean ∩- algebra S can be embedded in some omega algebra if and only if it lies in 〈3L〉∩ . Proof. In the variety 〈ℵ0L 〉∩, primitive images of omega algebras are isomorphic to omega algebras (by Lemma 4.5.13 below) and thus are copies of 3L. Hence nontrivial omega algebras are subdirect products of copies of 3L and must lie in 〈3L〉∩. In general a skew Boolean ∩-algebra lies in 〈3L〉∩ if and only if it can be embedded as an ∩-algebra in a power of 3L. But such powers are copies of omega algebras by Theorem 4.5.9. £ Skew Boolean covers of generalized Boolean algebras Recall that the center of a skew lattice S is the set ZS = { a ∈ S⎮ a∧ x = x∧ a for all x ∈ S} or equivalently, the set { a ∈ S⎮ a∨ x = x∨ a for all b ∈ S}. ZS is also the union of all singleton D-classes of S. (See Theorem 2.2.2.) For skew lattices ZS can be empty, but for a skew Boolean algebra S, at least {0} ⊆ ZS. If ZS = {0}, then S has a trivial center. The center of a skew Boolean algebra S is always an ideal of S. A skew Boolean cover of a generalized Boolean algebra B is any skew Boolean algebra S with trivial center such that S/D ≅ B. In this case, S is a minimal cover if the center ZS/ of S/θ is nontrivial for all nontrivial congruences θ ⊆ D. θ Theorem 4.5.12. For any nontrivial generalized Boolean algebra B, the derived omega algebra ω(B) forms a minimal, skew Boolean cover of B. 156 IV: Skew Boolean Algebras Proof. If a > 0 in B, then ( a, a) D ( a, 0) but ( a, a) ≠ ( a, 0) in ω(B). So let a nontrivial congruenec θ ⊆ D be given and suppose that ( a, aʹ) θ ( a, aʺ) with a ≥ aʹ, aʺ in B, but aʹ ≠ aʺ. Applying ∧ ( b, b) we get ( a∧ b, aʹ∧ b) θ ( a∧ b, aʺ∧ b) for all b ∈ B. Since aʹ ≠ aʺ, one of these does not equal aʹ ∧ aʺ in B, say aʹ. Setting b = aʹ we get ( aʹ, aʹ) θ ( aʹ, aʺ∧ aʹ) with aʹ > aʺ∧ aʹ. Thus we may assume that ( a, a) θ ( a, aʹ) with a > aʹ in B from the outset. Next, setting b = a \ aʹ, we get that a ≥ b > 0. Then ( a∧ b, a∧ b) θ ( a∧ b, aʹ∧ b), that is, ( b, b) θ ( b, 0). We show that the θ-class of ( b, b) is in fact its entire D-class. Since b > 0, this forces ω(B)/θ to have a singleton D-class other than {0}, making its center nontrivial. Applying ∨ ( c, c) to ( b, b) θ ( b, 0) for any c ≤ b, we get ( b, b) ∨ ( c, c) = ( b∨ c, ( b \ c) ∨ c) = ( b, b) and ( b, 0) ∨ ( c, c) = ( b∨ c, (0 \ c) ∨ c) = ( b, c). Thus ( b, b) θ ( b, c) for all c ≤ b, so that the θ-class of ( b, b) is precisely its D-class in ω(B). £ Minimal skew Boolean covers can be created without the ω-construction. The latter is an elegant way of achieving this since (1) it gives us minimal covers and (2) it provides a systematic way of carrying this out. While ω(B)/θ has a nontrivial center for every nontrivial congruence θ ⊆ D, a second class of congruences exists for which the quotient algebra always has a trivial center, namely the class of ∩- congruences β that also preserve intersections in that a β b implies a∩ c β b∩ c, in which case S/β also has finite intersections. As we have seen, such congruences arise from ideals of S. Given an ideal I, an ∩-congruence β I is defined by: a β I b if ( a \ ( a ∩ b)) ∨ ( b \ ( a ∩ b)) ∈ I. Conversely, every ∩-congruence β determines an ideal I given by the congruence class β[0] of 0. β The two assignment processes are reciprocal. (See Lemma 4.4.7 and Theorem 4.4.8.) Lemma 4.5.13. Given any ∩ -congruence β on ω(B), its image ω(B)/β is isomorphic to an omega algebra and thus has trivial center. Put otherwise ∩ -homomorphic images of omega algebras are copies of omega algebras. Proof. Let β be derived from ideal I. In turn let I 0 ≅ I/D be the ideal of B consisting of all left-coordinates of elements in I so that I = ω( I 0). If β0 is the congruence on B induced from I 0, then the map f: ω(B) → ω(B/β0) defined by f[( a, aʹ)] = (β0[ a], β0[ aʹ]) is a ∩ - homomorphism for which f–1{(0, 0)} is precisely I. Thus β is the congruence induced by f and ω(B)/β ≅ ω(B/β0). £ We can now decompose the congruence lattice of ω(B) as a subdirect product of (i) the interval [Δ, D] of all congruences (inclusively) between the trivial congruence Δ and D and (ii) the lattice Con (ω(B)) of all ∩-congruences on ω(B). Indeed Theorem 4.4.9 gives us: ∩ Theorem 4.5.14. The full lattice of congruences Con(ω(B)) is a subdirect product of the interval [Δ, D] and Con (ω(B)) under the embedding Con(ω(B)) → [Δ, D] × Con (ω(B)) given ∩ ∩ by θ → (θ∩D, βθ[0]). For all θ ∈[Δ, D] except Δ, the image ω(B)/θ has nontrivial center, while for all β ∈ Con (ω(B)) the image ω(B)/β has a trivial center. ∩ 157 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. This follows from Theorem 4.4.14 and the two previous results. £ Does being a minimal skew Boolean cover of its maximal lattice image characterize an ω-algebra? We briefly explore this question, beginning with a theorem. Its fairly straightforward proof is omitted. Theorem 4.5.15. Given an atomic skew Boolean algebra S with atomic D- classes Di for i in index set I, then S has a trivial center and hence is a skew Boolean cover of S/D if and only if | Di| ≥ 2 for all i. S is a minimal skew Boolean cover of S/D if and only if | Di| = 2 for i ∈ I. S ≅ ω(S/D) must occur whenever S is also left-handed and complete. £ Example 4.5.2. A skew Boolean algebra S that is a minimal skew Boolean cover of S/D but not isomorphic to any ω(S/D) is given by taking the subset lattice P(X) of an infinite set X and letting S be the atomic subalgebra of ω(P(X)) consisting of all pairs (A, Aʹ) for Aʹ finite. Given (A, Aʹ) for A infinite and Aʹ finite, the D-class of (A, Aʹ) has |A| many elements in S, but |P(A)| many elements in ω(P(X)). Since S and ω(P(X)) share P(X) as a maximal lattice image, and |A| < |P(A)| for all subsets of X, S cannot be isomorphic to any ω(B). The functor ω and its left adjoint In the following, 〈2〉 and 〈3L〉 denote the respective varieties of generalized Boolean algebras and left-handed skew Boolean algebras viewed as categories. The ω-construction is the object stage of a functor ω: 〈2〉 → 〈3L〉. Given homomorphism f: B → Bʹ in 〈2〉, its ω-image in 〈3L〉 is the homomorphism fω: ω(B) → ω(Bʹ) in 〈3L〉 given by fω( a, aʹ) = ( f( a), f( aʹ)). Theorem 4.5.16. ω: 〈2〉 → 〈3L〉 preserves limits. Thus it has a left adjoint, Ω: 〈3L〉 → 〈2〉 . Proof. Observe first that ω preserves products: indeed ω(∏ i ∈IB i) ≅ ∏ i ∈Iω(B i) under the map (〈 ai〉, 〈 aiʹ 〉) → 〈( ai), ( aiʹ)〉. Next, note that ω preserves equalizers: given f, g: B → Bʹ, equ( fω, gω) = ( equ( f, g))ω where equ( f, g) is the standard equalizer given by the inclusion of the maximal subalgebra of B on which f and g agree. Hence ω preserves limits. Next we show that every skew Boolean algebra S has a solution set of homomorphisms F = { fi: S → ω(B i)⎮ i ∈ ∩} such that every homomorphism f: S → ω(B) in 〈3L〉 can be written as a composite f = h ωo fi for some fi ∈ F and some homomorphism h:B i → B in 〈2〉. So given a homomorphism f: S → ω(B) of skew Boolean algebras, consider the subalgebra Bʹ of B that is determined by the subset union ∪{{ a, aʹ}⎮( a, aʹ) = f( x) ∈ ω(B) for some x ∈ S}. Clearly f[S] ⊆ ω(Bʹ). Moreover |Bʹ| ≤ max{ℵ0, |S|}. Hence a solution set for S is given by the set of all homomorphisms f: S → ω(B j) where {B j⎪ j ∈ J} is a maximal class of mutually 158 IV: Skew Boolean Algebras non-isomorphic generalized Boolean algebras of order |B j| ≤ max{ℵ0, |S/D|}. Since ω is limit-preserving and each S in 〈3L〉 has a solution set, Freyd’s Adjoint Functor Theorem implies that ω must have a left adjoint Ω. £ Thus given a skew Boolean algebra S, a homomorphism ηS: S → ω(Ω(S)) exists, called the universal morphism from S to ω, such that for any homomorphism µ: S → ω(Bʹ) in 〈3L〉, a unique homomorphism u: Ω(S) → Bʹ exists in 〈2〉 such that µ factors as uωo ηS. Due to Corollary 4.5.10, each universal morphism ηS is an embedding. Thus, given a homomorphism f: S → Sʹ, fΩ: Ω(S) → Ω(Sʹ) is the unique homomorphism in 〈2〉 making the following diagram commute. η S S ⎯ → ⎯⎯ ω(Ω(S)) Ω ω f ↓ ↓ ( f ) . η S′ S′ ⎯⎯ → ⎯ ω(Ω(S′)) What does ηS: S → Ω(S) look like for some typical left-handed skew Boolean algebras? Theorem 4.5.17. Let P be a primitive left-handed skew Boolean algebra with P\{0} = X. Then Ω(P) ≅ FBX, the generalized Boolean algebra reduct of the free Boolean algebra on X . In particular, Ω(P) ≅ 22|X| if |X| is finite. (See [Leech and Spinks, 2008] Theorem 5.2.) £ In particular, Ω(2) ≅ 22 and Ω(3L) ≅ 24. One can show that Ω(S1 × S2) ≅ Ω(S1) × Ω(S2). (See [Leech and Spinks, 2008] Theorem 5.3.) Ω(S) is thus easily determined when S/D is finite. Twisted product constructions were introduced by Kalman [1958] as a means of building De Morgan algebras from distributive lattices. The ω-construction given here is a variation of a construction due to Pagliani [1998] for producing Nelson algebras from Heyting algebras. For applications of twisted product constructions to both algebra and logic see Pagliani [1997]. Historical remarks Skew Boolean algebras in some form seem to have been studied first by Robert Bignall in his 1976 dissertation and then in a 1980 paper by his advisor, William Cornish. In 1990, a paper on skew Boolean algebras appeared among Jonathan Leech’s early papers on skew lattices. Bignall and Leech then published a joint paper on skew Boolean algebras with intersections in 1995. Much in Sections 4.1 and 4.4 appeared in these two papers. In Section 4.4, the material on the lattice of varieties appeared in the 2017 paper by Leech and Spinks. The material in Section 4.2 on finite algebras and the free case in particular appeared in the 2016 paper by Ganya Kudryatseva and Leech, as did the material in Section 4.4 on infinite free algebras. The material in Section 4.3 appeared in a 2013 paper by Karin Cvetko-Vah, Matthew Spinks and Leech, but the initial information about strongly distributive skew lattices appeared in Leech’s 1992 paper on normal skew lattices. The construction in Section 4.5 was due to Spinks and developed in a joint 159 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond project with Leech published in 2007. Further developments of that material occur in a 2013 paper by Kudryatseva, who is also publishing a paper on free skew Boolean intersection algebras, set to appear in 2017. Bignall, Cornish and Leech were not the only ones to initiate a study of noncommutative Boolean algebras in some form. As seen in Section 4.2, computer scientists (J. Berendsen et al) in studying the override and update operations introduced in 2010 a class of algebras that was shown by Cvetko-Vah, Leech and Spinks in their 2013 paper to be term equivalent to right-handed skew Boolean algebras. Indeed they are (∨, \, 0)-term reducts of the latter with a variant of ∧ obtained as a defined operation. In a 2011 paper, Janis Cirulis, studied near lattices (meet semilattices with joins existing for pairs of elements with common upper bounds) that were supplied with an override operation (reducing to the conditional commuting join when it existed). Under special assumptions Cirulis obtained a class of skew Boolean algebras with intersections, with ∩ being the near lattice meet. Skew Boolean algebras clearly form a natural class of objects to study. They play a significant role in the more general study of Boolean-like phenomena, with connections to discriminator varieties, iBCK algebras and their offspring which we meet in Chapter 7, and more recently, Church algebras. (See Spinks [2002] and Cvetko-Vah and Salibra [2015]) It is no coincidence that skew Boolean algebras have attracted the interest of some in computer science. Indeed, most of the individuals mentioned in these paragraphs have some degree of research interest in computer science. (In particular, see Bignall and Spinks [1996] – [1998] and Spinks and Veroff. [2006]) Topological representations of (generalized) Boolean algebras and distributive lattices have been studied since M. H. Stone’s papers in the 1930s. More recently this has been extended to skew Boolean algebras (possibly with intersections) and strongly distributive skew lattices. Here one is given a skew Boolean algebra S with maximal (generalized) Boolean image B = S/D. The latter is dual to a topological space B under standard Stone duality with S itself being dual to an étale covering π: X → B. (See the papers below by Bauer and Cvetko-Vah ( et al) and by Kudryatseva.) In Chapter 6 we will look at the skew Boolean algebras of idempotents in rings, and in particular, the case where the idempotents in a ring are closed under multiplication and thus naturally form a skew Boolean algebra. In Chapter 7 we will look at algebraic structures that support a skew lattice reduct, in some cases with intersections; that is, we will look at algebras where skew Boolean operations are term-defined using the given operations of the algebra. 160 IV: Skew Boolean Algebras References A. Bauer and K. Cvetko-Vah, Stone duality for skew Boolean intersection algebras, Houston Journal of Mathematics, 39 (2013), 73 – 109. A. Bauer and K. Cvetko-Vah, M. Gerkhe, G. Kudryatseva and S. van Gool, A noncommutative Priestly duality, Topology and Applications, 160 (2013), 1423 – 1438. J. Berendsen, D. Jansen, J. Schmaltz and F. Vaandrager, The axiomatization of overriding and update, J. of Applied Logic, 8 (2010), 141–150. R. J. Bignall, Quasiprimal varieties and components of universal algebras, Flinders University of South Australia Dissertation,1976. R. J. Bignall and J. Leech, Skew Boolean algebras and discriminator varieties, A. Universalis 33 (1995), 387 – 398. R. J. Bignall, and M. Spinks, Propositional skew Boolean logic, Proc. of the 26th International Symposium on Multiple-valued Logic (1996), IEEE Computer Society Press, 43 – 48. Multiple-valued logic as a programming language, Proc. of the 27th International Symposium on Multiple-valued Logic (1997), IEEE Computer Society Press, 227 – 232. Multiple-valued logics for theorem-proving in first order logic with equality, Proc. of the 28th International Symposium on Multiple-valued Logic (1998), IEEE Computer Society Press, 102-107. Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae Mathematicae Japonicae, 58 (2003), 629 – 638. Corrigendum: Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae Mathematicae Japonicae, 66 (2007), 387–390. J. Cirulis, Nearlattices with an overriding operation, Order, 28 (2011), 33 – 51. W. H. Cornish, Boolean skew algebras, Acta Math. Hungary 36 (1980), 281-291. K. Cvetko-Vah, J. Leech and M. Spinks, Skew Lattices and Binary Operations on Functions, J. Applied Logic, 11 (2013), 253- 161 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond 265. K. Cvetko-Vah and A. Salibra, The connection of skew Boolean algebras and discriminator varieties to Church algebras, Algebra Universalis, 73 (2015), 369-390. B. Davey, On the lattice of subvarieties, Houston Journal of Mathematics, 5 (1979), 183 – 192. J. A. Kalman, Lattices with involution, Trans. Amer. Math. Soc., 87 (1958), 485–491. G. Kudryavtseva, A refinement of Stone duality to skew Boolean algebras, Algebra Universalis 67 (2012), 397–416. A dualizing object approach to noncommutative Stone duality, J. Australian. Math. Soc ., 95 (2013), 383-403. Skew Boolean intersection algebras and set partitions, Order, first online. G. Kudryavtseva and J. E. Leech Free skew Boolean algebras, International J. of Algebra and Computation, 26 (2016), 1323-1348. J. E. Leech, Skew Boolean algebras, Algebra Universalis 27 (1990), 48-72. Normal skew lattices, Semigroup Forum 44 (1992), 1 – 8. J. E. Leech and M. Spinks, Skew Boolean algebras derived from generalized Boolean algebras, Algebra Universalis 58 (2008), 287–302. Varieties of skew Boolean algebras with intersections, J. Australian Math. Soc., 102 (2017), 290-306. P. Pagliani, Rough set systems and logico-algebraic structures, in E Orlowska (ed.), Incomplete Information: Rough Set Analysis, Physica-Verlag, 1997, pp. 109–190. Intrinsic co-Heyting boundaries and information incompleteness in rough set analysis, Rough Sets and Current Trends in Computing, Lecture Notes in Artificial Intelligence 1424, 1998, pp. 123–130. M. Spinks, Contributions to the Theory of Pre-BCK Algebras, Monash University Dissertation, 2002. M. Spinks and R. Veroff, Axiomatizing the skew Boolean propositional calculus, J. Automated Reasoning 37 (202006), 3 – 20. 162 V: FURTHER TOPICS IN SKEW LATTICES Skew lattices were introduced in Section 1.3 and then studied in Chapter 2. Two main sources of initial examples were encountered: i) Partial function algebras P(A, B) that form noncommutative variants of Boolean algebras defined on power sets P(A). (Example 2.6.1.) ii) Sets of idempotents in rings that are closed under multiplication and the (often called) circle operation: e ¡ f = e + f – ef. (Theorem 2.1.7.) Both types of examples satisfy the primary conditions for an algebraic system (S; ∨, ∧) to be a skew lattice: ∨ and ∧ are associative idempotent binary operations on S satisfying the absorption identities x∧( x ∨ y) = x = ( y ∨ x)∧ x and x∨( x ∧ y) = x = ( y ∧ x)∨ x that guarantee the basic dualities, x∧ y = y iff x∨ y = x and x∧ y = x iff x∨ y = y. These examples are also a source of optional conditions that any skew lattice might satisfy. In both cases commutation was symmetric in that x∧ y = y∧ x iff x∨ y = y∨ x, thus making instances of commutation unambiguous. Also in both cases the skew lattices were distributive in that both x∧( y ∨ z)∧ x = ( x∧ y∧ x) ∨ ( x∧ z∧ x) and x∨( y ∧ z)∨ x = ( x∨ y∨ x) ∧ ( x∨ z∨ x) hold. For partial function algebras a stronger pair of identities hold: x∧( y ∨ z) = ( x∧ y) ∨ ( x∧ z) and ( y ∨ z)∧ x = ( y∧ x) ∨ ( z∧ x). In Section 2.3 we saw that a chain of implications holds for all skew lattices: ∀ x, y, z ∈ S, x∧( y ∨ z) = ( x∧ y) ∨ ( x∧ z) & ( y ∨ z)∧ x = ( y∧ x) ∨ ( z∧ x) ⇒ ∀ x, y, z, w ∈ S, x∧( y ∨ z)∧ w = ( x∧ y∧ w) ∨ ( x∧ z∧ w) ⇒ ∀ x, y, z ∈ S, x∧( y ∨ z)∧ x = ( x∧ y∧ x) ∨ ( x∧ z∧ x) & x∨( y ∧ z)∨ x = ( x∨ y∨ x) ∧ ( x∨ z∨ x). In general these implications are strict, but if the skew lattice is symmetric the converse of the first implication holds; likewise if the skew lattice is normal in that x∧ y∧ z∧ w = x∧ z∧ y∧ w, the converse of the second implication holds. We also observed that x∧( y ∨ z)∧ x = ( x∧ y∧ x) ∨ ( x∧ z∧ x) and its dual are not equivalent, but given the assumption of symmetry we will prove they are. In addition, both types of algebras are categorical in that nonempty composites ψ¡ϕ of successive coset bijections from D-classes to say lower D-classes are also coset bijections. Indeed partial function algebras (and normal skew lattices in general) are strictly categorical in that all composites of coset bijections between successive D-classes are also nonempty. In this chapter we study these properties in greater detail. 163 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond In Section 5.1 we consider symmetric skew lattices where commutation is unambiguous as well as variations of this condition. Symmetry is first bisected into lower symmetry where commutation shifts “downward” ( x∨ y = y∨ x ⇒ x∧ y = y∧ x) and its dual, upper symmetry, where commutation shifts “upward” ( x∧ y = y∧ x ⇒ x∨ y = y∨ x). Characterizing identities for each are given in Theorem 5.1.1. Both types of partial symmetry are characterized in Theorem 5.1.2 by a pair of forbidden subalgebras (one right-handed and one left-handed). The absence of all four as subalgebras characterizes symmetric skew lattices. In Section 5.2 we prove the equivalence of x∧( y ∨ z)∧ x = ( x∧ y∧ x) ∨ ( x∧ z∧ x) and its dual x∨( y ∧ z)∨ x = ( x∨ y∨ x) ∧ ( x∨ z∨ x) in the presence of symmetry. In particular, Theorem 5.2.3 states that given lower symmetry, x∧( y ∨ z)∧ x = ( x∧ y∧ x) ∨ ( x∧ z∧ x) implies x∨( y ∧ z)∨ x = ( x∨ y∨ x) ∧ ( x∨ z∨ x), and dually, in the presence of upper symmetry, the converse implication holds. These results are due to Matthew Spinks, who had obtained in a very lengthy computer-generated proof, but eventually reduced the proof to one of a moderate size that was then “humanized” by Karin Cvetko-Vah. (All relevant references are given in the sections.) For lattices, not only is x∧( y ∨ z) = ( x∧ y) ∨ ( x∧ z) equivalent to x∨( y ∧ z) = ( x∨ y) ∧ ( x∨ z), both identities in turn are equivalent to a lattice not having copies of M3 or N5 as subalgebras. Another equivalent condition for lattices is this form of cancellation: x∨ z = y∨ z & x∧ z = y∧ z ⇒ x = y. For skew lattices, however, these conditions are mutually nonequivalent in the absence of other qualifying assumptions. Either a distributive identity or the cancellative implication will by itself rule out M3 or N5 occurring as subalgebras, but not conversely. A skew lattice is called cancellative if both the above implication (cancelling on the right) and its dual, x∨ y = x∨ z & x∧ y = x∧ z ⇒ y = z (cancelling on the left), hold. Cancellative skew lattices are always strongly symmetric. Since every skew chain is cancellative (Proposition 5.3.2), but not necessarily distributive (Example 5.3.3), being cancellative does not imply being distributive. Conversely the four forbidden algebras of Section 5.1 are distributive, but not cancellative. Cancellative skew lattices, and their close variants, are studied in Section 5.3. They all form varieties that are characterized by small sets of forbidden subalgebras (Theorem 5.3.8). Their coset structure also produces some interesting counting features. (See Theorems 5.3.10 and 5.3.11.) Both partial function algebras and skew lattices of idempotents in rings are cancellative. Like being symmetric, being categorical is desirable in a skew lattice as it makes cosets and their bijections well-behaved globally. Indeed, all distributive skew lattices are categorical (by Theorem 5.4.2). This condition is studied more closely in Section 5.4. Categorical skew lattices form a proper subvariety of skew lattices and in Theorem 5.4.4 we give a countable family of forbidden subalgebras that characterize this subvariety. Strictly categorical skew 164 V: Further Topics in Skew Lattices lattices are studied in the last part of the Section and Theorem 5.4.7 gives a number of equivalent conditions for a skew lattice to belong to this variety. Perhaps the most notable one is the unique midpoint condition on skew chains A > B > C in S: given a > c with a ∈A and c ∈C, a unique midpoint b ∈B exists such that a > b > c. This is equivalent to the condition: for each pair e ≤ f in a skew lattice, the interval subalgebra [ e, f] ={ x ∈S⎪ e ≤ x ≤ f} is a sublattice. As a consequence, any strictly categorical skew lattice S for which the lattice image S/D is distributive is a distributive skew lattice. (Theorem 5.4.9.) All normal skew lattices belong to this variety as do their duals, conormal skew lattices (where x∨ y∨ z∨ w = x∨ z∨ y∨ w holds). The subvariety of all skew lattices generated from these two classes of skew lattices turns out to be a proper subvariety of strictly categorical skew lattices. At the end of the section we ask if this generated subvariety is the variety of paranormal skew lattices (defined there). In general one has the following chain of subvarieties: 〈 Normal ∪ conormal〉 ⊆ Paranormal ⊂ Strictly categorical. In Section 5.5 we take a closer look at distributivity. Every distributive skew lattice S must be quasi-distributive in that every lattice image of S is distributive. Quasi-distributivity is a necessary, but not sufficient condition for distributivity, thanks to Spinks’ examples (Theorem 1.3.10). A possible complementary condition is for a skew lattice to be linearly distributive in that each subalgebra that is totally pre-ordered (by ≻) is distributive. Linearly distributive skew lattices form a variety. (See Theorems 5.5.1 and 5.5.5.) For skew lattices one has the strict chain of subvarieties: Strictly categorical ⊂ Linearly distributive ⊂ Categorical. In the strictly categorical case, quasi-distributivity suffices for the skew lattice to be distributive (Theorem 5.4.9). For linearly distributive algebras in general this does not occur (Spinks’ examples again). However: given symmetry, linearly distributive + quasi-distributive implies distributive. (See Theorem 5.5.11.) Being a vital part of distributivity, linear distributivity and distributive skew chains in particular are studied further in Section 5.6. Given a skew chain A > B > C with a > c for a ∈ A and c ∈ C, consider the set µ( a, c) = { b ∈ B⎪ a > b > c} of midpoints b of a and c in B. One has |µ( a, c)| ≥ 1, and unless A > B > C is strictly categorical, |µ( a, c)| ≥ 2 for all such pairs. In general, B decomposes into a disjoint union of AC- components B1, B2, … that induce sub-skew chains A > B i > C. (The definitions and relevant discussion occur prior to Example 5.4.3.) Each component B i contains at least one midpoint bi for a > c. In any case, A > B > C is distributive if and only if each A > B i > C is strictly categorical, and in particular, each component B i contains a unique midpoint bi for each such a > c. µ( a, c) thus parameterizes in a natural way the AC-components of B. In so doing, distributivity minimizes the number of midpoints any a > c can have in a skew chain A > B > C relative to the AC-component structure of B (Theorem 5.6.4). Here, and in the latter part of Section 5.4, the orthogonality of cosets from two D-classes (A and C) in a third D-class (B) occurs again. (Recall Lemma 2.4.8 and Theorem 2.4.9 in Section 2.4.) 165 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond In the final seventh section we present some combinatorial results about skew chains and skew diamonds. These counting theorems are a continuation of some results in Section 2.4, in particular Theorem 2.4.10 and its corollaries. Whereas the former are over 20 years old, the results in Section 7 were published in the past few years and are due primarily to Cvetko-Vah, Leech and Pita de Costa. The section concludes with some consequences for cancellative skew lattices. This is followed by a few remarks of a bibliographic nature. 5.1 Symmetric skew lattices Recall from Theorem 2.2.4 that symmetric skew lattices are characterized as skew lattices by the following identities with both sides respectively equaling x∧ y∧ x and x∨ y∨ x: x∧ y∧( x∨ y∨ x) = ( x∨ y∨ x)∧ y∧ x (5.1.1) and x∨ y∨( x∧ y∧ x) = ( x∧ y∧ x)∨ y∨ x. (5.1.2) Matthew Spinks in [Spinks 1998] observed that these identities can be trimmed to x∧ y∧( x∨ y) = ( y∨ x)∧ y∧ x, (5.1.3) and x∨ y∨( x∧ y) = ( y∧ x)∨ y∨ x (5.1.4) since x∧ y∧ x ≺ both x∧ y and x∨ y, so that x∨ y∨( x∧ y) = x∨ y∨( x∧ y∧ x)∨( x∧ y) = x∨ y∨( x∧ y∧ x) due to x∧ y∧ x R x∧ y holding in any skew lattice. The three other terms are handled similarly. Symmetry is parsed as follows. A skew lattice is lower symmetric if x∨ y = y∨ x implies x∧ y = y∧ x. Dually, a skew lattice is upper symmetric if x∧ y = y∧ x implies x∨ y = y∨ x. Theorem 5.1.1. Lower symmetry is characterized by identity (5.1.3). Dually, upper symmetry is characterized by identity (5.1.4). Hence both classes of skew lattices form varieties. A skew lattice S is thus upper [ lower] symmetric if and only if both S/L and S/R are. Proof. Indeed, if x∨ y = y∨ x, then (5.1.3) plus absorption gives x∧ y = y∧ x so that lower symmetry holds. Conversely, since x ≥ x∨ y∨ x, occurrences of x and ( x∨ y∨ x)∧ y∧( x∨ y∨ x) must ∨-commute. Hence if a skew lattice is lower symmetric, x and ( x∨ y∨ x)∧ y∧( x∨ y∨ x) also ∧-commute so that (5.1.1) and hence (5.1.3) follows. Similar remarks hold for (5.1.4) and upper symmetry. £ Consider the following pair of Hasse diagrams, each determining a right-handed skew diamond and its left-handed dual. (The dotted lines denote the natural partial order ≥.) 166 V: Further Topics in Skew Lattices j − 1 j 2 1      a − − a − a − b 1 a 2 b 1 b 2 1 2 b 1 2       0 m − 1 m 2 The induced right-handed algebras are denoted respectively by R,0 R,1 NS and . Their left- 7 NS7 handed duals by are denoted L,0 L,1 R,0 NS and . Cayley tables for are given by: 7 NS7 NS7 ∨ 0 an bn jn ∧ 0 an bn jn 0 0 an bn jn 0 0 0 0 0 am am am jm jm and am 0 an 0 an . bm bm jm bm jm bm 0 0 bn bn jm jm jm jm jm jm 0 an bn jn All four algebras can be obtained from any one by ∨-∧ duality, L-R duality or a combination of both. None is symmetric. Indeed, the tables above give a 1∧ b 2 = 0 = b 2∧ a 1 but a 1∨ b 2 ≠ b 2∨ a 1. The following theorems appeared in 2011 in a paper of Cvetko-Vah, Kinyon, Leech and Spinks. Theorem 5.1.2. A skew lattice is upper symmetric if and only if it contains no copy of R,0 L,0 R,1 L,1 NS or . It is lower symmetric if and only if it contains no copy of or . 7 NS7 NS7 NS7 Finally, it is symmetric if and only if it contains no copy of any of these skew lattices. Proof. Since neither R,0 L,0 NS nor is upper symmetric, any skew lattice containing a copy 7 NS7 of one of them cannot be upper symmetric. Conversely, consider first the case where S is a non- upper symmetric, right-handed skew lattice. Thus a, b ∈ S exist such that a∧ b = b∧ a but a∨ b ≠ b∨ a. Consider the subalgebra T generated by a and b. Since a∧ b∧ x = a∧ b = x∧ a∧ b, for x = a or b and hence for all x ∈ T, a∧ b is the zero element of T. Moreover a and b must be in incomparable R-classes for anti-symmetry to occur and thus one has the following configuration with seven distinct elements in four D-classes within a right-handed skew diamond: a ∨ b − b ∨ a     a − a ∧ ( b ∨ a) b ∧ ( a ∨ b) − b .   a ∧ b For if a = a∧( b∨ a), then b∨ a ≥ a, b forcing a∨ b = b∨ a by Theorem 2.2.1, contradicting our assumption on a and b. Thus a ≠ a∧( b∨ a) and similarly b ≠ b∧( a∨ b) giving us at least the seven 167 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond distinct displayed elements. We claim that this set is closed under ∨ and ∧. Each displayed D-class is closed. We consider representatives of the less trivial remaining cases. a∧( b∨ a) ∨ ( a∨ b) = [ a∧( b∨ a) ∨ a]∨ b = a∧( b∨ a) ∨ b = b∨ a since a∧( b∨ a) R a and both a∧( b∨ a) and b are ≤ b∨ a. Likewise, a∨[ b∧( a∨ b)] = a∨[ b∧( a∨ b)]∨( a∨ b) = a∨( a∨ b) = a∨ b = [ b∧( a∨ b)]∨ a∨ b = [ b∧( a∨ b)]∨ a and [ a∧( b∨ a)]∨[ b∧( a∨ b)] = [ a∧( b∨ a)]∨[ b∧( a∨ b)]∨ a∨ b = [ a∧( b∨ a)]∨ a∨ b = b∨ a by a prior calculation. Thus, T is as in the diagram and it is a copy of R,0 NS . Dual remarks 7 involving L,0 NS arise in the left-handed case. 7 Suppose next that S is any non-upper symmetric skew lattice with a, b ∈ S such that a∧ b = b∧ a, but a∨ b ≠ b∨ a. Again, a and b generate a subalgebra Sʹ that forms a non-upper symmetric subalgebra for which a∧ b is the zero element. Moreover, one of Sʹ/R or Sʹ/L must be non-upper symmetric and thus a copy of L,0 R,0 NS or . Whichever case it is, a copy also 7 NS7 exists in Sʹ by Theorem 2.2.9. The lower symmetric case follows by duality. Both cases combine to characterize full symmetry. £ Upper and lower symmetry can each be parsed further in left-right fashion to give a four- fold partition of symmetry that will prove useful in Section 5.3. Theorem 5.1.3. Given a skew lattice S: i) S/L being upper symmetric is equivalent to either of the following: a) S contains no copy of R,0 NS . 7 b) S satisfies x∨ y∨ x = ( y∧ x)∨ y∨ x. ii) S/R being upper symmetric is equivalent to either of the following: a) S contains no copy of L,0 NS . 7 b) S satisfies x∨ y∨ x = x∨ y∨( x∧ y). iii) S/R being lower symmetric is equivalent to either of the following: a) S contains no copy of L,1 NS . 7 b) S satisfies x∧ y∧ x = ( y∨ x)∧ y∧ x. iv) S/L being lower symmetric is equivalent to either of the following: a) S contains no copy of R,1 NS . 7 b) S satisfies x∧ y∧ x = x∧ y∧( x∨ y). The above four conditions hence determine four varieties of skew lattices. 168 V: Further Topics in Skew Lattices Proof. Notice that S/L is upper symmetric if and only if S/L itself contains no copy of R,0 NS . 7 Thus the equivalence of (i)(a) with the upper symmetry of S/L follows from Theorem 2.2.9. To show the equivalence with (i)(b), we begin with x, y ∈ S, set u = u( x, y) = ( y∧ x)∨ y. Since x∧ y∧ x L y∧ x, we have ( x∧ y∧ x) ∨ y ∨ ( x∧ y∧ x) L ( y∧ x) ∨ y ∨ ( y∧ x) = ( y∧ x)∨ y = u. Setting w = ( x∧ y∧ x) ∨ y ∨ ( x∧ y∧ x), since x∧ w = w∧ x = x∧ y∧ x we get ( x∧ u)∧( u∧ x) = x∧ u∧ x = x∧ u∧ w∧ x = ( x∧ u)∧( x∧ w) = x∧ u and ( u∧ x)∧( x∧ u) = u∧ x∧ u = u∧ w∧ x∧ u = u∧ x∧ w∧ u = u∧ x∧ w = u∧ w∧ x = u∧ x. Thus, x∧ u L u∧ x. Moreover, if x∧ y L y∧ x, then u( x, y) must reduce to y. S/L is thus upper symmetric if and only if for all x, y ∈ S, x ∨ u( x, y) L u( x, y) ∨ x. Using the x∧ y = y∨ x identity on D-classes we get, ( y∧ x) ∨ y ∨ x ∨ ( y∧ x) ∨ y = x ∨ ( y∧ x) ∨ y and x ∨ ( y∧ x) ∨ y ∨ ( y∧ x) ∨ y ∨ x = ( y∧ x) ∨ y ∨ x which reduce to ( y∧ x) ∨ y ∨ x ∨ y = x ∨ y and x ∨ y ∨ x = ( y∧ x) ∨ y ∨ x by Lemma 2.1.4, with the left identity being redundant in the presence of the right identity. (The lemma states that in any skew lattice, a, c ≻ b ⇒ a∨ b∨ c = a∨ c while a, c ≺ b ⇒ a∧ b∧ c = a∧ c.) The three other cases are similar. The final assertion is now clear. £ 5.2 Distributive identities in the symmetric case We give Cvetko-Vah’s 2006 proof of Spinks’ Theorem (Spinks [1998]) stating that for symmetric skew lattices, the following identities are equivalent: x ∧ ( y ∨ z) ∧ x = ( x ∧ y ∧ x) ∨ ( x ∧ z ∧ x) (5.2.1) and x ∨ ( y ∧ z) ∨ x = ( x ∨ y ∨ x) ∧ ( x ∨ z ∨ x). (5.2.2) Recall also the following results that are used in further computations often without reference Lemma 5.2.1 A band S is regular if and only if axb = ab holds for all a, b ≺ x ∈ S. £ Lemma 5.2.2 A skew lattice S satisfies any identity or equational implication satisfied by both its left factor S/R and its right factor S/L. £ 169 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond If S is a right-handed skew lattice then the two symmetry identities of (5.1.1) and (5.1.2) above simplify as x∧ y∧( x∨ y) = y∧ x (5.2.3) and x∨ y = ( y∧ x)∨ y∨ x. (5.2.4). Likewise for right-handed skew lattices, identities (5.2.1) and (5.2.2) above simplify as ( y ∨ z) ∧ x = ( y ∧ x) ∨ ( z ∧ x) (5.2.5) and x ∨ ( y ∧ z) = ( x ∨ y) ∧ ( x ∨ z). (5.2.6) We are ready to prove Spinks’ Theorem, doing so first for the right-handed case. Theorem 5.2.3. For any skew lattice S the identities (5.2.1) and (5.1.1) imply (5.2.2), and dually, the identities (5.2.2) and (5.1.2) imply (5.2.1). Proof. Assume S is a right-handed skew lattice satisfying (5.2.3) and (5.2.5). Set α = x ∨ ( y ∧ z) and β = ( x ∨ y) ∧ ( x ∨ z). By (5.2.5) and absorption one also has β = x ∨ ( y ∧ ( x ∨ z)). (5.2.7) Equality (5.2.3) yields y ∧ ( x ∨ z) = ( x ∨ z) ∧ y ∧ ( x ∨ z ∨ y). Thus by (5.2.7), β = x ∨ (( x ∨ z) ∧ y ∧ ( x ∨ z ∨ y)) = x ∨ ( x ∧ y ∧ ( x ∨ z ∨ y)) ∨ ( z ∧ y ∧ ( x ∨ z ∨ y)) by (5.2.5) = x ∨ ( z ∧ y ∧ ( x ∨ z ∨ y)) (5.2.8) by absorption. Apply ∧ ( x ∨ y ∨ z) on the right of (5.2.8). By (5.2.5) and absorption, β ∧ ( x ∨ y ∨ z) = x ∨ ( z ∧ y ∧ ( x ∨ y ∨ z)) (5.2.9) On the other hand, α ∨ ( x ∨ z ∨ y) = x ∨ ( y∧ z) ∨ x ∨ z ∨ y = x ∨ ( y∧ z) ∨ z ∨ y = x ∨ z ∨ y. Hence α ≤ x ∨ z ∨ y and thus α ∧ ( x ∨ z ∨ y) = α. Thus by (5.2.5) and absorption again, x ∨ ( y ∧ z) = x ∨ (( y ∧ z) ∧ ( x ∨ z ∨ y)). (5.2.10) Switch y and z in (5.2.10) to get x ∨ ( z ∧ y) = x ∨ (( z ∧ y) ∧ ( x ∨ y ∨ z)) and use (5.2.9) to obtain β ∧ ( x ∨ y ∨ z) = x ∨ ( z ∧ y). (5.2.11) Apply ∧ ( x ∨ y) on the right to obtain β ∧ ( x ∨ y ∨ z) ∧ ( x ∨ y) = ( x ∨ ( z ∧ y)) ∧ ( x ∨ y) = x ∨ ( z ∧ y ∧ ( x ∨ y)). (5.2.12) 170 V: Further Topics in Skew Lattices On the other hand regularity and right-handedness with (5.2.5) imply β ∧ ( x ∨ y ∨ z) ∧ ( x ∨ y) = β ∧ ( x ∨ y) = ( x ∨ y) ∧ ( x ∨ z) ∧ ( x ∨ y) = ( x ∨ z) ∧ ( x ∨ y) = x ∨ ( z ∧ ( x ∨ y)). (5.2.13) Together (5.2.12) and (5.2.13) give the identity x ∨ ( z ∧ y ∧ ( x ∨ y)) = x ∨ ( z ∧ ( x ∨ y)). (5.2.14) Notice that α ∨ x ∨ z = x ∨ ( y ∧ z) ∨ z = x ∨ z. Hence α ≤ x ∨ z and thus α ∧ ( x ∨ z) = α. Finally, α = α ∧ ( x ∨ z) = ( x ∨ ( y ∧ z)) ∧ ( x ∨ z) = x ∨ ( y ∧ z ∧ ( x ∨ z)) by (5.2.5) = x ∨ ( y ∧ ( x ∨ z)) by (5.5.16) with y and z switched = β by (5.2.7). A similar argument shows that a right-handed skew lattice satisfying (5.2.4) and (5.2.6) must also satisfy (5.2.5). Duality yields that the assertion follows for left handed-skew lattices. The theorem now follows from Lemma 5.2.2. £ Corollary 5.2.4 (Spinks [1998] and [2000]) For symmetric skew lattices, the distributive identitie s (5.2.1) and (5.2.2) are equivalent. £ 5.3 Cancellation in skew lattices A skew lattice is cancellative if both x ∨ z = y ∨ z and x ∧ z = y ∧ z imply x = y, (5.3.1) 17 and x ∨ y = x ∨ z and x ∧ y = x ∧ z imply y = z. (5.3.2) 18 A skew lattice satisfying (5.3.1) [or (5.3.2)] is right [ left] cancellative. Lattices are cancellative precisely when they are distributive. For skew lattices, we at least have: Lemma 5.3.1. Left [right, fully] cancellative skew lattices are quasi-distributive. Proof. All forms of cancellation prevent M3 or N5 from being subalgebras. £ Proposition 5.3.2 Any skew chain is cancellative. Proof. Assume that S is a right-handed skew chain with x, y, z ∈ S satisfying x ∨ z = y ∨ z and x ∧ z = y ∧ z. Then x D y since S/D is totally ordered and thus cancellative. Assume that 171 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond D x ≤ D z. From x ∧ z = y ∧ z we get x ∧ u = y ∧ u for all u in D z. Thus to prove that x = y it thus suffices to find u ∈ D z such that x ≤ u and y ≤ u. Obviously, u = x ∨ z = y ∨ z does the job. On the other hand, D z ≤ D x implies x = x ∨ z ∨ x = x ∨ z = y ∨ z = y ∨ z ∨ y = y. Hence S satisfies (5.3.1). A similar argument, but with ∨ and ∧ interchanged, shows that (5.3.2) holds. The left-handed case follows by duality. Since any skew chain S for which S/D is finite is a subalgebra of S/R × S/L, the general case must follow. £ This leads us to the observation that cancellative skew lattices need not be distributive. Example 5.3.3. Consider the following right-handed skew chain S on 8 elements. ∧ z′ z z′ − − z     y′ y′ y′ ′ ∨ y′ y′ y y′′ y′ − − y′ − − y − − y′′ y′ y′ y x y′ y′ y′ y′     y y′ y x′ y y′ ′ y y′ ′ x − − x′ y′ ′ y′ y′ ′ The D-relation is denoted by – –, while ,  and  denote ≥. The partial tables indicate those operation outcomes not determined simply by S being right-handed or by ≥. By the previous proposition, S is cancellative. But S does not satisfy (5.2.1): ( x ∨ y) ∧ z = yʹ∧ z = yʹʹʹ and ( x ∧ z) ∨ ( y ∧ z) = xʹ ∨ y = y. £ Theorem 5.3.4. Cancellative skew lattices are strongly symmetric. Proof. An easy check shows that none of R,0 L,0 R,1 L,1 NS , , or is cancellative. 7 NS7 NS7 NS7 Likewise it is easily verified that none of the S m,n or T m,n for mn ≥ 2 can be cancellative. The theorem follows from Theorems 5.1.2 and 5.1.7. £ Corollary 5.3.5 (Cvetko-Vah [2006b]) The distributive identities (5.2.1) and (5.2.2) are equivalent for all cancellative skew lattices. Proof. This is an immediate consequence of the theorem above and Corollary 5.2.4. £ Conversely, distributivity does not imply cancellativity. Minimal examples of distributive skew lattices that are neither left, right nor fully cancellative are given by a dual pair of skew lattices with the common Hasse diagram below. We denote the right-handed case by R NC and 5 its left-handed dual by L R NC . The Cayley tables for are also given. Transposing them 5 NC5 gives the tables the left-handed variant. 172 V: Further Topics in Skew Lattices 1 ∨ 0 a b c 1 ∧ 0 a b c 1   0 0 a b c 1 0 0 0 0 0 0 a a a 1 1 1 a 0 a 0 0 a a b − c b b 1 b b 1 b 0 0 b c b   c c 1 c c 1 c 0 0 b c c 0 1 1 1 1 1 1 1 0 a b c 1 One might ask whether the absence of M3, N5 and the above pair of 5-element algebras as subalgebras of a skew lattice S will insure that S is cancellative. While this is not the case (the four 7-element algebras are counterexamples), the question does lead to a form of cancellation the underlies the three forms of cancellation above. A skew lattice S is simply cancellative if for all x, y, z ∈ S, x∨ z∨ x = y∨ z∨ y and x∧ z∧ x = y∧ z∧ y together imply x = y. (5.3.3) This is equivalent to a second implication, x∨ z = y∨ z, z∨ x = z∨ y, x∧ z = y∧ z and z∧ x = z∧ y imply x = y. (5.3.4) Indeed both sets of conditions imply each other. From x∨ z∨ x = y∨ z∨ y we get x∨ z∨ x∨ z = y∨ z∨ y∨ z, that is x∨ z = y∨ z, and similarly, z∨ x = z∨ y. Conversely from x∨ z = y∨ z and z∨ x = z∨ y, x∨ z∨ x = y∨ z∨ y must follow. Similar remarks hold for equalities involving ∧. Thus, simple cancellativity is implied separately by left, right and full cancellativity. Lemma 5.3.6. A skew lattice S is simply cancellative if and only if it is quasi-distributive and all skew diamonds within S are simply cancellative. Given a skew diamond T with incomparable D- classes A and B, join class J and meet class M, the following are equivalent: i) T is simply cancellative ii) Given e > f with e ∈ J and f ∈ M, unique a ∈A and b ∈B exist such that both e > a >f and e > b > f. iii) Given e > f with e ∈ J and f ∈ M , unique a ∈ A and b ∈ B exist such that e = a∨ b = b∨ a and f = a∧ b = b∧ a. iv) T contains no copy of L R NC or . 5 NC5 173 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. Simply cancellative skew lattices are quasi-distributive since M3 and N5 cannot occur as subalgebras. Assume S is at least quasi-distributive. If a, aʹ, b ∈ S exist such that a∨ b∨ a = aʹ∨ b∨ aʹ and a∧ b∧ a = aʹ∧ b∧ aʹ, then a D aʹ since a and aʹ have the same image in the distributive lattice S/D. If a and aʹ are comparable to b ( a, aʹ ≻ b or b ≻ a, aʹ), then either a∨ b∨ a = aʹ∨ b∨ aʹ or a∧ b∧ a = aʹ∧ b∧ aʹ will reduce to a = aʹ. Thus we need only consider the case where b is incomparable to a and aʹ. In any case, the first assertion is by now clear. As for the equivalence of (i) through (iv) for a skew diamond T, (ii) and (iii) are trivially equivalent by Theorem 2.2.1. Clearly these conditions imply (iv). Conversely, suppose (iv) holds. Then given any c ∈ A and d ∈ B, setting a = f∨( e∧ c∧ e)∨ f and b = f∨( e∧ d∧ e)∨ f gives a pair a, b satisfying these inequalities. Suppose e > a, aʹ > f where a D aʹ. Then a R a∧ aʹ L aʹ. Thus if a ≠ aʹ then either a ≠ a∧ aʹ or a∧ aʹ ≠ aʹ. Moreover e > a∧ aʹ > f so that either { a, a∧ aʹ, b, e, f} or { a L R ʹ, a∧ aʹ, b, e, f} is a copy of NC or in T contrary to (iv). Thus a is unique and 5 NC5 similarly so is b. Hence (iv) implies (ii). Suppose that (i) holds and let e ∈ J and f ∈ M be as in (ii) and (iii). As we have seen, a ∈ A and b ∈ B exist such that e > a, b > f. If aʹ ∈ A is such that e > aʹ > f also, then a∨ b∨ a = e = aʹ∨ b∨ aʹ and a∧ b∧ a = f = aʹ∧ b∧ aʹ. Simple cancellation yields a = aʹ. Similarly b is unique and (ii) follows. Conversely suppose that (ii) – (iv) hold and let a∨ b∨ a = aʹ∨ b∨ aʹ and a∧ b∧ a = aʹ∧ b∧ aʹ in T. Since T is quasi-distributive, at least a D aʹ. Again, if a and b are comparable, then a = aʹ, as seen above. So we may assume they are incomparable and hence, say a ∈ A and b ∈ B. Setting e = a∨ b∨ a = aʹ∨ b∨ aʹ in J and f = a∧ b∧ a = aʹ∧ b∧ aʹ in M, we have e ≥ a, aʹ > f. By (ii), a = aʹ and (i) follows. £ Theorem 5.3.7. A skew lattice S is simply cancellative if and only if it contains no copy of M L R 3, N5, NC or . Collectively, these skew lattices form a variety. 5 NC5 Proof. The first assertion is clear from the preceding discussion. An equational base for this class of skew lattices is given by the identity for quasi-distributivity and the identity f ∨ ( e∧ y∧ z∧ y∧ e) ∨ f = f ∨ ( e∧ z∧ y∧ z∧ e) ∨ f where e = e( x, y, z) = x∨ y∨ z∨ x and f = f( x, y, z) = x∧ e∧ y∧ z∧ y∧ e∧ x = x∧ y∧ z∧ y∧ x. On either side are two typical D-related elements having common commuting joins and meets with a third element (here x). The implication defining simple cancellativity (where x, y and z assume different roles!) equates these elements. £ Returning to left [right, full cancellativity] we have: 174 V: Further Topics in Skew Lattices Theorem 5.3.8. 1) The following are equivalent for a skew lattice S . i) S is left cancellative. ii) None of M L R R,0 L,1 3, N5, NC , , nor are subalgebras of S. 5 NC5 NS7 NS7 iii) S is simply cancellative, S/L is upper symmetric, and S/R is lower symmetric. 2) The following are equivalent for a skew lattice S . i) S is right cancellative. ii) None of M L R R,1 L,0 3, N5, NC , , nor are subalgebras of S. 5 NC5 NS7 NS7 iii) S is simply cancellative, S/L is lower symmetric, and S/R is upper symmetric. 3) The following are equivalent for a skew lattice S . i) S is cancellative. ii) None of the above 5 or 7 -element algebras occur as subalgebras of S. iii) S is simply cancellative and symmetric. 4) Left [right, fully] cancellative skew lattices form a variety. Proof. We begin with (1). If S is left cancellative, then S cannot contain copies of any of the algebras listed in (ii) since none of them are left cancellative. Thus (i) implies (ii). The equivalence of (ii) and (iii) follows from Theorems 5.3.7 and 5.1.2. Now assume S satisfies (iii), and suppose a, b, c ∈ S satisfy a ∨ b = a ∨ c and a ∧ b = a ∧ c. Since S/L is upper symmetric and S/R is lower symmetric, Theorem 5.1.2 gives b ∨ a ∨ b = ( a ∧ b) ∨ a ∨ b = ( a ∧ c) ∨ a ∨ c = c ∨ a ∨ c and b ∧ a ∧ b = ( a ∨ b) ∧ a ∧ b = ( a ∨ c) ∧ a ∧ c = c ∧ a ∧ c. By simple cancellativity, b = c. Therefore S is left cancellative and (i) holds. The proof of (2) is similar. Indeed (2) follows from (1) via horizontal duality (new x ∨ y = old y ∨ x; new x ∧ y = old y ∧ x). (3) is a consequence of (1) and (2) combined, together with Theorems 5.3.7 and 5.1.1. Finally (4) now follows from (1) – (3) and Theorems 5.1.1, 5.1.2 and 5.3.7. £ Equational bases. A basis for left cancellative skew lattices is thus as follows. 1) [ x∧( y∨ z)] ∧ [( x∧ y) ∨ ( x∧ z)] ∧ [ x∧( y∨ z)] = x∧( y∨ z). 2) f∨( e∧ y∧ z∧ y∧ e)∨ f = f∨( e∧ z∧ y∧ z∧ e)∨ f where e = x∨ y∨ z∨ x and f = x∧ y∧ z∧ y∧ x. 3) x∨ y∨ x = ( y∧ x)∨ y∨ x and x∧ y∧ x = ( y∨ x)∧ y∧ x. 175 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Identities (1) and (2) insure that a skew lattice S is simply cancellative (per Theorem 5.3.7) and (3) ensures that S/L is upper symmetric and S/R is lower symmetric (per Theorem 5.1.2). An equational base for right cancellative skew lattices is obtained by replacing (3) with the left-right dual identities. An equational base for cancellative skew lattices is given by replacing (3) with the two identities for symmetry. Consider the lattice of varieties below. Simply cancellative skew lattices   Left cancellative Right cancellative . skew lattices skew lattices   Cancellative skew lattices While the bottom variety is the inter-section of the middle varieties, we do not know if the top variety is generated from the middle varieties and thus is their join in the lattice of all skew lattice varieties. The bottom variety is, of course, the intersection of each variety with the variety of symmetric skew lattices. Using Mace4, four distinct minimal cases of order 12 exist that are simply cancellative, but are neither left nor right cancellative. They turn out to be the fibered product L,0 L,1 L,0 L,1 NS of and over their maximal lattice image 2 7 ×2 ×2; the splice ×2 NS7 NS7 NS7 of L,0 L,1 L,0 NS with obtained by identifying the join class of with the meet class of 7 NS7 NS7 L,1 NS ; their two right-handed duals. 7 Cancellation is often used to compare the sizes of sets. Indeed much of our discussion of simple cancellation in Lemma 5.3.6 can be recast as follows. Given a skew diamond with incomparable D-classes A and B, join class J and meet class M, set: ω(J, M) = {( j, m) ∈ J × M⎪ j > m} and Comm 2(A, B) = {( a, b) ∈A × B⎪ a∨ b = b∨ a & a∧ b = b∧ a}. ω(J, M) is the natural partial order between J and M, while Comm 2(A, B), consists of all pairs, one from each class, that commute under both operations. Define ξ: Comm 2(A, B) → ω(J, M) by ξ( a, b) = ( a∨ b, a∧ b). ξ is at least well defined. In the proof of Lemma 5.3.6 we saw that ξ is surjective. The unique parallel commuting factorization of Lemma 5.3.6 (iii) gives us the first half of: 176 V: Further Topics in Skew Lattices Theorem 5.3.10. Given a simply cancellative skew diamond {J > A, B > M}, the function ξ: Comm 2(A, B) → ω(J, M) defined by ξ( a, b) = ( a∨ b, a∧ b) is a bijection. Conversely, given any skew diamond {J > A, B > M} the map ξ as stated is always well-defined, but it is a bijection only if the skew diamond is simply cancellative. Proof. If {J L R > A, B > M} is not simply cancellative, then either a copy of NC or occurs 5 NC5 which gives ξ a properly many-to-one instance. £ The above results in this section are from the 2011 paper of Cvetko-Vah, Kinyon, Leech and Spinks. We conclude this section with several observations: Theorem 5.3.11. Skew lattices in rings are cancellative. Proof. So let ab = ac and a∇ b = a∇ c. Expanding and making the obvious reductions reduces the latter to b + ba –bab = c + ca – cac = c + ca – cab. Thus b(1 – a – ab) = c(1 – a – ab). But (1 – a – ab)2 = 1 – a + aba and so b = b(1 – a + aba) = c(1 – a + aba). Similarly c = c(1 – a + aca) = c(1 – a + aba) and b = c follows. Right cancellation is shown similarly. £ Theorem 5.3.12. Strongly distributive skew lattices are cancellative. Proof. Strongly distributive skew lattices are clearly quasi-distributive, and by Theorem 2.3.4 also symmetric. By the same theorem they are also normal, so that neither L R NC nor can 5 NC5 be a subalgebra. Hence none of the 5- or 7-element forbidden algebras can be a subalgebra, making any strongly distributive skew lattice cancellative. £ Primitive skew lattices and skew chains in general are trivially cancellative: they are clearly symmetric and quasi-distributive and it is impossible for either variant of NC5 to be a subalgebra. Thus we have: Proposition 5.3.13. Every skew lattice in the variety of skew lattices generated from the class of all primitive skew lattices is cancellative. More generally, every skew lattice in the variety of skew lattices generated from the class of all skew chains is cancellative. £ Queries: Are the varieties generated by these two classes of algebras the same? Is the second variety in fact the variety of all cancellative skew lattices? 177 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond The answer to the first question is, no. Primitive skew lattices are trivially categorical, and thus all skew lattices in the first variety are categorical. We shall see in the next section that skew chains A > B > C of length 2 exist that are not categorical. The first variety is thus a proper subvariety of the second. 5.4 Categorical skew lattices We continue our study begun in Section 2.4 of categorical skew lattices, where all nonempty composites of coset bijections are coset bijections. This reduces to the implication: if a > b > c with aʹ D a and cʹ D c such that cʹ = aʹ∧ c∧ aʹ in C and aʹ = cʹ∨ a∨ cʹ in A (making both a > c and aʹ > cʹ part of a common coset bijection from A to C), then aʹ∧ b∧ aʹ = cʹ∨ b∨ cʹ in B. a – a′ = c′ ∨ a ∨ c′   b – b′ = a′ ∧ b ∧ a′ = c′ ∨ b ∨ c′ (where denotes >)   c – c′ = a′ ∧ c ∧ a′ For if χ, the unique coset bijection from A to C taking a to c, factors as ψοϕ, where ϕ and ψ are the unique coset bijections from A to B and from B to C taking a to b and b to c respectively, then one has a –1 ʹ∧ b∧ aʹ = ϕ[ aʹ] = ψ [ cʹ] = cʹ∨ b∨ cʹ. Contraposition gives the following criterion for a skew lattice S to not be categorical: given a > b > c in S with aʹ D a and cʹ D c being such that aʹ∧ c∧ aʹ = cʹ in D c and cʹ∨ a∨ cʹ = aʹ in D a, but aʹ∧ b∧ aʹ ≠ cʹ∨ b∨ cʹin D b. We have seen that categorical skew lattices form a variety. In the left-handed skew case: Lemma 5.4.1. A left-handed skew lattice is categorical if either of the following pair of dual identities hold: x ∧ [ y ∨ ( x ∧ y ∧ z)] = x ∧ y (5.4.1) or [( x ∨ y ∨ z) ∧ y] ∨ z = y ∨ z. (5.4.2) Proof. Let S be a left-handed non categorical skew lattice. Thus a > b > c in S exist with aʹ L a and cʹ L c such that aʹ ∧ c = cʹ in C, a ∨ cʹ = aʹ in A but aʹ ∧ b ≠ b ∨ cʹ in B. This creates the following configuration A: a − a′    B: b − a′ ∧ b − b ∨ c′ .    C: c − c′ 178 V: Further Topics in Skew Lattices Being left-handed, both aʹ > aʹ∧ b and b∨ cʹ > cʹ are clear. Trivially cʹ∧( aʹ∧ b) = cʹ and ( b∨ cʹ)∨ aʹ = aʹ. On the other hand, ( aʹ∧ b)∧ cʹ = ( aʹ∧ b)∧( aʹ∧ c) = aʹ∧( b∧ c) = aʹ∧ c = cʹ also so that aʹ∧ b > cʹ. In dual fashion aʹ > b∨ cʹ. Notice that both aʹ ∧ b ≠ b ∨ cʹ = aʹ ∧ ( b ∨ cʹ) = aʹ ∧ [ b ∨ ( aʹ ∧ b ∧ cʹ)]. and b ∨ cʹ ≠ aʹ ∧ b = ( aʹ ∧ b) ∨ cʹ = [( aʹ ∨ b ∨ cʹ) ∧ b] ∨ cʹ. The lemma follows by contraposition. £ Comment: The converse (in the left-handed case) has been shown using Prover9. These identities do not imply left-handedness. As a consequence of this lemma we have: Theorem 5.4.2. Any skew lattice satisfying either distributive identity (5.2.1) or (5.2.2) is categorical. In particular, all distributive skew lattices are categorical. Proof. To begin, if a left-handed skew lattice satisfies (5.2.1), then (5.4.1) follows: x ∧ [ y ∨ ( x ∧ y ∧ z)] = ( x∧ y) ∨ [ x ∧ ( x ∧ y ∧ z)] = ( x∧ y) ∨ ( x ∧ y ∧ z) = x∧ y. Likewise a left-handed skew lattice satisfying (5.2.2) must satisfy (5.4.2). Dually, a right-handed skew lattice satisfying either (5.2.1) or (5.2.2) are categorical. In general, if a skew lattice S satisfies either distributive identity, then so do its factors S/R and S/L. Hence each factor is categorical and thus so is S, since it is isomorphic to a subalgebra of S/R × S/L. £ Forbidden subalgebras Given a comparable D-classes, A > B, if a, aʹ ∈ A lie in a common B-coset, we denote this by a –B aʹ; likewise b –A bʹ in B if b, bʹ lie in a common A-coset. Of interest here are skew chains A > B > C, since a skew lattice is categorical if and only if all its skew chains are thus. Two elements b and bʹ in the middle class B are AC- connected if a finite sequence b = b 0, b 1, b 2, … , bn = bʹ exists such that bi –A bi+1 or bi –C bi+1 for all i ≤ n – 1. Clearly this defines an equivalence relation on B. An AC- component of B (or just component when the context is clear) is an equivalence class for this relation, that is, a maximally AC-connected subset of B. (Connectedness is actually a congruence on the rectangular algebra B, making the components subalgebras of B. Indeed it is the join-equivalence of the congruence partitions given by the A-cosets and by the B-cosets.) In the examples below, B is connected. Given a component B1 in the middle class B, a sub-skew chain is given by A > B1 > C. Indeed if A1 is a B-coset in A and C1 is a B-coset in C, then A1 > B1 > C1 is also a sub-skew chain; moreover, a skew chain is categorical if and only if all such sub-skew chains are categorical. Our classification of forbidden subalgebras rests on the next lemma and its right-handed dual. 179 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Lemma 5.4.3. Given a left-handed skew chain A > B > C, let a > c and aʹ > cʹ be input-output pairs for a common coset bijection between A and C where a ≠ aʹ in A and c ≠ cʹ in C. Upon setting A* = { a, aʹ}, B* = { x ∈ B⎪ a > x > c or aʹ > x > cʹ} and C* = { c, cʹ}, one obtains a sub-skew chain: A* > B* > C*. In particular, i) aʹ > x > cʹ for x in B* implies: a > both a∧ x and x∨ c > c with a∧ x –A* x –C* x∨ c. ii) a > x > c for x in B* implies: aʹ > both aʹ∧ x and x∨ cʹ > cʹ with aʹ∧ x –A* x –C* x∨ cʹ . All A*- cosets and all C*- cosets in B* are of order 2. An A*C*- component in B* is either a subset { b, bʹ} that is simultaneously an A* and C*- coset in B* or else it is a larger subset having the alternating coset form: … –A* • –C* • –A* • –C* • –A* • –C* … Only the former can occur if the skew chain is categorical. Proof. Being left-handed, we need only check the mixed outcomes, say a∧ x, x∧ a, c∨ x and x∨ c where aʹ > x > cʹ for case (i). Trivially x∧ a = x = c∨ x. As for a∧ x, a∧( a∧ x) = a∧ x = ( a∧ x)∧ a, due to left-handedness, so that a > a∧ x; likewise c∧( a∧ x) = c, while ( a∧ x)∧ c = a∧ x∧ a∧ cʹ = a∧ x∧ cʹ = a∧ cʹ = c by left-handedness and the common coset bisection context. Hence a∧ x > c also, so that a∧ x is in B*. The dual argument gives a > x∨ c > c, so that x∨ c ∈ B* also. Similarly (ii) holds and we have a sub-skew chain. Clearly the A*-cosets in B* either all have order 1 or all have order 2. If they have order 1, then a, aʹ > all elements in B*, and by transitivity, a, aʹ > both c, cʹ, so that a > c belongs to a different coset bijection than aʹ > cʹ. Thus all A*-cosets in B* have order 2 and likewise all C*-cosets in B* have order 2. In an A*C*-component in B*, if the first case does not occur, a situation x –C* y –A* z with x, y and z distinct develops. Since A* and C*-cosets have size 2, it extends in an alternating coset pattern in both directions, either doing so indefinitely or eventually connecting to form a cycle of even length. £ This leads to: Example 5.4.4: Consider the class of skew chains A > B n > C for 1 ≤ n ≤ ω, where A = { a 1, a 2}, B n = { a 1, a 2, a 3, … , a 2 n} or {… , b–2, b–1, b 0, b 1, b 2, …} if n = ω and C = { c 1, c 2}. The natural partial ordering given by a 1 > bodd > c 1 and a 2 > beven > c 2. Both A and C are full B-cosets as well as full cosets of each other. A- and C-cosets in B are given respectively by the partitions: 180 V: Further Topics in Skew Lattices b { } { } 1, b 2 b 3, b 4 ... b 2 n–1, b 2 n and b 2 n, b 1 b 2, b 3 ... b 2 n−2, b 2 n−1 and { b } and { } when n = ω. 2 k+1, b 2 k+2 b 2 k, b 2 k+1 Clearly B n is a single component. We denote the left-handed skew chain thus determined by X n and its right-handed dual by Y n for n ≤ ω. Their shared Hasse diagrams are as follows. a − 1 a 2 a − a   1 2    b − b 1: 1 2 2: b − b − b − b − b  1 2 3 4 1 A C A C c −  1 c 2   c − 1 c 2 a − − 1 a 2 a 1 a 2           n: b − − − − − ( ) − − − − − − 1 b 2 .. b 2 n–1 b 2 n b 1 ω: ... b− 2 b−1 b 0 b 1 b 2 ... A C C A C A C A C A C           c − c − 1 c 2 1 c 2 In line with our remarks above, instances of left-handed operations on X2 are given by a 1 ∨ c 2 = a 1 ∨ a 2 = a 2, a 1 ∧ b 4 = b 3 ∧ b 4 = b 3 and b 1 ∨ c 2 = b 1 ∨ b 4 = b 4. Except for X1 and Y1, none of these skew lattices is categorical. Indeed, a 1 > b 1 > c 1, a 2∧ c 1∧ a 2 = c 2, c 2∨ a 1∨ c 2 = a 2, but a 2∧ b 1∧ a 2 = b 2, while c 2∨ b 1∨ c 2 is either b 2 n or b 0. Thus, except for X1 and Y1, none of these is distributive. Note also that each X n is generated from a 1, c 2 and any bi. Thus no X n contains a copy of a lower X m as a subalgebra. Similar remarks hold for the Y n. Theorem 5.4.5. A left-handed skew lattice is categorical if and only if it contains no copy of X n for 2 ≤ n ≤ ω. Dually, a right-handed skew lattice is categorical if and only if it contains no copy of Y n for 2 ≤ n ≤ ω. In general, a skew lattice is categorical if and only if it contains no copy of any of these algebras. Proof: We begin with a left-handed noncategorical skew chain S. Without loss of generality we may assume that A is a full B-coset in itself and that C is a full B-coset in itself. Let a > b > c in S with aʹ L a and cʹ L c being such that aʹ ∧ c = cʹ in C, a ∨ cʹ = aʹ in A so that a, aʹ correspond to c, cʹ under a coset bijection between A and C, but aʹ ∧ b ≠ b ∨ cʹ in B where A, B, C are respective L-classes. The first new elements formed are aʹ∧ b and b∨ cʹ in B. Note that 181 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond a∨( aʹ∧ b) = aʹ since both are images of aʹ∧ b in A, which is the unique B-coset within A. Likewise, ( b∨ cʹ)∧ c = cʹ. No new elements are created thus far from { a, aʹ, b, c, cʹ} giving us: A: a − a′    B: b − a′ ∧ b − b ∨ c′    C: c − c′ At this stage it follows that the subalgebra formed from { a, aʹ, b, c, cʹ} only has { a, aʹ} in the top L-class and { c, cʹ} in the bottom L-class. Continuing, step-by-step, in both directions we get − a′ − a − a′ − a − a′ − a − a′ −        – [ a ∧ ( b ∧ c′)] ∨ c′ – a ∧ ( b ∧ c′) – b ∨ c′ – b – a′ ∧ b – ( a′ ∧ b) ∨ c – a ∧ [( a′ ∧ b) ∨ c] – A C A C A C A C        . − c′ − c − c′ − c − c′ − c − c′ − Expansion thus continues in B. If repetition never occurs, we obtain a copy of X . Otherwise, a ω cyclic closure arises and we have obtained a copy of some X n. The left-handed case follows from this. The right-handed case is similar. Clearly, a categorical skew lattice contains none of these algebras. Conversely, if a skew lattice S contains copies of none of them, then neither does S/R or S/L since every skew chain with finitely many D-classes in either S/R or S/L can be lifted up into S. Thus both S/L and S/R are categorical, and hence so it S which is embedded in their product. £ Corollary 5.4.6. Distributive skew lattices contain no copies of X n or Y n for n ≥ 2. £ Comment: If these skew chains seem familiar it is because they are precisely the maximal skew chains within the skew diamonds arising in the classification of all symmetric skew lattices. X n and Y n for all n ≥ 2 in particular arise in the various non-categorical cases. Strictly categorical skew lattices We turn to categorical skew lattices that are strictly categorical in that for each skew chain of D-classes A > B > C, each A-coset in B has nonempty intersection with each C-coset in B, making both B an entire AC-component and empty coset bijections unnecessary. Examples are: 182 V: Further Topics in Skew Lattices a) Normal skew lattices characterized by the identity x∧ y∧ z∧ w = x∧ z∧ y∧ w; equivalently, each subset [ e]↓ = { x ∈S⎪ e ≥ x} = { e∧ x∧ e⎪ x ∈S} is a sublattice. b) Conormal skew lattices that satisfy x∨ y∨ z∨ w = x∨ z∨ y∨ w; equivalently, every subset [ a]↑ = { x ∈S⎪ x ≥ e} = { e∨ x∨ e⎪ x ∈S} is a sublattice. c) Primitive skew lattices consisting of two D-classes: A > B and rectangular skew lattices. Any algebra in the variety that primitive skew lattices generate. Their significance is due in part to skew Boolean algebras being normal as skew lattices. Theorem 5.4.7. Let A > B > C be a strictly categorical skew chain. Then: i) For any a ∈A, all images of a in B lie in a unique C- coset in B. ii) For any c ∈C, all images of c in B lie in a unique A- coset in B. iii) Given a > c with a ∈A and c ∈C, a unique b ∈B exists such that a > b > c. This b lies jointly in the C- coset in B containing all images of a in B and in the A-coset in B containing all images of c in B. (It is the midpoint of a and c in B.) Proof. (i) Without loss of generality we assume that C is a full B-coset within itself. If a∧C∧ a = { c ∈C⎪ a > c} is the image set of a in C parameterizing the A-cosets in C and if a > b in B, then the set { c∨ b∨ c⎪ c ∈ a∧C∧ a} of all images of a in the C-coset C∨ b∨C in B, parameterizes the A-C cosets in B lying in C∨ b∨C (that is, the coset intersections A∧ b∧A ∩ C∨ b∨C in C∨ b∨C) that are inverse images of the A-cosets in C relative to the coset bijection of C∨ b∨C onto C. (See Theorem 2.4.14 and its preceding discussion.) By assumption, all A-cosets X in B are in bijective correspondence with these A-C cosets under the map X → X ∩ C∨ b∨C. Thus each x in { c∨ b∨ c⎪ c ∈ a∧C∧ a} is the (necessarily) unique image of a in the A-coset in B to which x belongs and as we traverse through these x’s, every such A-coset occurs as A∧ x∧A. Thus all images of a in B lie in the C-coset C∨ b∨C in B. In like fashion one verifies (ii). Finally, given a > c with a ∈A and c ∈C, a unique AC-coset U exists that is the intersection of the A-coset containing all images of c in B and the C-coset containing all images of a in B. In particular U contains any b in B such that a > b > c. Such a b exists in B, e.g., b = a∧( c∨ u∨ c)∧ a for any u in B. But being in a single AC-coset in B, at most one such b is in U. £ In the terminology of Section 2.4, the A-cosets in B are orthogonal to the C-cosets in B. All this leads to the following multiple characterization of strictly categorical skew lattices: 183 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 5.4.8. The following seven conditions on a skew lattice S are equivalent. i) S is strictly categorical. ii) Given both a > b > c and a > bʹ > c in S with b D bʹ, b = bʹ must follow. iii) Given both a ≥ b ≥ c and a ≥ bʹ ≥ c in S with b D bʹ, b = bʹ must follow. iv) S has no subalgebra isomorphic to either of the following 4- element skew chains: a a     b – – L b′. b – – R b′ .     c c v) Given a > b in S, the interval subalgebra [ a, b] = { x ∈ S⎪ a ≥ x ≥ b} is a sublattice. vi) Given any a ∈ S, [ a]↑ = { x ∈ S⎪ x ≥ a} is a normal subalgebra of S and [ a]↓ = { x ∈ S⎪ a ≥ x} is a conormal subalgebra of S. vii) S is categorical and given any skew chain A > B > C of D- classes in S, for each coset bijection ϕ: A → C, unique coset bijections ψ: A → B and χ: B → C exist such that ϕ = χψ. Proof. Theorem 5.4.7(iii) gives us (i) ⇒ (ii). Conversely, if S satisfies (ii) then no subalgebra of S can be one of the forbidden algebras in Theorem 5.4.5, making S categorical. We next show that given x, y ∈ B, there exist u and v in B such that x –A u –C y and x –C v –A y. This guarantees that in B, every A-coset meets every C-coset. Indeed, pick a ∈ A and c ∈ C so that a > x > c. Note that a > a∧( c∨ y∨ c)∧ a, c∨( a∧ y∧ a)∨ c > c. But by assumption x is the unique element in B between a and c under >. Thus a∧( c∨ y∨ c)∧ a = x = c∨( a∧ y∧ a)∨ c so that both x –A c∨ y∨ c –C y and x –C a∧ y∧ a –A y in B, which gives (ii) ⇒ (i). Next let S be categorical with A > B > C as stated in (vii). The unique factorization in (vii) occurs precisely when (ii) holds, making (ii) and (vii) equivalent. Finally, (iii) – (vi) are easily seen to be equivalent variants of (ii) .£ Corollary 5.4.9. Strictly categorical skew lattices form a variety of skew lattices. 184 V: Further Topics in Skew Lattices Proof. Consider the following identity or its dual: x ∨ ( y ∧ z ∧ u ∧ y) ∨ x = x ∨ ( y ∧ u ∧ z ∧ y) ∨ x. (5.4.2) Note that x∨ y∨ x ≥ x ∨ ( y ∧ z ∧ u ∧ y) ∨ x & x ∨ ( y ∧ u ∧ z ∧ y) ∨ x ≥ x by (1.1) and (1.6) with the middle expressions being D-equivalent, since z ∧ u D u ∧ z. Hence, if a skew lattice S is strictly categorical, then (5.4.2) holds by Theorem 5.4.8(iii). Conversely, let (5.4.2) hold in S and suppose that a ≥ both b, bʹ ≥ c in S with b D bʹ. Assigning x → c, y → a, z → b∧ bʹ and u → bʹ∧ b reduces (5.4.2) to b = b∧ bʹ∧ b = bʹ∧ b∧ bʹ = bʹ making S strictly categorical by Theorem 5.4.8(iii) again. £ While distributive skew lattices are categorical, they need not be strictly categorical; but: Theorem 5.4.10. A strictly categorical skew lattice S is distributive if and only if it is also quasi-distributive. Proof. Any distributive skew lattice is quasi-distributive. Conversely, in any strictly categorical skew lattice both a ≥ a∧( b∨ c)∧ a and a ≥ ( a∧ b∧ a) ∨ ( a∧ c∧ a). In turn, a∧ c∧ b∧ a ≤ both a∧( b∨ c)∧ a and ( a∧ b∧ a) ∨ ( a∧ c∧ a). Indeed, regularity and absorption give, e.g., ( a∧ c∧ b∧ a)∧[ a∧( b∨ c)∧ a] = a∧ c∧ b∧( b∨ c)∧ a = a∧ c∧ b∧ a and ( a∧ c∧ b∧ a) ∧ [( a∧ b∧ a) ∨ ( a∧ c∧ a)] = a∧ c∧ a∧ b∧ a ∧ [( a∧ b∧ a) ∨ ( a∧ c∧ a)] = a∧ c∧ a∧ b∧ a = a∧ c∧ b∧ a In any quasi-distributive skew lattice, a∧( b∨ c)∧ a D ( a∧ b∧ a) ∨ ( a∧ c∧ a). Thus if S is quasi-distributive and strictly categorical, Theorem 5.4.8(iii) implies that both (5.2.1) and dually (5.2.2) must hold. The converse is clear. £ Theorems 5.4.8 and 5.4.10 can also be used to show that a distributive, strictly categorical skew lattice S is simply cancellative. It is (fully) cancellative when S is also symmetric. Corollary 5.4.11. A skew lattice is strictly categorical and distributive if and only if no subalgebra is a copy of lattices M3 or N5 or either of the skew chains in Theorem 5. 4.7(iv). £ Order-closure and paranormal skew lattice Both normal skew lattices and conormal skew lattices are proper subvarieties of the variety of strictly categorical skew lattices. It is reasonable to ask if these subvarieties jointly generate the larger variety. This turns not to be the case since both types of algebras belong to another variety of skew lattices that excludes many primitive skew lattices, all of which must be strictly categorical. 185 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond A primitive skew lattice A > B is order-closed if for a, aʹ ∈ A and b, bʹ ∈ B, a, aʹ > b and a > b, bʹ imply aʹ > bʹ. a − − a′    ?    ? b − − b′ A primitive skew lattice A > B is simply order-closed if a > b holds for all a ∈ A and all b ∈ B. In this case the cosets of A and B in each other are precisely the singleton subsets. The following characterization of order-closed primitive skew lattices is easily verified: a primitive skew lattice is order-closed if and only if it factors as the product T × D of a simply order-closed primitive skew lattice T with a rectangular skew lattice D. A skew lattice is order-closed if all its primitive subalgebras are order-closed. Order-closed skew lattices form a subvariety of skew lattices. Using variables x, y, u and v, with w = x∧ y∧ u∧ v∧ x∧ y, one has the following generic situation between two D-classes (  denoting >): x ∧ y – – – w ∨ ( y ∧ x) ∨ w    .    w – – – x ∧ y ∧ v ∧ u ∧ x ∧ y Thus [ w∨( y∧ x)∨ w] ∧ ( x∧ y∧ v∧ u∧ x∧ y) ∧ [ w∨( y∧ x)∨ w] = x∧ y∧ v∧ u∧ x∧ y characterizes these skew lattices and we have: Proposition 5.4.12. Order-closed skew lattices are a variety that includes both normal skew lattices and conormal skew lattices. Proof. That we have a variety is clear. That it includes all normal skew lattices is due to the fact that given a primitive normal skew lattice A > B, for any a ∈ A only one b ∈ B exists such that a > b, thus making A > B satisfy the defining condition for order-closure in a trivial manner. £ The variety join of the varieties of normal skew lattices and conormal skew lattices is thus included in the intersection of the varieties of strictly categorical and order-closed skew lattices. Moreover, primitive skew lattices not satisfying the order-closed criterion above are easily designed. See, e.g., the example of Theorem 2.4.3. Thus strictly categorical skew lattices are not the join of the normal and conormal skew lattice varieties. 186 V: Further Topics in Skew Lattices A skew lattice is paranormal if is it is an order-closed, strictly categorical skew lattice. We can now pose the following question: Is the variety of paranormal skew lattices jointly generated by the varieties of normal and conormal skew lattices? Otherwise put, is it their join in the lattice of all skew lattice varieties? One can also consider conditioned versions of the question by asking if, say, the variety of symmetric paranormal skew lattices is the join of the varieties of symmetric normal and symmetric conormal skew lattices. “Symmetric,” of course, could be replaced by “distributive” or by “distributive and symmetric.” In the latter case, a related question is: Is the variety of distributive, symmetric, paranormal skew lattices generated from the class of all order-closed primitive algebras? The motivation for this is the fact that distributive, symmetric, normal skew lattices are generated from 3R and 3L. (See Theorem 2.6.12.) Indeed, returning to strictly categorical skew lattices in general, one may ask: Is the variety of distributive, symmetric, strictly categorical skew lattices generated from the class of all primitive algebras? 5.5 Distributive skew lattices We begin with a broader class of skew lattices. A skew lattice S is linearly distributive if every subalgebra T that is totally preordered under ≻ is distributive. Since totally preordered skew lattices are trivially symmetric, a skew lattice S is linearly distributive if and only if each totally preordered subalgebra satisfies either (5.2.1) or equivalently (5.2.2). Indeed, S is linearly distributive if it is distributive on each skew chain A ≥ B ≥ C in S. Since skew chains need not be even categorical, they need not be distributive! However: Theorem 5.5.1. Linearly distributive skew lattices form a variety of skew lattices. Thus a skew lattice S is linearly distributive if and only if both S/R and S/L are. Proof. Consider the terms x, y∧ x∧ y and z∧ y∧ x∧ y∧ z. Clearly x ≻ y∧ x∧ y ≻ z∧ y∧ x∧ y∧ z holds for all skew lattices. Conversely given any instance a ≻ b ≻ c in some skew lattice S, the assignment x → a, y → b, z → c will return this particular instance. Thus a characterizing set of identities for the class of all linearly distributive skew lattices is given by taking the basic identity u∧( v ∨ w)∧ u = ( u∧ v∧ u) ∨ ( u∧ w∧ u) and forming all the identities possible in x, y, z by making bijective assignments from the variables { u, v, w} to the terms { x, y∧ x∧ y, z∧ y∧ x∧ y∧ z}. £ 187 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond We proceed with several lemmas, the first of which is evident. Lemma 5.5.2. Left-handed skew lattices that satisfy (5.2.1) are characterized by: x∧ y∧ x = x∧ y, x∨ y∨ x = y∨ x and x ∧ ( y ∨ z) = ( x ∧ y) ∨ ( x ∧ z). (5.5.1L) Dually, right-handed skew lattices that satisfy (5.2.1) are characterized by: x∧ y∧ x = y∧ x, x∨ y∨ x = x∨ y and ( y ∨ z) ∧ x = ( y ∧ x) ∨ ( z ∧ x). (5.5.1R) Lemma 5.5.3. In a left-handed totally preordered skew lattice, if a∧( b∨ c) ≠ a∧ b ∨ a∧ c, then a ≻ b ≻ c. Thus, for a left-handed skew lattice S, the following are equivalent: a) S is linearly distributive. b) a∧( b ∨ c) = ( a ∧ b) ∨ ( a ∧ c) for all a ≻ b ≻ c in S. c) a∧( b ∨ c) = ( a ∧ b) ∨ ( a ∧ c) for all a ≻ b ≻ c in S. Left-handed linearly distributive skew lattices are thus characterized by: x∧[( y∧ x) ∨ ( z∧ y∧ x)] = ( x∧ y) ∨ ( x∧ z∧ y) . (5.5.2L) Dually, right-handed linearly distributive skew lattices are characterized by: [( x∧ y∧ z) ∨ ( x∧ y)]∧ x = ( y∧ z∧ x) ∨ ( y∧ x) . (5.5.2R) Proof. If say b ≻ a, then a∧( b∨ c) = a and ( a∧ b) ∨ ( a∧ c) = a ∨ ( a∧ c) = a. If c ≻ a, then a∧( b∨ c) = a again, and ( a∧ b) ∨ ( a∧ c) = ( a∧ b) ∨ a = ( a∧ b∧ a) ∨ a = a. Thus inequality only occurs when a ≻ b, c. But even here, a ≻ c ≻ b gives us a∧( b∨ c) = a∧ c and ( a∧ c) ≻ ( a∧ b) so that ( a∧ b) ∨ ( a∧ c) = a∧ c also. Thus, to completely avoid a∧( b∨ c) = a∧ b ∨ a∧ c we are only left with a ≻ b ≻ c. £ In particular primitive skew lattices are distributive. It turns out that linear distributivity is characterized succinctly by either of the dual pair of identities in the following lemma. 188 V: Further Topics in Skew Lattices Lemma 5.5.4. Identities (5.2.1) and (5.2.2) respectively imply x ∧ [( y∧ x∧ y) ∨ ( z∧ x∧ z)] ∧ x = ( x∧ y∧ x) ∨ ( x∧ z∧ x) (5.5.3) and x ∨ [( y∨ x∨ y) ∧ ( z∨ x∨ z)] ∨ x = ( x∨ y∨ x) ∧ ( x∨ z∨ x). (5.5.4) For left-handed skew lattices, these identities simplify to: x ∧ [( y∧ x) ∨ ( z∧ x)] = ( x∧ y) ∨ ( x∧ z) (5.5.3L). and [( x∨ y) ∧ ( x∨ z)] ∨ x = ( y∨ x) ∧ ( z∨ x). (5.5.4L). Dually, in the right-handed case they simplify to: [( x∧ y) ∨ ( x∧ z)] ∧ x = ( y∧ x) ∨ ( z∧ x) (5.5.3R). and x ∨ [( y∨ x) ∧ ( z∨ x)] = ( x∨ y) ∧ ( x∨ z) . (5.5.4R). Proof. Since x∧ y∧ x∧ y∧ x = x∧ y∧ x by regularity, (5.2.1) ⇒ (5.5.3) and also (5.2.2) ⇒ (5.5.4). £ Theorem 5.5.5. For all skew lattices, (5.5.3) and (5.5.4) are equivalent. A skew lattice satisfies either and hence both if and only if it is linearly distributive. Proof. For left-handed skew lattices, (5.5.3L) clearly implies (5.5.2L), while (5.5.3R) implies (5.5.2R). Thus if a skew lattice S satisfies (5.5.3), so do both S/R and S/L, making them linearly distributive, and hence S also by Theorem 5.5.1 Conversely, assume that S is linearly distributive. First, let S be left-handed also. Then left-handedness gives the first, third and sixth equalities below. x ∧ [( y∧ x) ∨ ( z∧ x)] = x ∧ [( z∧ x) ∨ ( y∧ x) ∨ ( z∧ x)] = { x ∧ [( z∧ x) ∨ ( y∧ x)]} ∨ [ x ∧ ( z∧ x)] = { x ∧ [( y∧ x) ∨ ( z∧ x) ∨ ( y∧ x)]} ∨ [ x ∧ ( z∧ x)] = { x ∧ [( y∧ x) ∨ ( z∧ x)]} ∨ [ x ∧ ( y∧ x)] ∨ [ x ∧ ( z∧ x)] = [ x ∧ ( y∧ x)] ∨ [ x ∧ ( z∧ x)] = ( x ∧ y) ∨ ( x∧ z). Linear distributivity implies the second and fourth equalities, since e.g., x ≻ ( z∧ x)∨( y∧ x) ≻ ( z∧ x). The fifth equality follows upon observing that x ∧ [( y∧ x)∨( z∧ x)] and ( x∧ y∧ x) ∨ ( x∧ z∧ x) are L-related, since they are equal in any lattice, and in particular in S/D = S/L. Thus (5.5.3L) holds. Similarly (5.5.3R) holds in the right-handed linearly distributive case. Again the embedding of S into S/R × S/L guarantees that all linearly distributive skew lattices satisfy (5.5.3). Thus linear distributivity is characterized by (5.5.3). 189 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Since any totally pre-ordered context is symmetric, (5.2.1) and (5.2.2) are equivalent in such contexts. Thus linear distributivity is also characterized by (5.5.4) by the dual argument. £ There is more to linear distributivity than just what occurs in totally preordered contexts. Indeed (5.5.3) implies that for each a in a skew lattice S, the map x |→ a∧ x∧ a defines a homomorphic retraction of the principal ideal S∧ a∧S onto the set a∧S∧ a of all x ≤ a. The identity states directly that this map preserves joins, with meets preserved due to regularity. Likewise, (5.5.4) implies that x |→ a∨ x∨ a defines a homomorphic retraction of the principal filter S∨ a∨S onto the set a∨S∨ a of all x ≥ a. Thus all three aspects of distributivity are equivalent for any skew lattice. There is more. While skew diamonds need not be distributive, linearly distributive skew diamonds are. Corollary 5.5.6. Linearly distributive skew diamonds are distributive. Proof. Given a linearly distributive skew diamond T with D-classes J > A, B > M, we check out (5.2.1) on T. Given x ∈J, then y∧ x∧ y = y and z∧ x∧ z = z so that (5.5.3) reduces to (5.2.1). For x in M, (5.2.1) immediately reduces to x = x ∨ x. So let x be in an intermediate D-class, say say A. We consider several possible cases. 1) y D z. Here y ∨ z = z ∧ y so that regularity gives x∧( y ∨ z)∧ x = x∧( z ∧ y)∧ x = ( x∧ z∧ x) ∧ ( x∧ y∧ x) = ( x∧ y∧ x) ∨ ( x∧ z∧ x). 2) y or z is in A or J, say y. Here y ∨ z ≻ x and our equation reduces to the absorption identity x = x ∨ ( x∧ z∧ x). Thus we may assume that y and z are in distinct D-classes, but other than A and J. 3) So let, say y ∈ M and z ∈ B. Then ( y ∨ z ∨ y) D z and Case (1) gives x∧( y ∨ z)∧ x = x∧[( y ∨ z ∨ y) ∨ z]∧ x = ( x∧( y ∨ z ∨ y)∧ x) ∨ ( x ∨ z ∧ x). The corollary will follow if we can show that x∧( y ∨ z ∨ y)∧ x = x∧ y∧ x. Since x∧( y ∨ z ∨ y), y and ( y ∨ z ∨ y)∧ x are in M, x∧( y ∨ z ∨ y)∧ x = x ∧ ( y ∨ z ∨ y) ∧ ( y ∨ z ∨ y) ∧ x = [ x∧( y ∨ z ∨ y)] ∧ y ∧ [( y ∨ z ∨ y)∧ x] = x∧ y∧ x, with the equalities due respectively to ∧ being idempotenct, uvw = uw holding in any rectangular band and absorption. (5.2.2) is seen in dual manner. £ Since distributive skew lattice are both linearly distributive and quasi-distributive, a natural question is: Does linear distributivity plus quasi-distributivity imply distributivity? While true for skew diamonds, it is not true in general. Spinks’ examples (in Theorem 1.3.10) each 190 V: Further Topics in Skew Lattices satisfy just one of (5.2.1) or (5.2.2). Since totally pre-ordered subalgebras are trivially symmetric, these examples are linearly distributive. The following lattice of varieties is thus strictly ordered by inclusion. LDist  QDist   ∧ – Dist ∨ – Dist .   Dist Dist is the variety of distributive skew lattices, ∧- Dist and ∨- Dist are the varieties of skew lattices satisfying (5.2.1) and (5.2.2) respectively, while LDist and QDist are the respective varieties of linearly distributive and quasi-distributive skew lattices. While Dist is the intersection of ∧- Dist and ∨- Dist, it is unclear if LDist ∩ QDist is their join in the lattice of all skew lattice varieties. All this leads us to modify our question and ask: Does symmetry plus linear distributivity and quasi-distributivity imply distributivity? This is indeed the case. This was first shown using Prover 9, which has also shown that linearly distributivity and distributivity are equivalent for simply cancellative skew lattices. We will first justify the implication in the case of left-handed, symmetric skew lattices. Recall that each of the following identities characterizes left-handedness: x ∧ y ∧ x = x ∧ y; x ∨ y ∨ x = y ∨ x; (5.5.5) x ∧ ( y ∨ x) = x; ( x ∧ y) ∨ x = x. (5.5.6) We use them freely in what follows. Dual identities characterize the right-handed case. Lemma 5.5.7. For left handed skew lattices, the following identities hold: (1) [ x ∨ ( y ∧ x)] ∧ x = x ∨ ( y ∧ x). (5.5.7) (2) [ x ∨ ( y ∧ x)] ∧ y = y ∧ x. (5.5.8) Proof. Clearly x ∨ ( y ∧ x) D x holds for all skew lattices. Since x ∧ [ x ∨ ( y ∧ x)] = x by absorption, we get x ∨ ( y ∧ x) L x for all skew lattices, so that (5.5.7) follows in general. For (5.5.8), we have: [ x ∨ ( y ∧ x)] ∧ y = [ x ∨ ( y ∧ x)] ∧ x ∧ y ∧ x = [ x ∨ ( y ∧ x)] ∧ y ∧ x = y ∧ x. The first equality follows from (5.5.7) and left-handedness, (5.5.1L). The second and third equalities follow respectively from (5.5.7) again and absorption. £ 191 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond We continue with a further characterization of quasi-distributivity in the left-handed case. Lemma 5.5.8. A left-handed skew lattice is quasi-distributive if and only if the following identity holds: x ∧ [( y∧ x) ∨ z] = x ∧ ( y∨ z). (5.5.9) Proof. Sufficiency is clear since neither M3 nor N5 can satisfy (5.5.9). For necessity, observe first that (5.5.9) holds in any distributive lattice. Hence in a quasi-distributive skew lattice one has at least x ∧ [( y∧ x) ∨ z] D x ∧ ( y∨ z). If we can show that x ∧ [( y∧ x) ∨ z] ≤ x ∧ ( y∨ z) in the left-handed case, then equality will hold, given the rectangular context of both expressions. Left- handedness clearly gives y∧ x ≤ y since y∧ x = y∧ x∧ y. Thus ( y∧ x) ∨ z ≤ y∨ z follows since both ( y∧ x) ∨ z ∨ y ∨ z = ( y∧ x∧ y) ∨ y ∨ z = y ∨ z and y ∨ z ∨ ( y∧ x) ∨ z = y ∨ ( y∧ x) ∨ z = y ∨ z by left-handedness and absorption. Left-handedness plus the inequality gives x∧[( y∧ x)∨ z] ∧ x∧( y∨ z) = x∧[( y∧ x)∨ z] ∧ ( y∨ z) = x∧[( y∧ x)∨ z] and similarly x∧( y∨ z) ∧ x∧[( y∧ x)∨ z] = x∧[( y∧ x)∨ z]. Thus indeed x∧[( y∧ x)∨ z] ≤ x∧( y∨ z) within a D-class context and equality follows. £ Besides quasi-distributivity and linear distributivity, a third consequence of distributivity is the following pair of identities. Combined, all three consequences together guarantee that a skew lattice is distributive. Lemma 5.5.9. The following identities are respective consequence of (5.2.1) and (5.2.2): x ∧ [( y∧ x∧ y) ∨ z ∨ y ∨ z ∨ ( y∧ x∧ y)] ∧ x = x ∧ ( y ∨ z ∨ y) ∧ x. (5.5.10). x ∨ [( y∨ x∨ y) ∧ z ∧ y ∧ z ∧ ( y∨ x∨ y)] ∨ x = x ∨ ( y ∧ z ∧ y) ∨ x. (5.5.11). Their left are right-handed variants are: x∧[ y ∨ z ∨ ( y ∧ x)] = x∧( z ∨ y) and [( x ∧ y) ∨ z ∨ y]∧ x = ( y ∨ z)∧ x. (5.5.10L and R) [( x ∨ y) ∧ z ∧ y]∨ x = ( y ∧ z)∨ x and x∨ [ y ∧ z ∧ ( y ∧ x)] = x∨( z ∧ y). (5.5.11L and R) Proof. (5.2.1) plus regularity first gives x ∧ [( y∧ x∧ y)∨ z∨ y∨ z∨( y∧ x∧ y)] ∧ x = ( x∧ y∧ x) ∨ ( x∧ z∧ x) ∨ ( x∧ y∧ x) ∨ ( x∧ z∧ x) ∨ ( x∧ y∧ x) which reduces to ( x∧ y∧ x) ∨ ( x∧ z∧ x) ∨ ( x∧ y∧ x) and then x ∧ ( y∨ z∨ y) ∧ x by (5.2.1) again. The derivation of (5.5.11) from (5.2.2) and regularity is similar. £ Both (5.5.10) and (5.5.11) can be given interpretations regarding the behavior of cosets and 192 V: Further Topics in Skew Lattices coset bijections within certain configurations of D-classes. (See the final section of [Kinyon, Leech, Pita Costa].) For our purposes, however, their importance lies in the following “umbrella” result and in the fact that both are direct consequences of symmetry. Theorem 5.5.10. A quasi-distributive, linearly distributive skew lattice is ∧- distributive if and only if it satisfies (5.5.10) . Likewise, it is ∨- distributive if and only if it satisfies (5.5.11) . In general, a skew lattice is distributive if and only if it is quasi-distributive, linearly distributive and satisfies both (5.5.10) and (5.5.11). Proof. Clearly, (5.2.1) and (5.2.2) imply respectively (5.5.10) and (5.5.11). Conversely, suppose that we have a skew lattice that is quasi-distributive and linearly distributive, and also satisfies (5.5.10). In the left-handed case we have: ( x ∧ y) ∨ ( x ∧ z) = x ∧ [( y ∧ x) ∨ ( z ∧ x)] by (5.5.3L) = x ∧ [ y ∨ ( z ∧ x)] by (5.5.9) = x ∧ [( z ∧ x) ∨ y ∨ ( z ∧ x)] by left-handedness = x ∧ [ z ∨ y ∨ ( z ∧ x)] by (5.5.9) = x ∧ ( y ∨ z) by (5.5.10L). Thus (5.2.1L) holds. Likewise, the right-handed case must hold. The general case for (5.5.10) implying (5.2.1) now follows as usual. The argument equating (5.5.11) with ∨-distributivity, given that the skew lattice is both quasi-distributive and linearly distributive, is dual. £ A consequence of the above results is the following theorem. Recall that a skew lattice is lower symmetric if x∨ y = y∨ x implies x∧ y = y∧ x. Dually, a skew lattice is upper symmetric if x∧ y = y∧ x implies x∨ y = y∨ x. Recall that both types of partial symmetry were characterized respectively by (5.1.3) x∧ y∧( x∨ y) = ( y∨ x)∧ y∧ x and by (5.1.4) x∨ y∨( x∧ y) = ( y∧ x)∨ y∨ x. Theorem 5.5.11. An upper symmetric skew lattice is lower distributive if and only if it is both quasi-distributive and linearly distributive; dually, a lower symmetric skew lattice is upper distributive if and only if it is both quasi-distributive and linearly distributive. Thus a symmetric skew lattice is distributive if and only if it is both quasi-distributive and linearly distributive. Proof. The first statement follows from Lemma 5.5.9 and the following proposition. The second statement follows by duality and the third follows in the usual way from the first two. £ Proposition 5.5.12. Upper symmetric skew lattices satisfy (5.5.10). Dually, lower symmetric skew lattices satisfy (5.5.11). Proof. We need only prove the first assertion. We begin with the case of a left-handed skew lattice S, organizing the proof in this case in the following steps. i) For all x, y, z in S, x ∨ y ≥ [ x ∨ ( y∧ z)] ∧ [ z ∨ ( y∧ z)]. 193 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond To begin, set u = [ x ∨ ( y∧ z)] ∧ [ z ∨ ( y∧ z)]. Since [ z ∨ ( y∧ z)] ∧ y = y ∧ z by (5.5.8), u∧ y = [ x ∨ ( y ∧ z)] ∧ ( y ∧ z) = y ∧ z. with the second identity due to absorption. But then, y ∧ u = y ∧ u ∧ y = y ∧ y ∧ z = y ∧ z = u ∧ y so that y ∨ u = u ∨ y by upper symmetry. Thus: x ∨ y ∨ u = x ∨ ( y ∧ z) ∨ y ∨ u = x ∨ ( y ∧ z) ∨ u ∨ y = x ∨ ( y ∧ z) ∨ ([ x ∨ ( y ∧ z)] ∧ [ z ∨ ( y ∧ z)]) ∨ y = x ∨ ( y ∧ z) ∨ y = x ∨ y, where the first and fifth equalities are due to (5.5.6), the second equality is the established case of commutation, the third is replacing u by its full expression and the fourth is due to absorption. But since S is left-handed, x ∨ y ∨ u = x ∨ y establishes x ∨ y ≥ u. ii) For all x, y, z ∈ S, z ∧ [ x ∨ ( y ∧ z)] = z ∧ ( x ∨ y) ∧ [ x ∨ ( y ∧ z)]. By duality u = ( x ∨ y) ∧ u, and so z ∧ [ x ∨ ( y ∧ z)] = z ∧ [ z ∨ ( y ∧ z)] ∧ [ x ∨ ( y ∧ z)] = z ∧ [ z ∨ ( y ∧ z)] ∧ u = z ∧ [ z ∨ ( y ∧ z)] ∧ ( x ∨ y) ∧ u = z ∧ [ z ∨ ( y ∧ z)] ∧ ( x ∨ y) ∧ [ x ∨ ( y ∧ z)] = z ∧ ( x ∨ y) ∧ [ x ∨ ( y ∧ z)], using reverse absorption in the first equality, left-handedness in the second and fourth equalities, part (i) in the middle equality and absorption in the final equality. iii) For all x, y, z ∈ S, z ∧ [ x ∨ ( y ∧ x ∧ z)] = z ∧ [ x ∨ ( y ∧ x)]. Replace y with y∧ x in (ii) to get z ∧ [ x ∨ ( y ∧ x ∧ z)] = z ∧ [ x ∨ ( y ∧ x)] ∧ [ x ∨ ( y ∧ x ∧ z)] = z ∧ [ x ∨ ( y ∧ x)] ∧ x ∧ [ x ∨ ( y ∧ x ∧ z)] = z ∧ [ x ∨ ( y ∧ x)] ∧ x = z ∧ [ x ∨ ( y ∧ x)], using (5.5.7) in the second and fourth equalities and absorption in the third. iv) To conclude the left-handed case, replace x by y ∨ x in ( iii). On the left side, absorption 194 V: Further Topics in Skew Lattices gives z ∧ ( y ∨ x ∨ [ y ∧ ( y ∨ x) ∧ z]) = z ∧ [ y ∨ x ∨ ( y ∧ z)]. On the right side, absorption and (5.5.5) give z ∧ ( y ∨ x ∨ [ y ∧ ( y ∨ x)]) = z ∧ ( y ∨ x ∨ y) = z ∧ ( x ∨ y). Therefore z ∧ ( x ∨ y) = z ∧ [ y ∨ x ∨ ( y ∧ z)]. But this is just (5.5.10L) with permuted variables. The right-handed case for (5.5.10R) follows by left-right duality, and the general implication of (5.5.10) thus holds. That lower symmetry implies (5.5.11) is seen in a ∨-∧ dual fashion. £ Theorem 5.5.13. Strictly categorical skew lattices satisfy (5.5.10) and (5.511). Proof. Take (5.5.10). Both terms are D-related in all skew lattices. But in all skew lattices we also have x ≥ both terms ≥ x∧ y∧ x. The theorem follows by Theorem 5.4.8. £ Before proceeding to the next section, here are two consequences of Theorems 5.4.2 and 5.4.9. Both implications are strict. Proposition 5.5.14. All strictly categorical skew lattices are linearly distributive and all linearly distributive skew lattices are categorical. 5.6 Midpoints and distributive skew chains A skew lattice is linearly distributive if and only if each skew chain of D-classes in it is distributive. In this section we characterize distributive skew chains in terms of the natural partial order. Given a skew chain A > B > C of comparable D-classes, with a ∈ A, c ∈ C such that a > c, any element b ∈ B such that a > b > c is called a midpoint in B of a and c. We begin with several straightforward assertions. Lemma 5.6.1. Given a skew chain A > B > C, for all a ∈ A and c ∈ C with a > c: i) For all b ∈ B, both a∧( c∨ b∨ c)∧ a and c∨( a∧ b∧ a)∨ c are midpoints in B of a and c. ii) When b in B is already a midpoint of a and c, both midpoints in (i) reduce to b. iii) When A > B > C is a distributive skew chain, both midpoints in (i) agree: a > a∧( c∨ b∨ c)∧ a = c∨( a∧ b∧ a)∨ c > c. (5.6.1) Midpoints provide a key to determining the effects of (5.2.1) and (5.2.2) in this context. To proceed further, we recall several concepts. Given a skew chain A > B > C, recall that elements b and bʹ in B are AC- connected if a finite sequence b = b 0, b 1, b 2, … , bn = bʹ exists in B such that bi –A bi+1 or bi –C bi+1 for all i ≤ n – 1. AC-connectedness is a congruence on B. Its 195 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond congruence classes, the components, are thus subalgebras of B. Given a component Bʹ of B, a sub-skew chain is given by A > Bʹ > C. Since a∧( c∨ b∨ c)∧ a is the same for all b in a common C-coset and c∨( a∧ b∧ a)∨ c is the same for all b in a common A-coset, we can extend Lemma 5.6.1: Lemma 5.6.2. Given a distributive skew chain A > B > C, for any pair a > c with a ∈ A and c ∈ C, each AC -component Bʹ in B has a unique midpoint b of a and c. We next sharpen Lemma 5.5.3 as follows. Lemma 5.6.3. Let S be a categorical skew chain consisting of D-c lasses A > B > C. If S is left-handed, then (5.2.1L) holds if and only if a∧( b∨ c) = ( a∧ b)∨ c for all a ≻ b ≻ c where a > c. Dually, if S is right-handed, then (5.2.1R) holds if and only if ( c∨ b)∧ a = c∨( b∧ a) for all a ≻ b ≻ c where a > c. (These identities are left and right-handed cases of (5.6.1) above.) Proof. We consider the left-handed case. If S is indeed distributive with a, b, c as stated in the lemma, then a∧( b∨ c) = ( a∧ b)∨( a∧ c) = ( a∧ b)∨ c, since a > c. Conversely, given just a ≻ b ≻ c in the respective D-classes A > B > C, set cʹ = a∧ c. Then a > cʹ and ( a∧ b) ∨ ( a∧ c) = ( a∧ b) ∨ cʹ. Next, since c and cʹ lie in the same A-coset in C and S is categorical, both b∨ c and b∨ cʹ lie both in a common C-coset in B and in a common A-coset in B so that a∧( b∨ c) = a∧( b∨ cʹ). Hence: a∧( b∨ c) = a∧( b∨ cʹ) = ( a∧ b) ∨ cʹ = ( a∧ b) ∨ ( a∧ c) The lemma now follows from Lemma 5.5.3 and left-right duality. £ Theorem 5.6.4. Given a skew chain A > B > C, the following condition are equivalent: i) A > B > C is distributive ii) For all a ∈ A, b ∈ B and c ∈ C with a > c, a∧( c ∨ b ∨ c)∧ a = c∨( a ∧ b ∧ a)∨ c. iii) Given a ∈ A and c ∈ C with a > c, each component Bʹ of B contains a unique midpoint b of a and c. . iv) For each component Bʹ of B, A > Bʹ > C is strictly categorical. When these conditions hold, each coset bijection ϕ: A → C uniquely factors through each component Bʹ of B in that unique coset bijections ψ: A → Bʹ and χ: Bʹ → C exist such that ϕ = χψ under the usual composition of partial bijections. Proof. Clearly (i) implies (ii). Given a > c in (ii), for each element x in B, both b 1 = a∧( c∨ x∨ c)∧ a and b 2 = c∨( a∧ x∧ a)∨ c are midpoints of a and c in B Replacing x by any element in its C-coset, does not change the b 1-outcome. Likewise, replacing x by any element in its A-coset, does not change the b 2-outcome. Hence (ii) is equivalent to asserting that given a > c fixed, for all x in a common AC-component Bʹ of B, both a∧( c∨ x∨ c)∧ a and c∨( a∧ x∧ a)∨ c produce the same output b in Bʹ such that a > b > c. Conversely, for any b in Bʹ such that a > b > c we must have a∧( c∨ b∨ c)∧ a = b = c∨( a∧ b∧ a)∨ c. 196 V: Further Topics in Skew Lattices Thus (ii) and (iii) are equivalent. Their equivalence with (iv) follows from Theorem 5.4.7 above. Given (ii) – (iv), (iv) forces A > B > C to categorical, since for each component Bʹ in B, A > Bʹ > C is categorical. Denoting the skew chain by S, (ii) forces S/R and S/L to be distributive by Lemma 5.6.3 and thus S ⊆ S/R × S/L to be distributive. In light of Theorem 5.4.7, the final comment is clear. £ Given a > c as above, their midpoint b in component Bʹ depends on the interplay of the A-cosets and C-cosets within Bʹ. Indeed, given any a ∈ A, the set of images in Bʹ of a is the set a∧Bʹ∧ a = { a∧ b∧ a⎪ b ∈ Bʹ} = { b ∈ Bʹ⎪ a > b}. This set parameterizes the A-cosets in Bʹ since each possesses exactly one b such that a > b. Likewise, for c ∈ C the image set c∨Bʹ∨ c = { c∨ b∨ c⎪ b ∈ Bʹ} = { b ∈ Bʹ⎪ b > c} parameterizes all cosets of C in Bʹ. (See Theorem 2.4.1.) Both images sets are orthogonal in Bʹ in the following sense. For any a ∈ A, all images of a in Bʹ lie in a unique C- coset in Bʹ. Likewise for any c ∈ C, all images of c in Bʹ lie in a unique A- coset in Bʹ. Finally, given a > c with a ∈ A and c ∈ C, their unique midpoint b ∈ Bʹ lies jointly in the C- coset in Bʹ containing all images of a in Bʹ and in the A- coset in Bʹ containing all images of c in Bʹ. (See Theorem 5.4.6.) Of course, every b in Bʹ is the midpoint of some pair a > c. For a fixed pair a > c, the set µ( a, c) of all midpoints in B is a rectangular subalgebra that parameterizes the class of all AC-components in B: let b in µ( a, c) correspond to the component Bʹ containing b. ∗  A-coset → • b •  • • The A-coset of b contains all images ∗ ( • 's) of c in ′ B . The C-coset of b has     all images ( ∗ 's) of a in ′ B . Element b ∗ is the unique image of both a and c. ∗  C-coset ↑ Example 5.6.5. Using Mace 4, two minimal 12-element categorical skew chains have been found that are not linearly distributive, one left-handed and the other its right-handed dual. Below are the Cayley Tables in the left-handed case. Here i and j assume the values 1 and 2, and k assumes the values 3 and 4. 197 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond ∨ ∧ a a i bj bk d j dk c 1 c 2 i bj bk d j dk cj a a 1 ai aj ak–2 aj dk–2 a 1 a 1 1 a 1 b 1 b 3 d 1 d 3 cj a a 2 ai aj ak–2 aj dk–2 a 2 a 2 2 a 2 b 2 b 4 d 2 d 4 cj b b 1 ai bj bk d j dk b 1 d 3 1 b 1 b 1 b 1 b 1 b 1 c 1 b b 2 ai bj bk d j dk b 2 d 2 2 b 2 b 2 b 2 b 2 b 2 c 1 b b 3 b 3 b 3 b 3 b 3 b 3 c 1 3 ai bj bk d j dk b 3 d 1 b b . 4 b 4 b 4 b 4 b 4 b 4 c 1 4 ai bj bk d j dk b 4 d 4 d 1 d 1 d 1 d 1 d 1 d 1 c 2 d 1 ai bj bk d j dk b 3 d 1 d 2 d 2 d 2 d 2 d 2 d 2 c 2 d 2 ai bj bk d j dk b 2 d 2 d 3 d 3 d 3 d 3 d 3 d 3 c 2 d 3 ai bj bk d j dk b 1 d 3 d 4 d 4 d 4 d 4 d 4 d 4 c 2 d 4 ai bj bk d j dk b 4 d 4 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 ai bj bk d j dk c 1 c 2 c 2 c 2 c 2 c 2 c 2 c 2 c 2 c 2 ai bj bk d j dk c 1 c 2 Their common Hasse diagram is as follows, with bi –C dj iff i + j ≡ 0 (mod 4). a 1 – a 2 A          B b 1 – b b d d A 2 – b 3 – A 4 – d 1 – A 2 – d 3 – A 4          C c 1 – c 2 In both cases, a 1 > bodd, dodd and a 2 > beven, deven, all bi > c 1, all di > c 2, and a 1, a 2 > both c 1, c 2. Thus both skew chains are categorical since all cosets involving just A and C are trivial. We denote the left-handed skew lattice thus determined by U and its right-handed dual by V. Both U and V are not distributive. Indeed, given the coset structure on B, we get a 1 ∧ ( b 2 ∨ c 2) = a 1 ∧ d 2 = d 1, while ( a 1 ∧ b 2) ∨ ( a 1 ∧ c 2) = b 1 ∨ c 2 = d 3 ≠ d 1 in U. V is handled similarly. Note that in both U and V, B is a AC-connected, but a 1 > b 1, b 3 > c 1, and also a 2 > b 2, b 4 > c 1, etc. This example is the n = 2 case of a sequence of similar skew chains A > B n > C, where A = { a 1, a 2} and C = { c 1, c 2} as above, but B n = { b 1, b 2, b 3, … , b 2 n} ∪ { d 1, d 2, d 3, … , d 2 n} for n finite or{… , b–2, b–1, b 0, b 1, b 2, …} ∪ {… , d–2, d–1, d 0, d 1, d 2, …} for n = ω. In all cases, a 1 > b odd, d odd; a 2 > b even, d even; all bi > c 1; all di > c 2; and a 1, a 2 > both c 1, c 2. (That a 1, a 2 > both c 1, c 2 insures that these skew chains are categorical since all “outer” cosets involving just A and C are trivial.) An important aspect of these examples is the fact that all A-cosets and C-cosets in B n are of size 2 and are interconnected in the following cyclic fashion in the finite case, 198 V: Further Topics in Skew Lattices b b d d b b d d b  b b d d ( b 1 –A 2 –C 2 –A 1 –C 3 –A 4 –C 4 –A 3 –C 5 2 n–1 – A 2 n – C 2 n – A 2 n–1 – C 1) or for n = ω,  d b b d d b b d d b b d  . –3 – C –1 – A 0 – C 0 – A –1 – C 1 – A 2 – C 2 – A 1 – C 3 – A 4 – C 4 We denote the left-handed skew lattice thus determined by U n and its right-handed dual by V n. Our example is U2. All U n and V n for n ≥ 2 are nondistributive since A-cosets and C-cosets in the single component B n for n ≥ 2 need not have nonempty intersection. Must a categorical skew lattice containing no copies of U n or V n, be linearly distributive? Here is a counterexample. Example 5.6.6: The underlying set of S is U2 × &2. ( a 1, 0) – ( a 1,1) – ( a 2, 0) – ( a 2,1)     ...     ( b 1, 0) – ( b ( b ( b ( d ( d ( d A 1,1) – A 2 , 0) – A 2 ,1) – ... – ( d 3, 0) – A 3,1) – A 4 , 0) – A 4 ,1)     ...     ( c 1, 0) – ( c 1,1) – ( c 2, 0) – ( c 2,1) All cosets are just cartesian products of U2-cosets with &2. The coset bijections are as follows. 1) ( a 1, 0) – ( a 1,1) – ( a 2, 0) – ( a 2,1) → ( b 1, 0) – ( b 1,1) – ( b 2, 0) – ( b 2,1) 2) ( a 1, 0) – ( a 1,1) – ( a 2, 0) – ( a 2,1) → ( b 3, 0) – ( b 3,1) – ( b 4, 0) – ( b 4,1) 3) ( a 1, 0) – ( a 1,1) – ( a 2, 0) – ( a 2,1) → ( d 1, 0) – ( d 1,1) – ( d 2, 0) – ( d 2,1) 4) ( a 1, 0) – ( a 1,1) – ( a 2, 0) – ( a 2,1) → ( d 3, 0) – ( d 3,1) – ( d 4, 0) – ( d 4,1) 5) ( b 1, 0) – ( b 1,1) – ( d 3, 0) – ( d 3,1) → ( c 1, 0) – ( c 1,1) – ( c 2, 0) – ( c 2,1) 6) ( b 2, 0) – ( b 2,1) – ( d 2, 0) – ( d 2,1) → ( c 1, 0) – ( c 1,1) – ( c 2, 1) – ( c 2, 0) 7) ( b 3, 0) – ( b 3,1) – ( d 1, 0) – ( d 1,1) → ( c 1, 0) – ( c 1,1) – ( c 2, 0) – ( c 2,1) 8) ( b 4, 0) – ( b 4,1) – ( d 4, 0) – ( d 4,1) → ( c 1, 0) – ( c 1,1) – ( c 2, 1) – ( c 2, 0) For each bijection, the outcomes occur in the same order as the inputs are listed. These coset bijections collectively determine operations ∨ and ∧ on S making S a left-handed categorical 199 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond skew chain that is not distributive since its homomorphic image U2 is not distributive. The twisted outcomes for (6) and (8) prevent it from containing copies of any of the U n for n ≥ 2. 5.7 Counting theorems and cancellative skew lattices In this chapter we have thus far revisited properties that skew lattices might possess. We began by further studying symmetric skew lattices and then introducing strongly symmetric skew lattices. We then gave Karin Cvetko-Vah’s proof of Spinks’ Theorem on the equivalence of the distributive identities (5.2.1) and (5.2.2) for symmetric skew lattices. Symmetry played an important role in our analysis of cancellation in skew lattices. We saw that simply cancellative skew diamonds {J > A. B > M} are characterized by a naturally defined bijection ξ: Comm 2(A, B) → ω(J, M), given by the function ξ( a, b) = ( a∨ b, a∧ b), between the set Comm 2(A, B) of all pairs ( a, b) in A×B for which both a∨ b = b∨ a and a∧ b = b∧ a, and the set ω(J, M) of all pairs ( j, m) in J×M for which j > m. This is, of course, an instance of the type of counting theorem, where two related but distinct sets necessarily have the same size. Other counting theorems exist for (fully) cancellative – and hence symmetric – skew lattices. But first we recall some definitions: Given a primitive skew lattice A > B. Recall that for any b ∈B, its image set in A is the set b∨A∨ b = { b∨ a∨ b⎮ a ∈ A} which also coincides with { a ∈ A⎮ a ≥ b}. Dually, for any a ∈ A, its image set in B is a∧B∧ a = { a∧ b∧ a⎮ b ∈ B} which also coincides with { b ∈ B⎪ a ≥ b}. Both image sets are rectangular subalgebras. Recall from Section 2.4 that: i) Image sets of all elements from one D-class have the same size in its opposite D-class. E.g., given b, bʹ in B, | b∨A∨ b| = | bʹ∨A∨ bʹ|. ii) Image sets naturally parameterize the cosets of either class in the other. Thus b∨A∨ b is a cross-section of all B-cosets in A while a∧B∧ a is a cross-section of all A-cosets in B. The index of B in A, denoted [A: B], is the size | b∨A∨ b| that counts the number of B-cosets in A. Likewise, the index of A in B, denoted [B: A], is the size | a∧B∧ a| that counts counts the number of A-cosets in B. In general, no direct relationship need exist between [A: B] and [B: A]. The common size of all A-cosets in B and all B-cosets in A is denoted by ωAB, or equivalently, ωBA. Clearly the following basic equalities hold: |A| = [A: B]ωAB and |B| = [B: A]ωAB. (5.7.1) Given a skew diamond {J > A, B > M} one has five pairs of indices. If the skew diamond is cancellative, one has the following “Index Laws” that connect opposite pairs of indices. They are from Karin Cvetko-Vah’s Dissertation [Ref.]. (See also CKLS.) Theorem 5.7.1. Let S be a cancellative skew diamond {J > A, B > M}. Then: 200 V: Further Topics in Skew Lattices [M: A] = [B: J] and [A: M] = [J: B]. (5.7.2) [M: B] = [A: J] and [B: M] = [J: A]. (5.7.3) In detail, given j ∈ J, a ∈ A, b ∈ B and m ∈ M such that j > a, b > m, isomorphisms α: { x ∈ A⎪ x > m} ≅ { y ∈ J⎪ y > b} and β:{ u ∈ B⎪ u < j} ≅ { v ∈ M⎪ v < a} are defined by α( x) = x∨ b∨ x and β( u) = u∧ a∧ u. Isomorphisms between other pairs of image sets are defined similarly. Conversely, if all such maps in a skew diamond are isomorphisms, then it is cancellative. Proof. Since b > m and S is symmetric, b commutes under both ∧ and ∨ with all x ∈ A such that x > m so that α is well-defined. Since S is cancellative, α is also one-to-one. Lemma 2.1.4 implies α( x ∨ xʹ) = ( x ∨ xʹ) ∨ b ∨ ( x ∨ xʹ) = ( x ∨ b ∨ x) ∨ ( xʹ ∨ b ∨ xʹ) = α( x) ∨ α( xʹ). Thus α is a ∨-homomorphism that must be a ∧-homomorphism since index sets are rectangular subalgebras. To show it is onto and thus an isomorphism, let y > b be given in J and set c = y∧ a∧ y in A. Then m < c < y and y = α( c) by Theorem 2.2.1. The verification that β is an isomorphism is similar. Conversely, given a skew diamond, if all such maps are at least bijections, copies of R L R,0 R,1 L,0 L,1 NC , and the algebras , , and cannot occur as subalgebras. 5 NC5 NS7 NS7 NS7 NS7 Since any skew diamond is trivially quasi-distributive, it must be cancellative. £ What else can be said? First, observe: Theorem 5.7.2. A skew diamond is simply cancellative if and only if it is strictly categorical, in which case it is also distributive. It is cancellative if and only if it is symmetric and strictly categorical. Proof. A skew diamond is already quasi-distributive. To be simply cancellative it needs to exclude both NC5 subalgebras, and to be strictly categorical it needs to exclude both 4-element skew chains in Theorem 5.4.8(iv). But both constraints are equivalent for skew diamonds. Theorem 5.4.10 insures the addendum of distributivity. £ The following two results from João Pita da Costa’s dissertation are relevant. Theorem 5.7.3 Given a strictly categorical skew chain A > B > C, if A and C are finite, then so is B and ω ω |B| = AB BC = [B : A] ω ω AC [B : C]. AC In general, a strictly categorical skew lattice S is finite if and only if its maximal lattice image S/D is finite and both the maximal and minimal D- classes in S are finite. 201 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. B is partitioned by A-cosets, each of size ωAB, and also by C-cosets, each of size ωBC. Thus there is a double partition by coset intersections of the form X∩Y where X is an A-coset in B and Y is a C-coset in B. Since A > B > C is strictly categorical, each |X∩Y| = ωAC. • • • •  • • • • • •  • •     • • • •  • • Thus if A and C are finite, then so are ωAB, ωBC and ωAC giving ωBC/ωAC many A-cosets in B, ωAB/ωAC many C-cosets in B and thus (ωBC/ωAC)ωAC(ωAB/ωAC) = ωABωBC/ωAC < ∞ elements in B. One has |B| = [B:A]ωAC[B:C] also since this double partition has [B:A][B:C] coset intersections, all of size ωAC. Given a strictly categorical skew lattice S, the condition is clearly necessary for S to be finite. Conversely given that its maximal and minimal D-classes are finite, so are all intermediate D-classes. If there are only finitely many of them, then S is finite. £ Theorem 5.7.4. Given any skew chain A > B > C, [C: A] ≤ [C: B][B: A]. If the skew chain is strictly categorical and both A and C are finite, then [C: A] = [C: B] [B: A]. In general, given any skew chain A1 > A2 > ... > A n in a in a strictly categorical skew lattice S, if A1 and A n are finite then so are all intermediate D -classes and [A1: A n] = [A1: A2] [A2: A3] … [A n–1: A n]. Proof. The general inequality is a consequence of the fact that given a > c with a ∈ A and c ∈ C, there exists a b ∈ B such that a > b > c. Hence given a has [B: A] images in B, each of which has [C: B] images in C, so that a has at most [C: B][B: A] images in C. Assuming S is also strictly categorical, then B is finite. The following equalities thus hold and with them the first half of the theorem: ω |B| ω ω 1 ω [A : C] = |A| = [A : B] AB and [B:C] = = AB BC = AB . ω ω ω ω ω ω AC AC BC AC BC AC Given the chain A1 > A2 > ... > A n, the factorization of [A1:A n] proceeds from the special case: 202 V: Further Topics in Skew Lattices [A1: A n] = [A1: A2][A2: A n] = [A1: A2][A2: A3][A3: A n] =…= [A1: A2][A2: A3]…[A n–1: A n]. £ Returning to skew diamonds, the following restatement of Theorem 2.4.10 parallels to some extent the theorem above: Theorem 5.7.5. In a symmetric skew diamond {J > A, B > M} , [M: J] = [M: A] [M: B] and [J: M] = [J: A][J : B] . £ Lemma 5.7.6. Given finite D- classes X > Y in a skew lattice, |X|[Y: X] = |Y|[X: Y], or put |Y| otherwise, [Y: X] = [X: Y] . |X| Proof. |X|[Y: X] and |Y|[X: Y] expand to [X: Y] ωXY [Y: X] and to [Y: X] ωXY [X: Y]. £ This next result of relevance is from Pita da Costa’s dissertation. Theorem 5.7.7. Given a finite cancellative skew diamond {J > A, B > M}, |A| |B| = |J| |M|. |A| [A : J] [M : B] Proof. One has = = = |M| . The first and third equalities come from the |J| [J : A] [B : M] |B| previous lemma. The middle equality is from [B: M] = [J: A] and [M: B] = [A: J] in Theorem 5.7.1. The equality now follows by cross-multiplying. £ This outcome fails in the four simply cancellative NS variants and both symmetric NC 7 5 variants. (But see Corollary 2.4.11.) One thus has: Corollary 5.7.8. A skew lattice S is cancellative if and only if it is quasi-distributive and all of its skew diamonds are cancellative. The latter occurs if and only if |A| |B| = |J| |M| holds in all finite skew diamonds {J > A, B > M} of S. £ This situation is sharpened if the skew diamond {J > A, B > M} is pointed in that |J| = 1 or |M| = 1. If J has a unique element, it is often denoted by 1; if it is M, the single point is often denoted by 0. L,0 R,0 L,1 R,1 L R NS , , and are pointed, while and are 7 NS7 NS7 NS7 NC5 NC5 doubly pointed. Indeed, a skew diamond is simply cancellative if and only if all doubly pointed skew diamond subalgebras are sublattices (thus eliminating any possible NC5 subalgebras). What about full cancellation? The next two Theorems are from the 2011 paper of Cvetko-Vah, Kinyon, Leech and Spinks. Theorem 5.7.9. A quasi-distributive skew lattice is cancellative if and only if all pointed skew diamonds in it factor as products of primitive skew lattices. Proof. Given a skew lattice S, the condition on pointed skew diamonds in it excludes copies of L R L,0 R,0 L,1 R,1 NC , , , , and from being subalgebras since in each of 5 NC5 NS7 NS7 NS7 NS7 these six cases the order of the join or meet classes is inconsistent with such a factorization. This 203 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond insures that S is cancellative. Conversely, given a cancellative skew lattice, we must show that all pointed skew diamonds in it factor as stated. Our task quickly reduces to showing that a cancellative pointed skew lattice S factors as stated. So let S = {J > A, B > M} be such a skew diamond with say M = {0}. A pair of primitive subalgebras are A0 = A ∪ {0} and B0 = B ∪ {0}. Claim: An isomorphism σ: A0 × B0 ≅ S is given by σ[( x, y)] = x∨ y for all x ∈ A0 and y ∈ B0. Thanks to Theorem 2.2.1, σ is easily seen to be surjective. It is clearly bijective from {0}×{0} to {0}, from A×{0} to A and from {0}×B to B. Thus σ is bijective overall if it is bijective from A×B to J. Since S is symmetric, a∨ b = b∨ a for each pair ( a, b) in A × B since a∧ b = 0 = b∧ a. Thus the bijectivity of A×B with J is given by Theorem 5.3.10. Finally for all x 1, x 2 ∈A0 and y 1, y 2 ∈ B0, σ[( x 1, y 1)∨( x 2, y 2)] = σ[( x 1∨ x 2, y 1∨ y 2)] = x 1∨ x 2 ∨ y 1∨ y 2 = x 1∨ y 1 ∨ x 2∨ y 2 = σ[( x 1, y 1)] ∨ σ[( x 2, y 2)], since elements from A0 commute with elements from B0. Expanding σ[( x 1, y 1)∧( x 2, y 2)] and σ[( x 1, y 1)] ∧ σ[( x 2, y 2)], we get respectively: ( x 1∧ x 2) ∨ ( y 1∧ y 2) and ( x 1∨ y 1) ∧ ( x 2∨ y 2). Case 1) One of the xi and one of the yj is 0. Here both expressions above reduce to 0. Case 2) Neither xi is 0 but one of the yj is 0. We get x 1∧ x 2 on the left and on the right either ( x 1∨ y 1) ∧ x 2 or x 1 ∧ ( x 2∨ y 2). Since the xi and yj commute, normality plus absorption gives, ( x 1∨ y 1)∧ x 2 = ( y 1∨ x 1)∧ x 2∧ x 1∧ x 2 = ( y 1∨ x 1)∧ x 1∧ x 2 = x 1 ∧ x 2 and x 1∧( x 2∨ y 2) = x 1∧ x 2∧ x 1∧( x 2∨ y 2) = x 1∧ x 2∧( x 2∨ y 2) = x 1 ∧ x 2. Case 3) One of the xi is 0, but neither of the yj is 0. This is similar to Case 2. Case 4) None of the xi or y j are 0. By x-y commutation plus u∧ v = v∨ u on D-classes, both ( x 1∧ x 2)∨( y 1∧ y 2) and ( x 1∨ y 1)∧( x 2∨ y 2) are easily seen to reduce to x 2∨ y 2∨ x 1∨ y 1. Thus σ is an isomorphism. The case where J = {1} follows by the dual argument. £ There is more. In general, every skew diamond S (cancellative or not) is a union of maximal pointed subalgebras in two ways. S = ∪ m∈M m∨S∨ m = ∪ j∈J j∧S∧ j where m∨S∨ m = { m∨ x∨ m⎪ x ∈S} = { x ∈S⎪ x ≥ m} is pointed below with zero m, and j∧S∧ j = { j∧ x∧ j⎪ x ∈S} = { x ∈S⎪ j ≥ x} is pointed above with identity 1, Given m, mʹ ∈M, define f: m∨S∨ m → mʹ∨S∨ mʹ by f( x) = mʹ∨ x∨ mʹ and g: mʹ∨S∨ mʹ → m∨S∨ m by g( x) = m∨ x∨ m. Regularity and the fact x∨ y∨ x = x holds on M imply that f and g are a reciprocal bijections. Again regularity gives mʹ∨( x∨ y)∨ mʹ = ( mʹ∨ x∨ mʹ)∨( mʹ∨ y∨ mʹ). Thus f and 204 V: Further Topics in Skew Lattices g are reciprocal ∨-isomorphisms that must restrict to isomorphisms between corresponding D-classes (since x∧ y = y∨ x on D-classes), giving, e.g., m∨A∨ m ≅ mʹ∨A∨ mʹ. Thus f induces isomorphisms between pointed primitive algebras, m∨A∨ m ∪ { m} ≅ mʹ∨A∨ mʹ ∪ { mʹ} and m∨B∨ m ∪ { m} ≅ mʹ∨B∨ mʹ ∪ { mʹ}. If S is also cancellative, then the previous theorem gives m∨S∨ m ≅ mʹ∨S∨ mʹ. Theorem 5.7.10. If {J > A, B > M} is a cancellative skew diamond, then all pointed skew diamonds m∨S∨ m ⊆ S for m ∈M, are isomorphic. Dually, all pointed skew diamonds j∧S∧ j ⊆ S for j ∈J are isomorphic. £ Summing up much of the discussion about cancallative skew lattices we have: Theorem 5.7.11. For a skew lattice S the following are equivalent: i) S is cancellative. ii) S is quasi-distributive and all [finite]skew diamonds in S are cancellative. iii) S is quasi-distributive and symmetric, with all [finite]skew diamonds in it being strictly categorical. iv) S is quasi-distributive and all [finite]pointed skew diamonds in S factor as a product of two primitive skew lattices. v) S is quasi-distributive and |A||B| = |J||M| in any finite skew diamond {J > A, B > M} in S. While neither the classes of distributive skew lattices or cancellative skew lattices includes the other, all skew diamonds in cancellative skew lattices are distributive (being strictly categorical), and all skew chains in distributive skew lattices are cancellative (being true in general). Historical remarks The results in Section 5.1, on symmetry come from [Cvetko-Vah, Kinyon, Leech and Spinks, 2011]. The results on comparing distributive identities in Section 2 are due to [Spinks, 1998 and 2000] and [Cvetko-Vah, 2006]. The material in Section 3 on cancellation is mostly from [Cvetko-Vah, Kinyon et al, 2011] again, while the results in Section 4 on categorical behavior are from [Kinyon and Leech, 2013]. The material in Sections 5 and 6 on distributivity and its consquences comes from [Kinyon, Leech and Pita Costa, 2014?]. The various counting results in the final section are from the dissertations of Pita Costa [2012] and Cvetko-Vah [2005] as well as [Cvetko-Vah, Kinyon et al, 2011]. 205 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond References K. Cvetko-Vah, Skew lattices in rings, University of Ljubljana Dissertation, 2005. A new proof of Spinks’ Theorem, Semigroup Forum 76 (2006), 267 – 272. On strongly symmetric skew lattices, Algebra Universalis 66 (2011), 99-113. K. Cvetko-Vah, M. Kinyon, J. Leech, and M. Spinks. Cancellation in skew lattices, Order 28 (2011), 9 – 32. M. Kinyon and J. Leech, Categorical skew lattices, Order 30 (2013), 763-777. M. Kinyon, J. Leech and J. Pita Costa, Distributivity in skew lattices, Semigroup Forum 91 (2015), 378 - 400. J. Pita Costa, Coset laws for categorical skew lattices, Algebra Universalis 68 (2012), 75–89. On the coset structure of skew lattices, University of Ljubljana Dissertation, 2012. M. Spinks, Automated deduction in non-commutative lattice theory, Tech. Report 3/98, Monash University, Gippsland, June 1998. On middle distributivity for skew lattices, Semigroup Forum 61 (2000), 341-345. 206 VI: SKEW LATTICES IN RINGS From the initial research into skew lattices in the 1980s, skew lattices of idempotents in rings and their particular examples have provided fundamental ideas about the subject. The absorption identities came from observing that nonempty sets of idempotents in a ring that were closed under both multiplication and the circle operation ( x ○ y = x + y – xy) satisfied them. The significance of the distributive identities a∧( b ∨ c)∧ a = ( a∧ b∧ a) ∨ ( a∧ c∧ a) and its dual was due to the fact they hold for all skew lattices in rings, whereas say a∧( b ∨ c) = ( a∧ b) ∨ ( a∧ c) need not hold. Occurrences of being symmetric, cancellative or categorical were observed first in the ring context. Maximal left-regular (right-regular) multiplicative bands turned out to be maximal left- handed (right-handed) skew lattices under • and ○; and maximal normal bands formed skew Boolean algebras in their host rings (with ○ often replaced by ∇). In this chapter we look more closely at skew lattices (of idempotents) in rings. In Section 6.1 left- and right-handed skew lattice extensions of a lattice of idempotents S 0 in a ring are introduced. These are the uniquely largest left-handed and right-handed skew lattices containing S 0 as a lattice section. More generally, quadratic skew lattices (with join ○) are studied. Theorems about the center Z( S) of a quadratic skew lattice S and decompositions of S related to its center are given. Attention is given to what occurs in matrix rings over fields. Section 6.2 looks at ∇-bands, that is, multiplicative bands of idempotents that are closed also under the cubic join ∇ where x∇ y = ( x ○ y)2 = x + y + yx – xyx – yxy. Even when ∇ is not associative, ∇-bands share many properties of quadratic skew lattices. (Theorem 6.2.2.) ∇ is thus a nocommutative join. When it is associative, the ∇-band is a cubic skew lattice that is necessarily distributive and cancellative, and hence categorical and symmetric. (Quadratic skew lattices are seen as being trivially cubic.) We consider various criteria given by Cvetko-Vah and Leech [2007] for the ∇-operation in ∇-bands to be associative. These include the following: the L and R congruences relative to multiplication are also ∇-congruences; ∇ is associative on every primitive subalgebra of comparable D-classes A > B in the band. In particular, normal ∇-bands where multiplication is normal, are seen (again) to be normal skew lattices. For many ∇-bands S where ∇ is not associative, a closely related associative join ∨ exists making (S; ∨, •) a skew lattice. (See Theorems 6.2.12 and 6.2.13 and the preceding discussion.) This holds for ∇-bands in finite dimensional algebras over fields (in the ring-theoretic sense of “algebra”). While maximal left [right] regular bands and normal bands in a ring form skew lattices under • and ∇, maximal regular bands in rings need not form skew lattices or even ∇-bands. Equivalently, a regular band in a ring need not generate a ∇-band under • and ∇. In Section 6.3 we consider ∇-inductive conditions. These are conditions which guarantee that a regular band satisfying them will generate at least a ∇-band in the host ring. (See Theorems 6.3.1 and 6.3.2.) Being normal or being left [right] regular are cases of ∇-inductive conditions. We conclude with Cvetko-Vah’s nice theorem (6.3.5) stating that a regular band in a ring with totally ordered D- 207 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond classes must generate a ∇-band. In particular, maximal totally pre-ordered regular bands in rings are ∇-bands. Not only does a maximal normal band S form a skew Boolean algebra in its host ring (with e \ f = e – efe), it is also the full set of idempotents in the subring Rʹ that it generates: S = E( Rʹ). Conversely, if E( R) is multiplicatively closed in a given ring R, then E( R) forms a skew Boolean algebra. (See Theorem 2.3.7 or Theorem 6.4.2 below.) In Section 4 we begin our study of such idempotent-closed rings. Much of the focus is on the case of idempotent-dominated rings where R = Q( R), the ideal generated from E( R). In this case, if K R is the canonical nilpotent ideal { k ∈ R⎪ xky = 0 for all x, y ∈ R}, then R/K R is the maximal abelian image of R. (See Theorem 6.4.10. Recall that R is abelian if its idempotents commute.) When E( R) also has a lattice section, R is a semidirect sum A ⊕ K R, that is direct under addition with A being a maximal abelian subring of R that is necessarily isomorphic to R/K R. (Theorem 6.4.11.) This is the case for all idempotent-closed and dominated rings of n× n matrices. When R is not idempotent-dominated, these facts apply directly to Q( R); but upon setting K = K Q( R), then K is a nilpotent ideal of R also, with R being idempotent-closed if and only if R/K is abelian. (Theorem 6.4.15.) These and related facts are studied in the fourth section. Like Boolean algebras, skew Boolean algebras decompose almost at will. (See Theorem 4.1.4 or Theorem 6.5.1 below.) To what extent does this extend to idempotent-closed rings, especially if they are idempotent-dominated? In particular, given certain finiteness conditions (e.g. the ACC or DCC on idempotents), must ring decompose as a direct sum of subrings that in some sense are “atomic”? These questions are pursued for idempotent-dominated rings in the fifth section. The “atomic” rings turn out to be rectangular rings – idempotent-dominated rings whose non-0 idempotents form a rectangular band under multiplication. Theorem 6.5.6 states that each idempotent-closed and dominated ring R satisfying the DCC on idempotents is an orthosum of ideals Qi (that is, R = ∑ Qi with QiQj = {0} for i ≠ j) where each Qi is a rectangular ring. While the orthosum condition is a weakening of the direct sum condition, if the annihilator ideal of R reduces to {0}, the sum must be direct. Rectangular rings are characterized in Theorems 6.5.11 and 6.5.14. The results of Sections 6.4 and 6.5 are then “tested” in the context of matrix rings over fields in Sections 6.6 and 6.7. The former studies upper triangular representations of normal skew lattices and skew Boolean algebras in matrix rings, and Section 6.7 studies upper triangular representations of (maximal) idempotent-closed and dominated subrings of matrix rings. (See Theorems 6.7.4 – 6.7.6.) The chapter ends with historical comments and relevant references. 208 VI: Skew Lattices in Rings 6.1 Quadratic skew lattices in rings Recall that a quadratic skew lattice in a ring R is any multiplicative band in R that is also closed under the circle operation: x○ y = x + y – xy. Letting • denote multiplication, by Theorem 2.1.7 such a band S satisfies the following absorption identities that guarantee that (S, ○, •) is indeed a skew lattice: x • ( x ○ y) = x = ( y ○ x) • x. x ○ ( x• y) = x = ( y• x) ○ x. In particular both (S, ○) and (S, •) are regular bands. We typically identify ○ as the join ∨ and • as the meet ∧. Joins are generally higher than their constituent elements and this is certainly the case here. In particular, in matrix rings rank( e • f) ≤ rank( e), rank( f) ≤ rank( e ○ f). By Theorem 2.1.9 every maximal right [left] regular multiplicative band S in a ring is closed under ○, making (S, ○, •) is a maximal right-handed quadratic skew lattice in the ring. Our first result is the dual of this theorem. Theorem 6.1.1. Let R be a ring and let S be a subset of R forming a left regular band in R under the circle operation. If S is a maximal such ○- band in S, then S is also closed under multiplication and forms a maximal right-handed quadratic skew lattice in S. Proof. First suppose that R has an identity 1. Then γ( x) = 1 – x induces a bijection on E(R) such that γ( xy) = γ( x)○γ( y) and γ( x○ y) = γ( x)γ( y) regardless of the outcomes also being idempotent. Hence, if S is a maximal left regular band in R under ○, γ[S] must be a maximal left regular multiplicative band in R and thus along with ○ forms a maximal left-handed skew lattice in R. Clearly S = γγ[S] is indeed a maximal right-handed skew lattice in R. If R does not have an identity, then it can be embedded in a ring Rʹ with identity 1. In Rʹ we extend S to a maximal left regular ○-band Sʹ that forms a maximal right-handed skew lattice in Rʹ. Hence S itself must generate a right-handed skew lattice in R. But given maximal status of S in R, it is this skew lattice. £ Our main emphasis in this section is with classes of quadratic skew lattices. To begin, recall that for any e ∈E(R), its R- set R e = e + e R(1 – e) is the maximal right-zero semigroup in R containing e. Since x○ y = yx on R e, R e is a maximal right-rectangular skew lattice in R. Recall also that if e > f in E(R), then R e ∪ R f forms a maximal right-primitive skew lattice in R. This has an immediate generalization. 209 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 6.1.2. Given any naturally totally ordered set of idempotents T in a ring R, S =  R e∈T e is a band that together with ○ forms a right-handed skew lattice in R that is maximal with respect to being right-handed and containing T as a lattice section. £ This result is illustrated by the following chain T of length 4 and its induced right-handed skew lattice S in M4(/). The asterisks denote free variable positions in the matrices. I I ⎡1 0 0 0⎤ ⎡1 0 0 ∗⎤ ⎢0 1 0 0⎥ ⎢ ⎥ ⎢ ⎥ 0 1 0 ∗ ⎢ ⎥ 0 0 1 0 ⎢ ⎥ 0 0 1 ∗ ⎢ ⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎡1 0 0 0⎤ ⎡1 0 ∗ ∗⎤ ⎢ ⎥ ⎢ ⎥ T: 0 1 0 0 ⎢ ⎥ 0 1 ∗ ∗ → S: ⎢ ⎥ 0 0 0 0 ⎢ ⎥ 0 0 0 0 ⎢ ⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎡1 0 0 0⎤ ⎡1 ∗ ∗ ∗⎤ ⎢0 0 0 0⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 0 ⎢ ⎥ 0 0 0 0 ⎢ ⎥ 0 0 0 0 ⎢ ⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥ 0 0 Finding right-handed skew chains (skew lattices whose D-classes are totally ordered) is thus comparatively easy. We turn our attention to the general case where the D-classes need not be totally ordered. A simple strategy for finding right-handed skew lattices in rings is as follows: (1) Find a lattice T in a ring with ∨ = ○ and ∧ = • . (2) Consider the union  R e + e R(1 – e). e∈T e =  e∈T (3) Search for skew lattices S ⊆  R e ∈ T e containing T as a lattice section. This leads us to the following fundamental result: 210 VI: Skew Lattices in Rings Theorem 6.1.3. Given the lattice T as above, for each e ∈ T set S e = { eʹ ∈ R e⎮ ∀ f ∈T, eʹ f ∈ R ef }. Then S =  S e∈T e is the unique maximal right-handed skew lattice in R having section T. (S is called the right extension of T and each S e is called the S-set of e relative to T.) Proof. Given a right-handed skew lattice Sʹ in E(R) having T as a lattice section, clearly Sʹ ⊆ S. We need only show that S is closed under both skew lattice operations and thus is a right-handed skew lattice. To begin, given eʹ ∈ S e and fʹ ∈ S f, eʹ fʹ = eʹ ffʹ ∈ R ef since eʹ f ∈ R ef by definition of S e and thus ( eʹ f) fʹ ∈ R ef also. Next, given g ∈ T, fʹ g ∈ R fg with fʹ g = fgfʹ g. Thus ( eʹ fʹ) g = eʹ( fʹ g) = eʹ( fgfʹ g) = ( eʹ fg) fʹ g ∈ R efg because eʹ fg ∈R efg and R fg ∪ R efg is a right primitive skew lattice. Hence eʹ fʹ ∈ S ef and S is at least a right regular band under multiplication. As such S generates a right-handed skew lattice Sʹ. But since R is the D-congruence on Sʹ, T must be a lattice section on Sʹ forcing Sʹ ⊆ S, that is Sʹ = S. £ Examples 6.1.1 from M4(F) for any field F. The lattice section in each case is the lattice T of all diagonal matrices in the skew lattice, one from each D-class. Notice that any D- class that is comparable to all other classes is a full R- set. This is true for all right extensions of lattices in rings. Incomparable pairs of D-classes, however, in some way create “interference” with each other. Both classes are properly less than full R-sets. That this is always the situation for matrix rings over fields is justified later. 211 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond I I     ⎡ 1 0 0 0 ⎤ ⎡ 0 0 0 0 ⎤ ⎡ 1 0 0 0 ⎤ ⎡ 0 0 0 0 ⎤ ⎢ 0 1 0 ∗ ⎥ ⎢ ∗ ⎥ ⎢ 0 1 0 ∗ ⎥ ⎢ ∗ 1 0 0 ⎥ ⎢ 1 0 0 0 0 1 ∗ ⎥ ⎢ ∗ 0 1 0 ⎥ ⎢ 0 0 1 ∗ ⎥ ⎢ ∗ 0 1 0 ⎥ ⎣⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 0 ⎦ ⎣ 0 0 0 1 ⎦ ⎣ 0 0 0 0 ⎦ ⎣ 0 0 0 1 ⎦     ⎡ 0 0 0 0 ⎤ ⎡ 0 0 0 0 ⎤ ⎢ ⎥ ⎢ ⎥ (a) ∗ 1 0 ∗ ∗ 1 0 ∗ ⎢ (b) ∗ 0 1 ∗ ⎥ ⎢ ∗ 0 1 ∗ ⎥ ⎣⎢ ⎥ ⎢ ⎥ 0 0 0 0 ⎦ ⎣ 0 0 0 0 ⎦    ⎡ 0 0 0 0 ⎤ ⎡ 0 0 0 0 ⎤ ⎡ 0 0 0 0 ⎤ ⎢ ∗ 1 0 ∗ ⎥ ⎢ ⎥ ⎢ ∗ ⎥ ⎢ 0 0 0 0 1 ∗ ∗ 0 0 0 0 ⎥ ⎢ ∗ 0 1 ∗ ⎥ ⎢ 0 0 0 0 ⎥ ⎣⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 0 ⎦ ⎣ 0 0 0 0 ⎦ ⎣ 0 0 0 0 ⎦    0 0 In the next case something else of note occurs. First: complementary pairs of D-classes A and B exist where A ∨ B = {I} and A ∧ B = {0}. Here I and 0 denote respectively the unique maximum and unique minimum of S, although in these cases they also denote the identity and zero matrices of the matrix ring. In the case below, complementary classes are in bold type. I   ⎡ ⎡0 0 0 0⎤ 1 0 0 0 ⎤ ⎢ ⎢ ⎥ 0 1 0 ∗ ⎥ 0 1 0 0 ⎢ ⎢ ⎥ 0 0 1 ∗ ⎥ 0 0 1 0 ⎢ ⎥ ⎣⎢ ⎥ 0 0 0 0 ⎦ ⎢0 0 0 1 ⎣ ⎦⎥ (c)    ⎡1 0 0 0⎤ ⎡0 0 0 0⎤ ⎢0 1 ∗ ∗⎥ ⎢ ⎥ ⎢ ⎥ 0 1 0 ∗ ⎢ ⎥ 0 0 0 0 ⎢ ⎥ 0 0 1 ∗ ⎢ ⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥    ⎡1 0 0 0⎤ ⎡0 0 0 0⎤ ⎢0 0 0 0⎥ ⎢ ⎥ ⎢ ⎥ 0 1 ∗ ∗ ⎢ ⎥ 0 0 0 0 ⎢ ⎥ 0 0 0 0 ⎢ ⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥   0 212 VI: Skew Lattices in Rings Second: these complementary pairs are trivial D-classes and thus form part of the center of the skew lattice. Third: this case is centrally complemented in that each central (trivial) D-class has a complementary class. Finally, we seemingly have an internal direct product. Indeed, setting ⎡1 0 0 0⎤ ⎡0 0 0 0⎤ ⎡0 0 0 0⎤ ⎡0 0 0 0⎤ ⎢0 0 0 0⎥ ⎢0 1 0 0⎥ ⎢0 1 0 ∗⎥ ⎢0 1 ∗ ∗⎥ S1 = ⎢ ⎥  0 { } and S2 = ⎢ ⎥  ⎢ ⎥  ⎢ ⎥  0 { } 0 0 0 0 ⎢ ⎥ 0 0 1 0 ⎢ ⎥ 0 0 1 ∗ ⎢ ⎥ 0 0 0 0 ⎢ ⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 1 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥ ⎢0 0 0 0 ⎣ ⎦⎥ an isomorphism of µ: S1 × S2 ≅ S is given by setting µ( x, y) = x ∨ y = x + y (since xy = 0). This leads us to the following three results: Theorem 6.1.4. If S is a cancellative skew lattice with a unique maximum 1 and a unique minimum 0, then all pairs of complementary D- classes in S are trivial and thus lie in the center of S. In particular, this occurs for all skew lattices in rings with unity 1 that contain both 1 and 0. Proof. Let A, B be a complementary D-classes with say a, aʹ ∈ A and b ∈ B. From a ∨ b = aʹ ∨ b = 1 and a ∧ b = aʹ ∧ b = 0, cancellation implies a = aʹ. £ Theorem 6.1.5. Every maximal [right-handed] skew lattice ( under ○ and •) in a ring with identity 1 is (a) centrally complemented and (b) contains all central idempotents of the ring. Proof. (a) If e ∈ Z(S) but not 1 – e, then 1 – e ∉ S so that Sʹ = e S + (1 – e)S is a larger [right-handed] skew lattice in R. Indeed given x + xʹ, y + yʹ ∈ Sʹ, we get ( x + xʹ)( y + yʹ) = xy + xʹ yʹ and ( x + xʹ) ○ ( y + yʹ) = x○ y + xʹ○ yʹ. Thus Sʹ is a skew lattice in R containing both S and 1 – e. Given the maximal status of S, 1 – e ∈ S and hence 1 – e ∈ Z(S). (b) If e is a central idempotent that is not in S, then clearly Sʹ = e S + (1 – e)S contains both e and S. As in (a), Sʹ must be a skew lattice, which by the maximal status of S equals the latter, and e ∈ S follows. £ 213 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 6.1.6. Let S be a distributive, symmetric skew lattice with both maximal and minimal elements, 1 and 0. Let B be a finite Boolean lattice such that {1, 0} ⊆ B ⊆ Z(S). Then: 1) For each atom α of B, α∧S = {α∧ x⎮ x ∈ S} = { y ∈ S⎮0 ≤ y ≤ α} is a skew lattice with maximal element α. 2) For atoms α ≠ β, and all x ≤ α and y ≤ β, x∧ y = 0 = y∧ x and x∨ y = y∨ x. 3) Given atoms n α1, … ,α n of B, an isomorphism ϕ: S ≅ ∏ (α ∧ is defined by i S) 1 –1 ϕ( x) = (α1∧ x, … , α n∧ x) with ϕ ( x 1, … , xn) = x 1 ∨ … ∨ xn. Proof. Since B ⊆ Z(S), α∧ x∧α∧ y = α∧ x∧ y and (α∧ x)∨(α∧ y) = (α∧ x∧α)∨(α∧ y∧α) = α∧( x∨ y)∧α = α∧( x∨ y). Thus assertion (1) follows. Next, since α∧β = 0 for x and y as stated in (2), x∧ y = x∧α∧ y∧β = x∧ y∧α∧β x∧ y∧0 = 0. Similarly y∧ x = 0, so that x∨ y = y∨ x by symmetry and (2) follows. Moreover, ϕ is tat least a homomorphism. For all x ∈ S, x = x ∧ 1 ∧ x = x ∧ (α1 ∨ … ∨ α n) ∧ x = ( x ∧ α1 ∧ x) ∨…∨ ( x ∧ α n ∧ x) = (α1 ∧ x) ∨…∨ (α n ∧ x), which guarantees that n ϕ is also one-to-one. Finally, given any ( x 1, …, xn) in ∏ (α ∧ , each x i S) 1 i = α i∧ xi so that α j∧( x 1 ∨ … ∨ xn) = (α j∧ x 1) ∨ … ∨ (α j∧ xn) = (α j∧ xj) = xj. Hence ϕ( x 1 ∨ … ∨ xn) = ( x 1, …, xn), making the map ϕ surjective and hence an isomorphism. £ Our next goal is to refine our description of the right extension S of a lattice T is a ring. In this regard, given commuting idempotents e and f in a ring R, S( e⎮ f) will denote the D-class of e in the skew diamond that is the right extension S of the lattice T = { e, f, e∨f, e∧ f} in E(R). Theorem 6.1.7. Let T be a lattice in a ring R with identity 1 and let S be the right extension of T. If e ∈ T, then S e = ∩{S( e | f)⎮ f ∈T}. For e, f ∈T, moreover, S( e⎮ f) = e + ef R f(1 – e) + e R(1 – e)(1 – f). In particular, given e and f in T: i) if e and f are comparable, then S( e⎮ f) = R e. ii) if e and f are disjoint ( ef = 0), then S( e⎮ f) = e + e R(1 – e)(1 – f). 214 VI: Skew Lattices in Rings Proof. In light of Theorem 6.1.3, the assertion about S e is obvious. In general, for all x ∈ R e, x = e + ea(1 – e) = e + ea(1 – e) f + ea(1 – e)(1 – f ) for some a in R ( x itself will do). Thus xf = ef + ea(1 – e) f, so that xf ∈R ef only when ea(1 – e) f = efb(1 – ef) for some b in R which occurs precisely when ea(1 – e) f = efb(1 – e) f for some b since (1 – e) f = (1 – ef) f. Thus, S( e⎮ f) ⊆ e + ef R f(1 – e) + e R(1 – e)(1 – f) ⊆ R e. Since { e + ef R f(1 – e) + e R(1 – e)(1 – f)} f = ef + ef R f(1 – e) = ef + ef R f(1 – ef) ⊆ R ef, the reverse inclusion holds. Case (ii) follows immediately from the general case. For case (i), if f > e, then S( e⎮ f) = e + e R f(1 – e) + e R(1 – e)(1 – f) which reduces to e + e R(1 – e) = R e. If e > f, then both f(1 – e) = 0 and (1 – e)(1 – f) = 1 – e, so that S( e⎮ f) = e + e R(1 – e) = R e again. £ The general case and the two special cases are illustrated in the following block matrix diagrams where u, x, y and z hold arbitrary values. ⎡ 1 0 0 0 ⎤ ⎡ 0 0 0 0 ⎤ ⎡ 1 0 0 y ⎤ ⎢ ⎥ e = ⎢ ⎥ ⎢ ⎥ 0 1 0 0 0 1 0 0 ⎢ f = S( e⎮ f) = all 0 1 x z ⎢ ⎥ . 0 0 0 0 ⎥ ⎢ 0 0 1 0 ⎥ 0 0 0 0 ⎣⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 0 ⎦ ⎣ 0 0 0 0 ⎦ ⎣ 0 0 0 0 ⎦ ⎡ 1 0 0 0 ⎤ ⎡ 0 0 0 0 ⎤ ⎡ 1 0 u y ⎤ ⎢ ⎥ ⎢ ⎥ e = ⎢ 0 1 0 0 ⎥ 0 1 0 0 0 1 x z ⎢ f = ⎢ ⎥ S( e⎮ f) = all ⎢ ⎥ . 0 0 0 0 ⎥ 0 0 0 0 0 0 0 0 ⎣⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 0 ⎦ ⎣ 0 0 0 0 ⎦ ⎣ 0 0 0 0 ⎦ ⎡ 1 0 0 0 ⎤ ⎡ 0 0 0 0 ⎤ ⎡ 1 0 0 y ⎤ ⎢ ⎥ ⎢ ⎥ e = ⎢ 0 1 0 0 ⎥ 0 0 0 0 ⎢ f = ⎢ ⎥ S( e⎮ f) = all 0 1 0 z ⎢ ⎥ . 0 0 0 0 ⎥ 0 0 1 0 0 0 0 0 ⎣⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 0 ⎦ ⎣ 0 0 0 0 ⎦ ⎣ 0 0 0 0 ⎦ The case of matrix rings over fields Given a matrix ring R = M n(F) over a (possibly skew) field F, let Δ denote the maximal lattice of 0-1 diagonal matrices and let e 1, … , en denote the atoms of Δ. Thus ei is the matrix with 1 in the ii-position and 0 elsewhere. The support of any e ∈ Δ is supp( e) = { i ∈ *⎮ eii = 1}. 215 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond In what follows, without loss of generality we assume that T is a sublattice of Δ such that both matrices 0 and I lie in T. With this in mind, for each e ∈ Δ, the T- cover of e is s e = ∧{ f ∈ T⎮ f ≥ e}, and the T- interior of e is e o = ∨{ f ∈ T⎮ f ≤ e}. Clearly an idempotent e ∈ Δ belongs to T if and only if it equals either, and hence both, its T- cover and its T-interior. We next state a pair of elementary lemmas, the first of which describes an R-set in matrix form. Lemma 6.1.8. Given e ∈ Δ and x ∈ R e = e + e R(1 – e), then e and x have the same diagonal entries. Moreover, the nonzero diagonal entries of x occur only in those rows indexed by supp( e) and in those columns with support indexed by supp(1 – e) . £ Lemma 6.1.9. Given e ∈T and j ∈ supp(1 – e), e ∧ e = 0 iff j j ∈ supp(1 – e)o . £ We now state our principal result for right-handed skew lattices in matrix rings. Theorem 6.1.10. Let R be the ring of all n× n matrices over a (skew) field F, let Δ n be the lattice of all 0-1 diagonal matrices and let T be any sublattice of Δ n containing at least the zero and identity matrices, 0 and I. If the skew lattice S is the right extension of T in R, then for each e ∈ S its D-c lass S e in S is described as follows: i) All matrices in S e have the same diagonal as e. ii) The nonzero, non-diagonal entries of any matrix in S e occur only in those columns indexed by supp(1 – e). iii) For j in supp(1 – e), the only positions in the jth column of a matrix in S e that admit nonzero entries are given by supp( e ∧ e ). j iv) For any j ∈ supp(1 – e), the jth column only has 0 s when j ∈ supp(1 – e)o . v) No further restrictions are imposed on the matrices in S e. Proof. Letting e ∈ T as stated, the class of matrices in S e is the intersection of classes of matrices themselves obtained by appropriate juggling of the block designs given after Theorem 6.1.7. Thus to determine the matrices in S e, one need only discover which non-diagonal positions can hold nonzero entries in these matrices since these positions can hold any member of F. Since S e ⊆ R e, these “free” positions can only occur in rows indexed by supp( e) and columns indexed by supp(1 – e). Thus (i) and (ii) are seen. Next let j ∈ supp(1 – e) and let Γ( e, j) ⊆ supp( e) consist of those jth column positions admitting non-0 entries. Claim: Γ( e, j) = supp( e ∧ e ) for all j j ∈ supp(1 – e). 216 VI: Skew Lattices in Rings If j ∈ supp(1 – e)o then Γ( e, j) is empty by Theorem 6.1.7(ii), while e ∧ e = 0 by Lemma 5.1.9 j and assertion (iv) is seen. Otherwise, if j ∉ supp(1 – e)o then by the previous theorem Γ( e, j) = ∩{ supp( e∧ f)⎮ f ∈ T such that j ∈ supp( f) and e ∧ f ≠ 0} = supp( e ∧ ∧{ f ∈ T⎮ j ∈ supp( f) and e ∧ f ≠ 0}). But by Lemma 6.1.9, the latter case is equivalent to asserting that e ∧ e ≠ 0. Thus in this case, j e = ∧{ f j ∈ T⎮ e ∧ f ≠ 0 and j ∈ supp( f)}) so that Γ( e, j) = supp(e ∧ e ) here also. Thus ) indeed holds for all j Γ( e, j) = s upp(e ∧ ej j ∈ supp(1 – e) and assertion (iii) is finally seen. £ Given the above definitions in the context of a diagonal lattice T and the matrix ring R, the class space Γ of T in R is the vector subspace consisting of all matrices A ∈R such that for 1 ≤ i ≤ n, in the ith column of A the only nonzero entries occur in the supp( e ) positions. The class j space of T allows use to express S e for any e ∈ T in the following succinct manner. Theorem 6.1.11. Given T and R as above with right extension S of T, if Γ is the class space of T in R, then for each e ∈ T, S e = e + eΓ(1 – e) . Examples 6.1.2 The class space for each of the skew lattices of Example 6.1.1 are, in the same order, as follows: ⎡∗ 0 0 0⎤ ⎡∗ 0 0 0⎤ ⎡∗ 0 0 0⎤ ⎢∗ ∗ 0 ∗⎥ ⎢∗ ∗ ∗ ∗⎥ ⎢0 ∗ ∗ ∗⎥ a) ⎢ b) ` c) . £ ∗ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ∗ ∗ ⎢ ⎥ ∗ 0 ∗ ∗ ⎢ ⎥ 0 0 ∗ ∗ ⎢ ⎥ ⎢0 0 0 ∗ ⎣ ⎦⎥ ⎢0 0 0 ∗ ⎣ ⎦⎥ ⎢0 0 0 ∗ ⎣ ⎦⎥ Corollary 6.1.12. If R is a matrix ring over a skew field, T is a lattice in R containing both 0, 1 and S is the right extension of T, then Z(S) is the Boolean lattice of all complemented elements in T. Proof. By Theorem 6.1.4 we need only show that every central element in S comes from a complemented element in T. So let S e = { e}. Thus e ∧ e = 0 for all j j ∈supp(1 – e) so that 1 – e ≥ ∨{ ej ⎮ j ∈supp(1 – e)}. Since 1 – e ≤ ∨{ ej ⎮ j ∈supp(1 – e)} always holds, e is complemented in T. £ 217 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond A lattice T in a ring R with unity is centrally closed if the Boolean lattice of all central idempotents lies in T. The previous result can be generalized as follows. Corollary 6.1.13. Given a semisimple, Artinian ring R and a centrally closed lattice T in R , the center Z(S) of the right extension S of T in R is the Boolean lattice of all complemented elements in T. Proof. This follows from the Wedderburn structure theorem for semisimple, Artinian ring and the previous theorem. £ We conclude this section with a further consequence of the above results. Recall that a skew chain is any skew lattice whose D-classes are totally ordered. n Theorem 6.1.14. Given R = F n× : i) Every maximal right zero semigroup in R is a maximal rectangular band in R. ii). Every maximal right-handed skew chain in R is a maximal skew lattice in R. Proof. Consider a maximal right zero semigroup S given by all matrices of block form j × j ⎡ j × j I X⎤ ⎡ 0⎤ 0 0 ⎣⎢ ⎦⎥ . If S is not a maximal rectangular semigroup in R, then some β = IB 0 ⎣⎢ ⎦⎥ exists ⎡I X⎤ in E(R) \ S such that β together with S generates a properly larger rectangular band. But 0 0 ⎣⎢ ⎦⎥ ⎡ I 0⎤ ⎡I + XB 0⎤ B 0 ⎣⎢ ⎦⎥ = 0 0 ⎣⎢ ⎦⎥ is idempotent only if XB = 0 for all possible X. But this is not possible since β ∈ S and thus B ≠ 0. Hence (i) must follow. Suppose next that S is a maximal chain of right-rectangular skew lattices in R. Clearly S is a quadratic skew lattice. Suppose that S lies in a larger skew lattice Sʹ. The inclusion S ⊆ Sʹ cannot increase any D-class already in S. Hence a new D-class B exists in Sʹ that is incomparable to some D-class A in S. If T is a lattice section of Sʹ, then Theorem 6.1.7 guarantees that the right extension of T will not include the full R-set A. (Put succinctly, increasing the lattice shape of S decreases some of its D-class sizes.). Hence it is impossible to join new D-class to S. £ Query: Given a lattice T in a matrix ring R of maximal height, is its right extension S a maximal ( right-handed) skew lattice in R? Conversely, are all maximal (right handed) skew lattices in R of maximal height = n? Settling these questions even for matrix rings would be of interest. 218 VI: Skew Lattices in Rings 6.2 ∇-bands and cubic skew lattices Given a ring R and e, f, in E(R) recall that e∇ f = ( e ○ f)2 = ( e + f – ef)2. The latter expands to e + f + fe – ef – efe – fef + efef in general, but reduces to e + f + fe – efe – fef when ef is idempotent. Recall also that ∇ extends ○ in that: i) Every skew lattice (S; ○, •) in a ring is also a skew lattice under ∇ and • since in this case ( e ○ f)2 = e ○ f so that e∇ f reduces to e ○ f. ii) Whenever e, f, ef, fe ∈E(R), then so are efe, fef and e∇ f by Theorem 2.7.5. iii) Situations occur where e∇ f is idempotent, but not e ○ f; but ∇ need not be associative, even if idempotent closure occurs. (See Examples 2.3.1 and 2.3.2.) iv). Due to ( i), maximal right [left] regular bands in any ring form skew lattices under ∇ and •. Every right [left] regular band in a ring generates a skew lattice under ∇ and •. v). Every maximal normal band in a ring R is a normal skew lattice under ∇ and •; indeed it forms a skew Boolean algebra. Thus a normal band in R generates a strongly distributive skew lattice under ∇ and •. (Theorem 2.3.6) vi) Maximal regular bands in rings, however, need not be closed under ∇, much less be skew lattices under ∇. (See Example 2.3.5.) Recall that every band is naturally partially ordered by e ≤ f if ef = e = fe which refines the natural preorder given by e ≺ f if efe = e. The equivalence induced from ≺ is the Green’s relation D. Turning to s-bands proper, a band congruence θ on a s-band is a s- congruence if θ is also a congruence under s. The assertions in the following lemma, coming from a 2004 paper of Cvetko-Vah, are easily verified. Lemma 6.2.1. Given a s- band S in a ring, for all a, b ∈ S: i) Given b ≺ a in S, a∇ b = a + ba – aba and b∇ a = a + ab – aba. ii) a∇ b∇ a is unambiguous: ( a∇ b)∇ a = a∇( b∇ a) = a + b – bab. £ Even when s is not associative, s-bands are very much like skew lattices in rings. Theorem 6.2.2. For any s- band S in a ring the following hold. i) As a band, S is regular. That is, abaca = abca holds on S. ii) D is a s- congruence and S/D is a lattice with D x ∨ D y = D x s y. iii) a ( a s b) = a = ( b s a) a and a s( ab) = a = ( ba)s a. iv) ab = ba iff a s b = b s a. v) a ( b s c) a = aba s aca and a s( bc)s a = ( a s b s a)( a s c s a). vi) a∇ c∇ a = b∇ c∇ b and aca = bcb implies a = b. (S; s, •) is thus a distributive, symmetric, cancellative skew lattice when s is also associative. 219 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. (i) Suppose A and B are D-classes in S with A > B. Let a ∈ A and b ∈ B so that a ≻ b. Consider b s a s b = a + b – aba. From ( a + b – aba) a( a + b – aba) = a + b – aba and a( a + b – aba) a = a, b s a s b ∈A follows. On the other hand, clearly b s a s b ≥ b. Thus given D-classes A > B in S, for all b ∈ B there exists a ∈ A such that a > b. Hence S is regular by Theorem 1.2.18. (ii) Let a ≻ b and a ≻ c. Since S is regular, bab = b, bac = bc, cab = cb and cac = c. Hence ( b s c) a( b s c) = b s c so that a ≻ b s c. Thus the D-class of b s c is the join-class of D b and D c and D is a s-congruence. (iii) follows from routine calculations such as a s ( ab) = a + ab + aba – aaba – abaab wich immediately reduces to a + ab + aba – aba – ab = a. (iv) Just observe that a s b and b s a differ only by their third terms, ba and ab. (v) For the first part, regularity gives a( b s c) a = a( b + c +bc – bcb – cbc) a = aba + aca +abaca – abacaba – acabaca = aba s aca. For the other identity, first observe that [ b( c s a)]2 = ( bc + ba +bac – bcac – baca)2 expands as ( bc + bcba – bca) + ( babc + ba – baca) + ( babc + bacba – baca) – ( bcabc + bcaba – bca) – ( babc + bacba – baca) = bc + ba – baca + babc – bcabc which in turn must equal b( c s a) = bc + ba +bac – bcac – baca. Equating and canceling common terms gives babc – bcabc = bac – bcac, that is, the identity babc + bcac = bac + bcabc. (6.2.2) Thus ( a s b s a)( a s c s a) = ( a + b – bab)( a + c – cac) = a + bc – bcac – babc + bac = a + bc – bac – bcabc + bac = a + bc – bcabc = a s( bc) s a. (vi) Given aca = bcb, the regularity of • implies cac = c( aca) c = c( bcb) c = cbc. Cancelling in a + c – cac = b + c – cbc ( a∇ c∇ a = b∇ c∇ b) in turn gives a = b. £ We turn to other properties observed in skew lattices. The next result demonstrates the important role played by instances of commutativity, especially on the algebraic reducts (S, •) and (S, s) of any s-band S. 220 VI: Skew Lattices in Rings Theorem 6.2.3. Join classes and meet classes are given by commuting joins and meets. Thus given D- classes A and B in a s -band S , their join D -class J and meet D -class M are J = { a s b⎮ a ∈ A, b ∈ B & a s b = b s a} and M = { ab⎮ a ∈ A, b ∈ B & ab = ba}. Moreover, for every a ∈ A there exists b ∈ B such that a s b = b s a in J and ab = ba in M. Proof. Given v ∈ J, there exist a ∈ A and b ∈ B such that v ≥ a, b. (Set a = vaʹ v and b = vbʹ v for any aʹ ∈ A and bʹ ∈ B.) For such a and b we have a∇ b ∈ J and v = v( a s b) v = v( a + b + ba – aba – bab) v = a + b + ba – aba – bab = a s b. Similarly, b s a equals v also and the assertion about J is seen. The case for M is similar. For the final assertion, pick a in A and let v ∈ J be such that v ≥ a. That b ∈ B exists such that a s b = v = b s a is now clear. The rest follows from symmetry. £ Corollary 6.2.4. Given a s -band S and e ∈ S, the following are equivalent: i) D e = { e}. ii) For all x ∈ S, e s x = x s e and ex = xe. Proof. Clearly ( ii) implies ( i); and ( ii) implies ( i) due to the final assertion of Theorem 6.2.3. £ Corollary 6.2.5. A set of commuting elements in a s -band S generates a sublattice. £ We next turn to the question when s is associative, giving various criteria. When it is associative on a particular s-band, the latter is called a cubic skew lattice in the ring. Clearly quadratic skew lattices in a ring (studied in the previous section) are trivially cubic. The associativity of s : the role of the commutator [ x, y] Recall that the commutator of elements x, y in a ring R is [ x, y] = xy – yx. Clearly x and y commute if and only if [ x, y] = 0. For any pair of idempotents e and f in a band S in a ring R e s f – e○ f = ef + fe – efe – fef = [ e, f]2. Thus [ e, f]2 = 0 on a s-band S if and only if s = ○ as binary operations on S. In this section we show that a s-band S is associative if and only if for all e, f ∈ S, [ e, f]2 lies in the center of the subring of R generated from S. That is, s is associative if and only if g[ e, f]2 = [ e, f]2 g for all e, f, g ∈ S. We begin with a pair of somewhat technical lemmas. Lemma 6.2.6. ( a s b) c( a s b) = – ( abca – aca – bca – bcb + bcab). 221 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. Multiplying out ( a + b + ba – aba – bab) c( a + b + ba – aba – bab) and cancelling yields ( aca + acb – acab) + ( bca + bcb – bcab) – ( abca + abcb – abcab) = aca + bca + bcb – bcab – abca where the underlined terms vanish collectively by identity (6.2.2) . £ Lemma 6.2.7. Given a, b, c in a s -band S, a s( b s c) = ( a s b) s c if and only if [ b, c]2 a – a[ b, c]2 a = c[ a, b]2 – c[ a, b]2 c. (6.2.7) Proof. a s ( b s c) = a + ( b + c + cb – bcb – cbc) + ( ba + ca + cba – bcba – cbca) – ( aba + aca + acba – abcba – acbca) – ( b s c) a( b s c) while ( a s b)s c = ( a + b + ba – aba – bab) + c + ( ca + cb + cba – caba – cbab) – ( cac + cbc + cbac – cabac – cbabc) – ( a s b) c( a s b). Equating a s( b s c) with ( a s b)s c and then canceling common terms yields – bcb – bcba – cbca – aca – acba + abcba + acbca – ( b s c) a( b s c) = – bab – caba – cbab – cac – cbac + cabac + cbabc – ( a s b)c( a s b). Applying the previous lemma gives – bcb – bcba – cbca – aca – acba + abcba + acbca + ( bcab – bab – cab – cac + cabc) = – bab – caba – cbab – cac – cbc – cbac + cabac + cbabc + ( abca – aca – bca – bcb + bcab) which reduces to – bcba – cbca – acba + abcba + acbca – cab + cabc = – caba – cbab – cbac + cabac + cbabc + abca – bca. Adding cab + bca to both sides, grouping the aXa terms on the left and the cYc terms on the right and then factoring gives, bca – bcba – cbca – a( bc – cb)2 a = cab – caba – cbab – c( ab – ba)2 c, Adding cba to both sides, then grouping and factoring once again gives ( bc – cb)2 a – a( bc – cb)2 a = c( ab – ba)2 – c( ab – ba)2 c which is the statement of the lemma. £ Theorem 6.2.8. A s- band S is associative if and only if for all a, b, c ∈ S, a[ b, c]2 = [ b, c]2 a. 222 VI: Skew Lattices in Rings Proof. This identity implies that of Lemma 6.2.7, making s associative. On the other hand, replacing a by aba and b by bab in the latter gives [ babc – cbab]2 aba – aba[ bab – cbab]2 aba = c[ ab – ba]2 – c[ ab – ba]2 c. Regularity first gives aba[ babc – cbab]2 aba = aba( c – c) aba = 0 and then [ babc – cbab]2 aba = ( babc – cbab)( c – c) aba = 0. Thus c( ab – ba)2 c = c( ab – ba)2 and by (6.2.7), a( bc – cb)2 a = ( bc – cb)2 a. Permuting variables in c( ab – ba)2 c = c( ab – ba)2 gives a( bc – cb)2 a = a( bc – cb)2 from which a[ b, c]2 = [ b, c]2 a follows. £ The associativity of s thus reduces to cases of possible commutation: does [ x, y]2 always produce elements lying in the center of the subring S± generated from S? The associativity of s : the Green’s relations L and R We next consider connections between the associativity of s and the equivalences L and R that refine D. Recall that L is a right congruence on S in that a L b implies ac L bc for all c ∈S while R is a left congruence on S. If S is regular, then both L and R are full congruences. In particular, L and R are multiplicative congruences on all s-bands. We turn to the status of L and R as s-congruences on a s-band. But first let a R b denote the conjunction, a s b = b and s b s a = a, and similarly let a L b denote a s b = a and b s a = b. s Lemma 6.2.9. In a s-band, a L b if and only if a R b and similarly a R b if and only if s a L b. In general, a L b implies ( c s a)( c s b) = c s a for all c and a R b implies s ( a s c)( b s c) = b s c for all c. Proof. Expanding, a s b = b and b s a = a reduce to a = aba + bab – ba and b = aba + bab – ab. Multiplying on the left by a and b respectively, yields a = ab and b = ba, that is a L b. Conversely, if a L b under the ring multiplication, then a s b reduces to b and b s a reduces to a. In general, for all c ∈S, ( c s a)( c s b) = ( c + a + ac – aca – cac)( c + b + bc – bcb – cbc) which with the assistance of L as a congruence on the multiplicative band expands as c + ( ac + a – acb) + ( ac) – ( ac + aca – acb) – ( cac) = c + a + ac – aca – cac = c s a Similarly, ( c s b)( c s a) = ( c s b) so that c s a L c s b. The case for R is similar. £ 223 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond In general L and R need not be s-congruences. However: Lemma 6.2.10. L and R are both s- congruences on a s- band S if and only if for all a, b, c ∈S, i) a( bc – cbc) a = a( bc – cbc). ii) a( bc – bcb) a = ( bc – bcb) a. Proof. Given u L v, ( u s x)( v s x) = ( u + x + xu – uxu – xux)( v + x + xv – vxv – xvx). Distributing each term of the left factor over the right factor, then adding and simplifying gives ( u) + ( xv + x – xvx) + ( xu) – ( uxu) – ( xu) = u + x + xv – uxu – xvx. (Here we use various consequences of regularity. E.g., u( v + x + xv – vxv – xvx) = u + ux + uxv – uxv – uxvx = u + ux – ux = u.) Thus, ( u s x)( v s x) = u s x holds for u L v if and only if xv – xvx = xu – xux. Replacing u and v by generic values, uvu and vu, we see that L is a s-congruence if xuvu – xuvux = xvu – xvux holds for all u, x, v in S. Rearranging the terms gives xvux – xuvux = xvu – xuvu which is equivalent to (i) upon switching variables. In similar fashion, R being a s-congruence is equivalent to (ii). £ Theorem 6.2.11. Given a s- band S in a ring, both L and R are s- conguences if and only if s is associative and thus S is a skew lattice. Proof. If s is associative so that S is a skew lattice, then both L and R must be full skew lattice congruences. On the other hand, given that L and R are s-congruences, the identities of the above lemma imply that a[ b, c]2 = a( bc – cbc + cb – bcb) = a( bc – cbc + cb – bcb) a = a( bc – bcb + cb – cbc) a = ( bc – bcb + cb – cbc) a = [ b, c]2 a. Thus, the criterion of Theorem 3.2.8 is satisfied and hence s is associative. £ The associativity of s : the role of Kimura factorizations A closely related criterion involves a canonical factorization that can occur in s-bands. We begin with bands. Every subsemilattice T of any band S meets each D-class in S in at most once element. A semilattice section in a band S is a subsemilattice T of S that meets each D-class exactly once, thus making T is an internal copy of S/D, the maximal semilattice image of S. If S is also regular, then TR = ∪ t∈TR t is a maximal right regular band in S and TL = ∪ t∈TL t is a maximal left regular band in S with TR and TL being copies of the maximal right and left regular 224 VI: Skew Lattices in Rings images S/L and S/R of S. Each e ∈S factors as e L e R where e L = ete ∈ TL and e R = tee ∈ TR where te is the unique element in T∩D e. Put otherwise, e L and e R are the unique elements in D e related to e and te in the following D-class picture. e R e L = ete L L tee = e R R te Due to S being regular ef = ( e L e R)( f L f R) = ( e L f L)( e R f R) holds for all e, f ∈ S. Indeed, ef = efefef = ( e L e R)( f L f R) ef( e L e R)( f L f R) = e L f L efe R f R = ( e L f L)( e R f R) with both latter reductions due to regularly. Multiplication on S thus decomposes internally into separate products on TL and TR so that S is isomorphic to a sub-band of TL × TR. We call the factorization e = e L e R the internal Kimura factorization of e relative to T. Even more is true if S is a s-band in a ring. In this case T is a lattice section in that for all e, f in T, e s f = e ○ f is in T also making (T; ○, •) a lattice that meets each D-class exactly once, so that T ≅ S/D. (If u ∈ T is in the join D-class of e and f, then u ≥ both e, f and so u ≥ e ○ f follows. But Since u and e ○ f lie in a common D-class, u = e ○ f.) Given a s-band S with a lattice section T, a modified join operation ∨T can be defined such that (S, ∨T, •) is a distributive, symmetric skew lattice. To begin, TL and TR are in fact skew lattices. Indeed, suppose that a, b ∈ TL. Certainly a s b ∈ S, but since a and b lie in the left regular band TL, we also have a s b = a○ b. Let ta, tb ∈ T be such that a L ta and b L tb. By left regularity, we have abta = ab, atbta = atb, batb = ba and btatb = bta. Hence ( a○ b)( ta○ tb) = ( a + b – ab)( ta + tb – tatb) = a + b – ab = a○ b Similarly, ( ta○ tb)( a○ b) = ta○ tb. Thus a○ b ∈ TL, and TL is a skew lattice in S as claimed. Likewise, TR is a skew lattice in S. The internal Kimura decomposition of the band S with respect to T enables us to define an operation ∨T on S by setting e ∨T f = ( e L ○ f L)( e R ○ f R). Clearly ( e ∨T f)L = e L ○ f L and ( e ∨T f)R = e R ○ f R. It follows that (S, ∨T, •) is an “internal” fibered product of the skew lattices (TL, ○, •) and (TR, ○, •) over their common sublattice T and thus is a skew lattice. We will call ∨T the associative join on S induced from T. Cleary ∨T is dependent 225 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond on T and is somewhat more complex in design than s. Indeed we will soon see that distinct lattice sections of S can produce different associative joins. But first we have: Theorem 6.2.12. Given a s- band S with a lattice section T, the binary operation s is associative if and only if e s f = e ∨T f for all e, f in S, where ∨T is the T -induced associative join on S. Thus if s is associative, all lattice sections T of S induce a common associative join, namely s . Proof. If s equals ∨T for some lattice section T, then clearly s is associative. Conversely, let s be associative and thus S a skew lattice. The regularity of s plus the fact the x s y = yx in any D-class of S gives, e s f = ( e s te s f s tf) s ( te s e s tf s f) = ( te s e s tf s f) ( e s te s f s tf) = ( ete s f tf)( tee s tf f) = ( e L ○ f L)( e R ○ f R) = e ∨T f . The next-to-last equality is because s reduces to ○ on any left or right regular band in S. The final assertion of the theorem is clear. £ Consider the following pair of lattice sections for lattice sections for Example 2.3.2. ⎧⎡ ⎫ ⎧⎡0 0 0 0⎤ ⎡0 0 0 0⎤⎫ 0 0 0 0 ⎤ ⎡ 0 0 0 0 ⎤ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥⎪ Let T ⎢ ⎥ ⎢ ⎥ 0 1 0 0 0 1 1 0 = 0 1 0 0 ⎨⎢ > 0 1 0 0 ⎬ and Tʹ = ⎨⎢ ⎥ > ⎢ ⎥⎬ . 0 0 1 0 ⎥ ⎢ 0 0 0 0 ⎥ 0 0 1 0 0 0 0 0 ⎩⎪⎣⎢ ⎪⎢ ⎥ ⎢ ⎥⎪ 0 0 0 0 ⎦⎥ ⎣⎢ 0 0 0 0 ⎦⎥⎭⎪ 0 0 0 0 ⎩⎣ ⎦ 0 0 0 0 ⎣ ⎦⎭ If A and B as chosen as in that example, then ⎡ ⎤ ⎡ ⎤ 0 1 0 0 0 1 1 0 A ⎢ ⎥ ⎢ ⎥ ∨T B = 0 1 0 0 ⎢ while A∨T B = 0 1 0 0 . ʹ 0 0 1 0 ⎥ ⎢ 0 0 1 0 ⎥ ⎣⎢ 0 0 0 0 ⎦⎥ ⎣⎢ 0 0 0 0 ⎦⎥ Hence the choice of T affects the outcome of the induced join when s is not associative. The preceding theorem and example raise the question of whether having a unique associative join ∨ implies s is associative. A related question is that of when lattice sections must exist and in what abundance to make this question meaningful. Both issues are treated in the following theorem. Theorem 6.2.13. If S is a s- band such that S/D is at most countable, then S has a lattice section T. Such a lattice section can always be found to include any given finite subset T0 of pairwise commuting elements in S. Finally, if all lattice sections T of S induce a common associative join on S then s is this join and as such is associative. 226 VI: Skew Lattices in Rings Proof. Let D1, D2, … be a complete listing of the D-classes of S. Pick a 1 ∈ D1 and consider the set X = { x ∈S⎮ a 1 x = xa 1}. X is a band containing a 1 and meeting each D-class of S. Indeed, given any other D-class Dʹ with y ∈Dʹ we have a 1 ≥ a 1 ya 1 ≤ a 1 ya 1s y s a 1 ya 1 = b. It follows that a 1 b = a 1 ya 1 = ba 1 so that b ∈X. Moreover, X is a s-band. Indeed, given x, y ∈ X, then a 1 and x s y = x + y + yx – xyx – yxy clearly commute, so that x s y ∈X also. Note that D1 ∩ X = { a 1}. Next pick a 2 ∈ D2 ∩ X, set Y = { y ∈X⎮ a 2 y = ya 2}. Again Y is a s-band that meets each D-class of S and moreover D i ∩ X = { ai} for i = 1, 2. The process continues through the countable set of D-classes to produce the lattice section T = { a 1, a 2, a 3, …}. By placing the D-classes of the elements in T0 at the front of the list of D-classes, the second assertion follows. Finally, suppose that all lattice sections T induce a common associative join ∨ on S. Let e, f ∈ S be given. Then e and f generate a skew lattice S0 in S with at most 16 elements. (See Theorem 2.7.5 and the surrounding discussion.) Take a lattice section T0 of S0 and extend it to a full lattice section T of S, using the first part of this theorem. Then e ∨T f = e ∨T f and the latter equals e s f in S0 0 by Theorem 6.2.12. Thus s is the common associative join induced by all lattice sections. £ Thus while “really big” s-bands need not have lattice sections, thanks to Theorem 6.2.13 all s-bands in finite dimensional matrix rings over fields have them so that all of the theorem applies. Since s is associative if and only if it is thus on all finitely generated subalgebras of a given s-band, an alternative to Theorem 6.2.12 is given in the following corollary to the above theorems and its own following corollary in turn. Corollary 6.2.14. Given a s- band S, s is associative if and only if for all e D u and f D v situations in S where uv = vu, e s f is calculated as ( eu ○ fv)( ue ○ vf) . £ Corollary 6.2.15. s is associative on a s- band S if and only if it is associative on all sub-algebras generated from at most 2 D- classes. £ The associativity of s: the role of primitive algebras As with skew lattices, a s-band S is primitive if it has exactly two D-classes, A > B. In general, a band S is totally pre-ordered if either e ≻ f or f ≻ e for all e, f ∈ S. (Again, e ≻ f if fef = f, or equivalently for s-bands, e s f s e = e.) These bands are of interest because maximal, totally pre-ordered regular bands in a ring form s-bands. (See Corollary 6.3.5 below.) For such s-bands, the previous results imply that to check for the associativity of s one need only check for associativity on its primitive subalgebras. It is thus natural to ask if the “totally pre-ordered” condition can be removed from this observation? Put otherwise, if s is associative on all primitive subalgebras of a s- band, must it be associative on that band? Or does a s-band S exist with exactly four D-classes, two of which are mutually incomparable, such that s is associative on all five maximal primitive subalgebras but not on all of S? This leads us to: 227 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 6.2.16 If s is associative on all primitive subalgebras of a s -band S, then s is associative on S, and conversely. Proof. Granted the assumption, we need only show that s is associative when S is generated from two incomparable D-classes, say A and B. Let their meet class be M and join class be J. Let a lattice section of S be given by T = { a 0, b 0, m 0, j 0}. Take a ∈ A, b ∈ B. We show that a s b = a ∨T b. To begin a ∨T b = ( aa 0 + bb 0 − aa 0 bb 0)( a 0 a + b 0 b − a 0 ab 0 b) = a + am 0 b − ab 0 b + bm 0 a + b − bm 0 ab − abm 0 a − aa 0 b + ab = a + bm 0 a + b − bm 0 ab − abm 0 a since aa 0 b + ab 0 b − am 0 b = aj 0 b = ab by regularity. Of course, a s b = a + b + ba − aba − bab. Denoting the difference ( a ∨T b) − ( a s b) by Δ( a, b), we have Δ( a, b) = bm 0 a − bm 0 ab − abm 0 a − ba + aba + bab. The associativity of s on the primitive s-band M∪B implies that Δ = bm ( abm 0 a − bm 0 ab − abm 0 a − bam 0 a + abm 0 a + bab 0 a, b) = bm 0 a − bm 0 ab − bam 0 a + bab = 0. Subtracting from Δ( a, b) yields the refinement Δ( a, b) = bam 0 a + aba − abm 0 a − ba. Using the latter to calculate Δ( a, bam 0 b) in the associative context of M∪A gives: 0 = Δ = bam ( a, bam 0 a + aba − abm 0 a − ba = Δ( a, b) . 0 b) The converse is trivial. £ Since s is associative on a s-band only if it is thus on all primitive subalgebras, two of our earlier associativity criteria can be fine-tuned for the primitive case as follows. Corollary 6.2.17. Given a primitive s- band P with D- classes A > B, the following are equivalent: i) s is associative on S. ii) For all a ∈ A and b, c ∈ B, a[ b, c]2 = [ b, c]2 a. iii) For all a ∈ A and b, c ∈ B, both a( bc – cbc) a = a( bc – cbc) and a( bc – bcb) a = ( bc – bcb) a. 228 VI: Skew Lattices in Rings Proof. That (i) implies (ii) and (iii) follows from Theorem 6.2.8, Lemma 6.2.10 and Theorem 6.2.11. The identities of (ii) and (iii), in their unconditional form, conversely imply (i). Given a, b, c ∈ P, the only case where the identities need not hold, is the case where a ∈ A and b, c ∈ B. In all alternative cases these identities hold. We check the case where b ∈ A and c ∈ B. Here we have b ≻ a, c so that xby = xy whenever x and y are either a or c. Hence a[ b, c]2 = a[ bc + cb – bcb – cbc] = ac + acb – acb – ac = 0, a( bc – cbc) = abc – acbc = ac – ac = 0 = a( bc – cbc) a, and similarly [ b, c]2 a = 0 and ( bc – bcb) a = 0 = a( bc – bcb) a. The cases b, c ∈ A, or b ∈ B but c ∈ A are similarly verified, as is the case where a, b, c ∈ B. £ Given the matrices A ≻ B, C from Example 2.3.2, we have: ⎡ 0 0 0 0 ⎤ ⎡ ⎤ 0 0 1 0 A[B, C]2 = ⎢ 0 0 0 0 ⎥ ≠ ⎢ 0 0 0 0 ⎥ = [B, C]2A, and ⎢ 0 0 0 0 ⎥ ⎢ 0 0 0 0 ⎥ ⎣ 0 0 0 0 ⎦ ⎣⎢ 0 0 0 0 ⎦⎥ ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 0 1 0 A(BC – BCB)A = ⎢ 0 0 1 0 ⎥ ⎢ ⎥ ⎢ ≠ 0 0 1 0 = (BC – BCB)A. 0 0 0 0 ⎥ ⎢ 0 0 0 0 ⎥ ⎣⎢ 0 0 0 0 ⎦⎥ ⎣⎢ 0 0 0 0 ⎦⎥ The case of normal ∇ -bands That ∇ is associative for any normal ∇-band has been seen already. The various criteria in this section provide essentially new proofs: Corollary 6.2.18. A normal ∇- band S is a skew lattice. (That is, ∇ must be associative.) Proof 1. The identity xyzw = xzyw implies a[ b, c]2 = 0 = [ b, c]2 a. Hence ∇ is associative by Theorem 6.3.8. Proof 2. Again, xyzw = xzyw implies that the criteria of Lemma 6.3.10 is satisfied: a( bc – cbc) a = 0 = a( bc – cbc) and a( bc – bcb) a = 0 = ( bc – bcb) a. Proof 3. Finally, xyzw = xzyw implies xyz = xyxz = xzyz. Thus given any e D u and f D v in S with uv = vu, the product ( eu○ fv)( ue○ vf) = ( eu + fv – eufv)( ue + vf – uevf) must reduce to ( e○ f)2 = e s f, so that s is associative by the criterion of Corollary 6.2.14. £ 229 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond 6.3 The question of s-closure While every s-band in a ring is regular as a band, not every regular band in a ring need generate a s-band in that ring. Thus far no simple necessary and sufficient criteria for a regular band in a ring to generate a s-band are known, though cases exist where successful generation is guaranteed. (By contrast, every regular band S is isomorphic to a regular band Sʹ in some ring such that Sʹ generates a cubic skew lattice in that ring.) When a s-band is generated from a band S, it is called the s -closure of S. Our next theorem describes what must occur at any stage in a successful generation of a s-band from a regular band in a ring. Theorem 6.3.1. Given a regular band S in a ring R and elements e, f ∈ S, S ∪ { e s f} generates a (possibly larger) regular band Sʹ in R if and only if for all a, b, c ∈ S, I) ea( e � f) bf = eabf. II) a( e – ef) abc( f – ef) c = a( e – ef) b( f – ef) c. Proof. Observe first that since S is regular, ea( ef + fe – efe – fef) bf = 0 making ea(– ef) bf = ea( fe – efe – fef) bf, so that ea( e s f) bf = ea( e� f) bf. Thus if the semigroup Sʹ generated from S and e s f is a regular band, then since ea, bf ≺ e s f we have ea( e s f) bf = eabf so that (I) must follow. In general, elements of the semigroup Sʹ generated from S and e s f look like a 0( e s f) a 1( e s f) a 2( e s f)… an–1( e s f) an with a 0, a 1, … , an ∈ S1 for n ≥ 1. (The n = 0 case is just a 0) For Sʹ to be a regular band, both ( e s f) a( e s f) b( e∇ f) = ( e s f) ab( e s f) (6.3.1) and [ a( e s f) b( e s f) c]2 = a( e s f) b( e s f) c (6.3.2) or by (6.3.1) a( e s f) bcab( e s f) c = a( e s f) b( e s f) c (6.3.2ʹ) must hold for all a, b c ∈ S. Conversely, (6.3.1) guarantees that all elements in Sʹ have the form a( e s f) b( e s f) c, (6.3.2) then guarantees that Sʹ is a band, and together with the regularity of S they guarantee that Sʹ is a regular band. Note in passing that the regularity of S implies that (6.3.2ʹ) can be replaced by a( e s f) abc( e s f) c = a( e s f) b( e s f) c. (6.3.2ʺ) Returning to (6.3.1), upon setting Λ = { e, f, fe, – efe, – fef}, we get ( e s f) a( e s f) b( e s f) = ∑ u, v ua( e s f) bv. ∈ Λ 230 VI: Skew Lattices in Rings Except for the two cases, ea( e s f) bf and fa( e s f) be, all of the ua( e s f) bv terms reduce to uabv terms due to the regularity of S. For example, efa( e s f) bfef = efae( e s f) bfef = efaebfef = efabfef. Thus (6.3.1) reduces first to ea( e s f) bf + fa( e s f) be = eabf + fabe and then to ea( e � f) bf + fa( f � e) be = eabf + fabe. (6.3.1ʹ) Clearly (I) implies (6.3.1ʹ) and when Sʹ is regular, (I) holds so that (6.3.1ʹ) follows. Thus (6.3.1ʹ) can be replaced by the stronger assertion, (I). Next, expanding (6.3.2ʺ) gives a( e + f + fe – efe – fef) abc( e + f + fe – efe – fef) c = a( e + f + fe – efe – fef) b( e + f + fe – efe – fef) c. Using the identity aeabcec = aebec holding for all regular bands, multiplying out the left side of (6.3.2ʺ) creates a number of terms that reduce immediately to the corresponding terms on the right. Of course we have, aeabcec = aeaebecec = aebec and afabcfc = afafbfcfc = afbfec, but also cases such as afe( abc) fefc = afef( abc) fefc = afef( b) fefc = afe( b) fefc, and aef( abc) fefc = aef( abc) efc = aef( b) efc = aef( b) fefc. Where this matching fails, (6.3.2ʺ) first reduces to a( e) abc( f) c + a( e) abc( fe – efe – fef) c + a( fe – efe – fef) abc( f) c = a( e) b( f) c + a( e) b( fe – efe – fef) c + a( fe – efe – fef) b( f) c. Using regularity again on products involving the underlined terms, the above reduces to a( e) abc( f – ef) c – a( ef) abc( f) c = a( e) b( f – ef) c – a( ef) b( f) c or a( e) abc( f) c – a( e) abc( ef) c – a( ef) abc( f) c = a( e) b( f) c – a( e) b( ef) c – a( ef) b( f) c. Regularity gives a( ef) abc( ef) c = a( ef) a( ef) bc( ef) c = a( ef) b( ef) c. (6.3.2) is thus reduced to (II). £ A successful generation of a s-band from a regular band requires that at each stage of generation the new regular band Sʹ also satisfies (I) and (II) in Theorem 6.3.1 for all a, b, c, e, f ∈ Sʹ. While (I) and (II) generally need not be passed on to later bands, here is a strategy for generating s-bands. To this end, a condition C potentially satisfied by regular bands in rings is s- inductive if 231 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond (i) any regular band S in a ring that satisfies C must satisfy (I) and (II), and (ii) for all e, f ∈ S, the regular band Sʹ generated from S ∪ { e s f} also satisfies C. Clearly: Theorem 6.3.2. Any regular band of idempotents in a ring that satisfies a s- inductive condition C will generate a s -band (its s- closure) in that ring. £ We consider three conditions that are s-inductive (in this parlance). We give new proofs based on the concept of s-induction. Since the first two cases are already known, we omit their elementary proofs based on s-induction. Corollary 6.3.3. Left [right] regularity is a s- inductive condition. £ Corollary 6.3.4. Normality is a s- inductive condition. £ Recall that a band S is totally pre-ordered if for all e, f ∈S, either e ≺ f or f ≺ e, that is, either efe = e or fef = f. The following nice result is due to Karin Cvetko-Vah [2005a]. Theorem 6.3.5. Being totally pre-ordered is a s- inductive condition. Thus every maximal totally pre-ordered regular band in a ring is a s -band. Proof. In verifying (I), we use the fact that ea( e� f) bf multiplies out to eaebf + eafbf – eaefbf. There are two cases to consider. Case 1: the ≺-maximum is e or f. Say e. Then eaebf + eafbf – eaefbf reduces to eabf + eafbf – eafbf = eabf. The subcase for f is similar. Case 2: the ≺-maximum is a or b. Say a. Here eaebf + eafbf – eaefbf reduces to ebf + efbf – efbf = ebf = eabf. The subcase for b is similar. For (II), first suppose that e ≺ f, so that e = efe. Then both sides of (II) reduce to 0: a( e – ef) abc( f – ef) c = a( e – ef) fabc( f – ef) c = a 0 abc( f – ef) c = 0 and a( e – ef) b( f – ef) c = a( e – ef) fb( f – ef) c = a 0 b( f – ef) c = 0. Likewise, this occurs when f ≺ e, so that (II) follows. Hence S ∪ { e s f} generates a regular band Sʹ in R. Is Sʹ also totally pre-ordered? Suppose that e ≺ f in S. (The f ≺ e case is similar.) Here e s f = f + fe – fef. Thus f( e s f) f = f while ( e s f) f ( e s f) = f ( e s f) = f ( f + fe – fef) = e s f. Since e s f D f in Sʹ, every element uʹ in Sʹ must be D-equivalent to some element u in S. Since S is totally pre-ordered, so is Sʹ. £ 232 VI: Skew Lattices in Rings ∇-bands open the door to cubic skew lattices that are distinct from any possible quadratic skew lattice a ring. This observation, however, is tempered by the following result: Theorem 6.3.6. Every cubic skew lattice (S: ∇, •) having a lattice section T in a ring R is isomorphic to a quadratic skew lattice (Sʹ: ○, •) in some ring Rʹ. This always occurs when S/D is countable and in particular for any (S: ∇, •) in a matrix ring over a (skew) field, in which case Rʹ can also be a matrix ring. Proof. Given a lattice section T with left and right extensions SL and SR, the internal Kimura decomposition gives an embedding of S into a quadratic skew lattice S → SR × SL ⊆ R × R. £ 6.4 Idempotent-closed rings A ring R is idempotent-closed if its set of idempotents E( R) is multiplicatively closed. We begin with a result in basic ring theory. Proposition 6.4.1 Let ring R be idempotent-closed. If R has an identity, then its idempotents commute. In general, if the idempotents of R commute, then E( R) is in the center of R and forms a generalized Boolean algebra (E( R); ∨, ∧, \, 0) upon setting e ∧ f = ef, e∨ f = e ○ f = e + f − ef and e \ f = e − efe. In this case, given e ∈ E( R), both eR and ann( e) = { x ∈ R⎪ ex = 0} are ideals and R decomposes internally as eR ⊕ ann(e) under the map x → ex + ( x − ex). Proof. If 1 ∈ R, then e(1− e) = 0 for all e ∈ E(R). Hence ef(1− e) = [ ef(1− e)]2 = 0 and thus ef = efe for all e, f in E( R). Similarly, fe = efe and thus ef = fe for all e, f in E( R). Next, assuming all idempotents commute, choose e ∈ E( R) and x ∈ R. Observe first that if ex = 0, then e + xe is idempotent; likewise, if xe = 0, then e + ex is idempotent. Thus in general, since e( x − ex) = 0 one has that e + xe − exe is idempotent. Multiplying each expression with e and using commutation, gives e + xe − exe = e = e + ex − exe, from which xe = exe = ex follows. The remaining assertions are easy consequences of this. £ A ring whose idempotents commute, and thus lie in the center of the ring, is called abelian. Such rings are clearly idempotent-closed. We are less concerned here with the internal structure of these rings, than in their role in the class of idempotent-closed rings. Suffice it to say, under mild hypotheses abelian rings decompose into direct sums of rings with identity whose only idempotents are 0 and 1. (See Section 6.5.) Moving beyond the abelian case, first recall that a band S is normal if it satisfies any and hence all of the following equivalent identities: ( a) efge = egfe. ( b) efgh = egfh. ( c) efege = egefe. 233 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 6.4.2 If a ring R is idempotent-closed, then: i) E( R) is a normal band under multiplication. ii) E( R) is also closed under ∇ which is idempotent and associative on E( R). iii) (E( R); ∇, •) is a strongly distributive (distributive, normal and symmetric) skew lattice. iv) Upon setting e \ f = e – efe, (E( R); ∇, •, \, 0) is a skew Boolean algebra. Proof. (i) Given e ∈ E( R), the principal subring eRe has an identity e and thus eE( R) e = E( eRe) is commutative by the above proposition. Hence E( R) satisfies ( c) and is normal. (ii) and (iii) follow from Theorems 2.3.6 and 2.3.7. Finally, let e\ f as e – efe in R. As such, e\ f is the complement of e∧ f∧ e in E( eRe) and satisfies the characterizing identities in Section 4.1: e = ( e∧ f∧ e) ∨ ( e \ f) and ( e∧ f∧ e) ∧ ( e \ f) = 0. £ Recall that E( R) is partially ordered by e ≥ f if ef = f = fe with e > f denoting e ≥ f ≠ e. Recall also that an idempotent e > 0 is primitive if no idempotent f exists such that e > f > 0. E( R) is 0-primitive if all of its non-0 elements are primitive. In this case, M( R) denotes E(R)\{0}. Recall that a band is rectangular if efe = e, or equivalently, efg = eg holds. A 0- rectangular band M 0 consist of a rectangular band M and a distinct element 0 ∉ M, so that x 0 = 0 = 0 x. Abstractly viewed, every primitive skew Boolean algebra has operations induced from a 0-rectangular band M 0: given x, y ∈ M, 0∧ x = 0 = x∧0, 0∨ x = x = x∨0, x∧ y = xy = y∨ x, x\0 = x and 0\ x = 0 = x\ y. In this simple manner, 0-rectangular bands are in 1-1 correspondence to the class of primitive skew Boolean algebras. Theorem 6.4.2 If E( R) is 0-primitive and closed under multiplication, then under multiplication, E( R) is a 0-rectangular band with e∇ f = fe on M( R). Proof. Given the assumptions on E( R), let e ≠ f in M( R). Then e ≥ efe ≥ 0 in E(R) so that efe is either 0 or e. If efe = 0, then so are fef, ef and fe (since, e.g., ef = efef = 0). It follows that e + f is an idempotent greater than either e or f and hence not primitive. Thus efe = e for all e, f in M( R) making M( R) a rectangular band with e∇ f = e + f + fe – efe – fef reducing to fe. £ Idempotent-dominated rings How does E( R) being multiplicatively closed affect the behavior of R? To answer this one is pressed to find reasonable assumptions connecting E( R) to all of R in order to obtain consequences for all of R. To do so we begin with the following lemma. Theorem 6.4.3 Given a ring R, set Γ( R) ={ x⎪ ex = x = xf, for some e, f ∈ E( R)}. Then: i) Γ( R) is a multiplicatively closed set containing E( R) that also has negatives: x ∈ Γ( R) implies −x ∈ Γ( R). ii) When E( R) is idempotent-closed, Γ( R) = ∪{ eRe⎪ e ∈ E( R)}. iii) In general, the set Q( R) of all finite sums of elements in Γ( R) is the smallest ideal in R containing E( R). 234 VI: Skew Lattices in Rings Proof. (ii) is clear. If E( R) is multiplicative, x ∈ Γ( R) and e, f ∈ E( R) are such that ex = x = xf, then set g = f∇ e. By absorption, ge = e and fg = f. Hence gx = gex = ex = x and likewise xg = x so that gxg = x and (ii) holds. For (iii), first, let Qʹ be the least ideal of R containing E( R). Clearly Q( R) is a subring of Qʹ. Conversely, Qʹ consists of sums of terms of the form xe, ey and xey where e ∈ E( R) and x, y ∈ R. For such e, x, y, observe that f = e + xe − exe and h = e + ey − eye are both idempotent, so that xe, ey and thus xey must lie in Q( R). Indeed as defined, ef = e and fe = f so that ff = ( fe) f = f( ef) = fe = f. Similar observations hold for h. £ Elements in Γ( R) are said to be idempotent-covered. R as a whole is idempotent-covered if its elements are thus. Rings with identity and von Neumann regular rings are trivially idempotent-covered. For such rings we have an extension of Proposition 6.4.1. Theorem 6.4.4 Given an idempotent-covered ring R, E( R) is multiplicative if and only if R is abelian. Proof. Let R be idempotent-covered and idempotent-dominated. Given e, f in E( R), by Theorem 6.4.3 g ∈E( R) exists such that g( ef − fe) = ef − fe = ( ef − fe) g. By normality, ef − fe = g( ef − fe) g = gefg − gfeg = gefg − gefg = 0. Thus E( R) is commutative and R is abelian. The converse is clear. £ A ring R is idempotent-dominated if it is generated from the set Γ( R) of all idempotent-covered elements, or put otherwise, if R = Q( R). Clearly idempotent-covered rings are idempotent-dominated, but not conversely. Idempotent-covered abelian rings enter into the general case of idempotent-closed rings in at least two ways. First, for any idempotent e in an idempotent-closed ring R, the principal subring eRe is abelian. Secondly, we will see that each idempotent-closed and dominated ring has a maximal abelian image that in many cases arises as a major subring of the given ring. Being idempotent-dominated can have a side effect. For any idempotent-covered ring R, the annihilator ideal vanishes, ann( R) = {0}. If R is just idempotent-dominated, this need not be the case, even when E( R) is also multiplicative. Indeed, given e, f in a multiplicatively closed E( R), the following small rectangular band arises efe R ef L L fe R fef and the combination α( e, f) = ef + fe − efe − fef can lie in ann( R). This is of interest since for all idempotents e and f we have e∇ f = e ○ f + α( e, f). Indeed we have: 235 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Lemma 6.4.5 When R is idempotent-closed and dominated, then: i) For all x, y ∈ R and all e, f ∈ E( R), xefy = xfey. ii) In particular, if e D f then xey = xfy. Consequently, α( e, f) ∈ ann( R) for all e, f ∈ E( R) so that e∇ f = e ○ f when ann( R) = {0}. Proof. Given that R is idempotent-dominated and E( R) is normal, xefy = xfey must hold as stated. If e D f then xey = xeffey = xfeefy = xfy. Thus xα( e, f ) = x( ef + fe − efe − fef) = x( ef + fe − fe − ef) = 0 for all x ∈ R, and similarly, α( e, f) x = 0. £ Given the above assumptions it is easy to see that ann( R) = {0} when E( R) is either left-handed ( efe = ef) or right-handed ( efe = fe). While ann( R) need not vanish, this is not the full story. Lemma 6.4.6 If R is idempotent-dominated, then i) ann( R) = { x ∈ R⎪ xe = 0 = ex for all e ∈ E( R)} ii) ann[ R/ ann( R)] = {0}. Proof. In general, ann( R) = { x ∈ R⎪ xy = 0 = yx for all y ∈ R} and if π: R → R/ ann( R) is the induced homomorphism, then ann( R) ⊆ π−1{ ann[ R/ ann( R)]} with π−1{ ann[ R/ ann( R)]} = { x ∈ R⎪ xyz = 0 = yzx = yxz for all y, z ∈ R}. But if R is idempotent-dominated, then ann(R) = { x ∈ R⎪ xe = 0 = ex for all e ∈ E( R)}. Hence π−1{ ann[ R/ ann( R)]} ⊆ ann( R) and equality follows and ann[ R/ ann( R)] vanishes in R/ ann( R). Thus, if R is idempotent-dominated, the description of ann( R) in the lemma insures that π−1{ ann[ R/ ann( R)]} ⊆ ann( R). Equality follows, as does the lemma. £ Lemma 6.4.7 If R is a ring with ideal I ⊆ ann( R), then the induced epimorphism π: R → R/I restricts to a bijection πE: E( R) → E( R/I). If E( R) is also multiplicative, so is E( R/I) and πE is an isomorphism of skew Boolean algebras. Proof. πE is a well-defined map between the stated sets. If π( e) = π( f) for e, f ∈ E( R), then e = f + a for some a ∈ ann(R) and squaring gives e = f. Thus πE is at least injective. Given x + I ∈ E( R/I), x 2 = x+a for some a ∈ ann( R) and hence x 4 = x 2 so that x 2 ∈ E( R) and πE is bijective. Since π is a ring homomorphism, the rest of the lemma follows. £ Since eRe ∩ ann( R) = {0} for all idempotents e in any ring R, we have: 236 VI: Skew Lattices in Rings Theorem 6.4.8 If R is both idempotent-dominated and idempotent-closed, then R/ann( R) has both properties with ann(R/ ann(R)) = {0} so that e∇ f = e○ f in R/ ann( R). The natural epimorphism π: R → R/ ann( R) induces a skew Boolean algebra isomorphism πE: E( R) ≅ E( R/ ann(R)) and ring isomorphisms πE: eRe → π( e)( R/ ann( R))π( e) between corresponding principal subrings. £ When ann(R) ≠ {0}, R/ ann( R) provides a cleaner, trimmer version of R, sharing many of its characteristics, but without a non-vanishing annihilator ideal. The canonical ideal K R The annihilator ideal is a part of generally larger canonical nilpotent ideal KR that all rings possess. We begin with an example. Example 1 For n ≥ 1, M n( F) is the ring of n × n matrices over a field F. M n( F) is trivially idempotent-covered since it has an identity, but E(M n( F)) is never multiplicative unless n = 1. For n ≥ 2, given fixed integer parameters i, j, k ≥ 0 subject to i + j + k = n and 1 ≤ j ≤ n, consider the subring Rn of all matrices with the following block design satisfying the added i, j, k restriction that D be a diagonal matrix: ⎡ ⎤ 0 i× i Ai× j Ci× k ⎢ ⎥ Dijk = ⎢ ⎥ . 0 j× i D j× j B j× k ⎢ ⎥ ⎢ 0 k× i 0 k× j 0 k× i k ⎣ ⎦⎥ Rn is idempotent-dominated and idempotent-closed. Rn and E( Rn ) are noncommutative i, j, k i, j, k i, j, k when j < n and commutative when j = n. The idempotents of Rn are the matrices where D has i, j, k only 0-1 entries in the diagonal, AD = A, DB = B and AB = C (for cases where A, B or C do not disappear.) E( Rn ) is right-handed when i = 0 and left-handed when k = 0, with C vanishing in i, j, k either case along with A or B, respectively. For i, k > 0 so that j ≤ n − 2, ann( Rn ) is nontrivial i, j, k consisting of all matrices for which blocks D, A and B are 0-submatrices. The strictly upper triangular matrices (where D = 0) form a nilpotent ideal, K. Considering just addition, the additive group of the ring is the direct sum of the group of all diagonal matrices (a maximal idempotent-covered subring with central idempotents) and the group of strictly upper triangular matrices (the nilpotent ideal K). To what extent does such a direct decomposition characterize idempotent-dominated rings with multiplicative sets of idempotents? This leads us to the following considerations. £ 237 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Given a ring R, K R denotes the ideal { x ∈ R⎪ uxv = 0 for all u, v ∈ R}. Clearly K R K R K R = {0} so that K R is a nilpotent ideal of index 3. Lemma 6.4.9 If R is idempotent-dominated, then K R = { x ∈ R⎪ exf = 0 for all e, f ∈ E( R)}. If R is also idempotent-closed, then K R = { x ∈ R | exe = 0 for all e ∈ E( R)}. Proof. The first statement is clear. If R is also idempotent-closed and exe = 0 for all e ∈ E( R), then for all e, f ∈ E( R), exf = e( e∇ f) x( e∇ f) f = e 0 f = 0 and the second statement follows. £ Theorem 6.4.10 If R is an idempotent-dominated and idempotent-closed ring, then the ring R/K R is the maximal abelian image of R. It is also idempotent-covered with E( R/K R) ≅ E( R)/D. Conversely, if R is an idempotent-dominated ring for which R/K R is abelian, then R is idempotent-closed. (In general, if R is any ring with an ideal K such that RKR = {0}, then R is idempotent-closed if and only if R/K is idempotent-closed, in which case E( R/K) is a homomorphic image of E( R) with both sharing a common maximal lattice image.) Proof. The quotient ring R/K is automatically idempotent-dominated. Suppose x + K ∈ E( R/K) so that x + K = xn + K for all n ≥ 1. Then x 2 = x + k for some k ∈ K. From this we get x 4 = x 3 + xkx = x 3, and hence x 6 = x 3 so that x + K = e + K for some e ∈ E( R). Thus all idempotents in R/K come from idempotents in R, making E( R/K) multiplicative. By Lemma 6.4.5, ef − fe ∈ K for all e, f ∈ E( R), making E( R/K) commutative. Being idempotent-dominated, this forces R/K to be idempotent-covered. Indeed, given any idempotent-dominated abelian ring S, for any x = e 1 x 1 e 1 + · · · + enxnen ∈ S, upon setting f = e 1∇ e 2∇ . . . ∇ en we have f ≥ all ei so that fxf = f. Next, let α: R → A be a homomorphism onto a ring A having only central idempotents. A is automatically idempotent-dominated and hence idempotent-covered so that KA = {0}. Given k ∈ K, since α is surjective, α( k) is in K A so that α( k) = 0. Thus K ⊆ ker(α) and the maximality of the abelian image R/ K follows. The converse follows from the remark following the theorem. So let R be a ring with an ideal K such that RKR = {0}, and let e, f ∈ E( R) be given. If R/K is idempotent-closed, then at least efef = ef + k for some k ∈ K. But then efef = e( efef) f = e( ef + k) f = ef + ekf = ef, so that R is idempotent-closed. Likewise, if R is idempotent-closed, then so is R/K by the above argument for R/K with E( R/K) again a homomorphic image of E( R). Given idempotent closure, 238 VI: Skew Lattices in Rings if e > f in E( R), then e + K ≠ f + K in E( R/K), for otherwise e = f + k for some k ∈ K so that e = e( f + k) e = efe = f in R. This forces images of distinct D-classes in E( R) to remain distinct in E( R/K), so that E( R)/D ≅ E( R/K)/D. £ In general, ann( R) ⊆ K R with the inclusion often proper; but thanks to Theorems 6.4.8 and 6.4.10, for an idempotent-closed and dominated ring R, ann( R) = K R precisely when both ideals vanish and R idempotent-covered with central idempotents. When E( R) has a lattice section, the ideal K R has a natural near-complement in the ring R. Theorem 6.4.11 Let R be idempotent-dominated and idempotent-closed. If E( R) has a lattice section E 0, then setting A = { x ∈ S⎪∃ e ∈ E 0, x = exe}, we have the following: i) A is an idempotent-covered abelian subring of R. ii) As additive groups, R = A ⊕ K R. iii) The natural epimorphism R → R/K R induces a ring isomorphism A ≅ R/ K R. Conversely, let S be any ring with an abelian subring A and an ideal K such that SKS = {0} and as additive groups, S = A ⊕ K so that as rings, S/K ≅ A. Then S is idempotent-closed and E( A) is a lattice section of E( S); moreover, for all a ∈ E( A), the D- class D a in E( S) consists of all elements of the form ( a + ka)( a + ak) with k ∈ K. Proof. Suppose exe = x and fyf = y for e, f ∈ E 0. Setting g = e∇ f = e o f in E 0, g ≥ e, f so that gxg = x and gyg = y also. This gives g(x ± y) g = gxg ± gyg = x ± y and g( xy) g = ( gx)( yg) = xy. Thus A must be an idempotent-covered subring for which E(A) is multiplicative. By Theorem 6.4.4, (i) follows. Suppose that we are given x = e 1 x 1 e 1 + · · · + enxnen ∈ R. For each ei in E( R) let fi ∈ E 0 be such that ei D fi and set y = f 1 x 1 f 1 + · · · + fnxnfn ∈ A. We claim that x − y is in K R. We need only show that each eixiei − fixifi ∈ K R. But due to Lemma 6.4.5(ii), for all u, v in R: u( eixiei − fixifi) v = ueixieiv − ufixifiv = ueixieiv − ueixieiv = 0. Hence x − y ∈ K R so that x = y + ( x − y) ∈ A + K R. Thus R = A + K R and clearly A ∩ K R = {0} so that (ii) follows. From (i) and (ii), (iii) follows. Conversely, that S is idempotent-closed follows from the remark after Theorem 6.4.10. Given a ∈ A and k ∈ K, ( a + k)2 = a 2 + ( ka + ak + k 2) ∈ A⊕ K. Thus a + k is idempotent if and only if a 2 = a in A and k = ka + ak + k 2 in K. The latter gives k 2 = kak so that when idempotent, a + k = ( a + k)2 = ( a + ka)( a + ak). 239 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Here a( a + k) a = a + aka = a and ( a + k) a( a + k) = ( a + ka)( a + ak) = a + k. Thus a D ( a + k) making E( A) is a lattice section of E( S). It is easily seen that any element ( a + ka)( a + ak) is idempotent whenever a ∈ E( A) and k ∈ K. £ Thus the observations of Example 1 generalize to all idempotent-closed and dominated rings R such that E( R) has a lattice section. For rings satisfying the chain conditions of the next section, this is the case. (The latter include all finite-dimension matrix ring examples.) E( R) also has a lattice section if it has a maximal D-class M, since for any e ∈ M, { f ∈ E( R)⎪ f ≤ e} is a lattice section. While K R generally exceeds ann( R), this is not the whole story. K R also contains two related canonical ideals, the left and right annihilator ideals of R: annL( R) = { x ∈ R⎪ xy = 0 for all y ∈ R} and annR( R) = { y ∈ R⎪ xy = 0 for all x ∈ R}. It happens that annL( R) + annR( R) = K R for all idempotent-dominated and idempotent-closed rings. We prove this when E( R) is bounded with a maximal D-class. The general case follows from the next proposition and Theorems 6.5.1 and 6.5.3 below. An idempotent-dominated and closed ring R is bounded if E( R) has a maximal D-class, M, consisting of all m in E( R) such that m∇ e∇ m = m and eme = e for all e ∈E( R). Such an m is called maximal in E( R) and xmy = xy for all x, y in R. Every element x ∈ R is a sum of elements miximi where each mi is in M Proposition 6.4.12 Let R be a bounded idempotent-dominated and closed ring and let m be maximal in E( R). Then as an additive group, K R = K Rm ⊕ ann( R) ⊕ m K R where K Rm = { km⎪ k ∈ K R} and m K R = { mk⎪ k ∈ K R}. In particular, annR( R) = K Rm ⊕ ann( R) and annL( R) = ann( R) ⊕ m K R. Finally, µ: R → mRm defined by µ( x) = mxm is a ring homomorphism onto the abelian subring mRm, with kernel K R. Proof. Setting K = K R, the identity k = km + ( k – km − mk) + mk gives K = K m + ann( R) + m K. Let that k, k′ ∈ K and a ∈ ann( R) be such that km + a + mk′ = 0. Then mk′ = m( km + a + mk′) = m 0 = 0 and similarly km = 0, leaving a = 0 also. Thus the sum is direct. Next let x ∈ annL( R). Being in K, x has the form km + a + mk′. Thus 0 = xm = km, so that x = a + mk′, giving annL( R) ⊆ ann( R) ⊕ m K. The reverse inclusion is trivial. That annR( R) = K m ⊕ ann( R) is seen in similar fashion. Since xmy = xy in R, the final statement is clear. Thus as additive subgroups, R = mRm ⊕ K m ⊕ ann( R) ⊕ m K. £ 240 VI: Skew Lattices in Rings Returning to Example 1, blocks D, A, C and B correspond respectively to mRm, K m, ann( R) and m K. We briefly consider a class of rings that are always idempotent-closed. A ring is weakly commutative if the identity xyzw = xzyw holds. Such a ring R has a nil radical N R consisting of all nilpotent elements in R. N R is indeed an ideal and R/N R is commutative with a vanishing nil radical. Given a commutative ring A and a normal band S, the semigroup ring A[ S] is weakly commutative. This makes idempotent-closed rings easy to find. Indeed, all examples in this section are weakly commutative. In any idempotent-closed ring, E( R) generates a weakly commutative subring, denoted if Q 0( R). Theorem 6.4.13 If R is a weakly commutative ring, then E( R) is multiplicative and the subring eRe is commutative for each e ∈ E(R) . The converse also holds if R is idempotent-dominated. Finally, for any ring R, E( R) is multiplicative if and only if Q 0( R) is weakly commutative. Proof. Given e, f ∈ E( R), ( ef)2 = efef = eeff = ef. Also, given exe, eye in eRe, we have ( exe)( eye) = e( exe)( eye) e = e( eye)( exe) e = ( eye)( exe). Conversely let R be idempotent-dominated with E( R) being multiplicative and each subring eRe, for e ∈ E( R), being commutative. Let eae, fbf, gcg, hdh in Γ( R) be given with e, f, g, h ∈ E( R). Since E( R) is multiplicative, as in the proof of Lemma 6.4.7, e′, f′, g′, h′ in E( R) exist such that e′ ≥ e, f′ ≥ f, g′ ≥ g, h′ ≥ h with e′, f′, g′ and h′ being D-related. Thus we may assume at the outset that e, f, g and h are D-related. This plus the assumption that each eRe be commutative gives ( eae)( fbf)( gcg)( hdh) = ( eae) e( fbf) e( gcg) e( hdh) = ( eae) e( gcg) e( fbf) e( hdh) = ( eae)( gcg)( fbf)( hdh) holding in Γ( R). Distribution extends the identity xyzw = xzyw from Γ( R) to all of R. If we just assume E( R) is normal, then this property extends via distribution to weak commutativity of the generated subring Q 0( R). The converse is clear. £ Idempotent-closed rings in general What can be said about an arbitrary idempotent-closed ring R where Q( R) could be a proper ideal? We begin by quoting a standard result in ring theory. Lemma 6.4.14 Given a ring R with a nil ideal N, every idempotent in R/N is of the form e + N for some idempotent e in R. £ As an immediate consequence we have: 241 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 6.4.15 In any ring R, K = K Q(R) is a nil ideal and Q( R/ K) = Q( R)/K. Thus R is idempotent-closed if and only if R/ K is abelian. (Note that R K R need not be {0} here.) Proof. Indeed, for any ideal I in R, K I is a nil ideal of R. Thus for K = K Q(R), each idempotent in R/K has the form e + K for some idempotent e in R, so that Q( R/ K) = Q( R)/K. Suppose R/ K is abelian. Then Q( R)/K = Q( R/ K) is also, in which case Q( R) is idempotent-closed by Theorem 6.4.10, and so is R. Conversely, if R is idempotent-closed, then Q( R)/K is abelian, and since Q( R) contains all the idempotents of R, Lemma 6.4.14 above assures that all idempotents in R/ K commute, making it abelian. £ What can one say about R/ Q( R) in general? In particular, is E( R/ Q( R)) = {0}? This would be an extreme case of idempotent-closure. The answer is affirmative when R is idempotent-closed. We begin with a special case: Lemma 6.4.16 If a ring R is abelian, then E( R/ Q( R)) = {0}. Proof. Letting Q = Q( R), suppose that x + Q is idempotent in R/Q. Thus x 2 − x ∈ Q. It follows that e in E(R) exists such that x 2 − x = e( x 2 − x) = ( ex)2 − ex. Moreover, x decomposes as ex + y where y = x − ex so that ey = 0. Then 2 x − x = ( ex + y)2 − ( ex + y) = ( ex)2 + y 2 − ( ex + y) = [( ex)2 − ex] + [ y 2 − y]. But since x 2 − x = ( ex)2 – ex, y 2 − y = 0 in R, that is, y is in E( R), so that x ∈ Q and x + Q is the zero element in R/Q. £ Theorem 6.4.17 If R is idempotent-closed, then E( R/ Q( R)) = {0} = Q( R/ Q( R)). Proof. Again, Q = Q( R) is an ideal and K = K Q is a nil ideal of R. From R/Q ≅ ( R/K)/( Q/K) and the preceding two results, E( R/ Q( R)) = {0}, and hence Q( R/ Q( R)) = {0} follows. £ Thus squeezing Q( R) to a point eliminates any nonzero idempotents in R/ Q( R) provided R is idempotent-closed. In general, this need not be so. In any case, every idempotent-closed ring R is a ring extension of an idempotent-dominated subring Q by a ring T = R/Q for which E( T) = {0}. From the standpoint of the idempotents, Q and T are polar opposites: the idempotents are maximally engaged in Q, while they are minimally engaged in T. In turn Q is an extension of a nilpotent ring K for which E(K) = {0} by an idempotent-covered abelian ring A. Indeed, in many cases Q is a “subdirect sum” of K and an internal copy of A. We consider other decompositions in the following section. 242 VI: Skew Lattices in Rings 6.5 Decomposing E( R) and R Recall that, every skew Boolean algebra is a subdirect product of primitive algebras, thanks largely to the following restatement of Theorem 4.1.4. Theorem 6.5.1 Given a D- class A of a skew Boolean algebra S, set S 1 = { e ∈ S⎪ e∧ a∧ e = e for some ( and hence all) a ∈ A}, and S 2 = { f ∈ S⎪ f∧ a = a∧ f = 0 for some ( and hence all) a ∈ A}. Then both S 1 and S 2 are subalgebras of S, all elements of S 1 commute with all elements of S 2 and the map µ: S 1 × S 2 → S defined by µ( e 1, e 2) = e 1 ∨ e 2 is an isomorphism of skew Boolean algebras. The inverse isomorphism is given by µ−1( e) = ( e∧ a∧ e, e \ e∧ a∧ e) . £ Described otherwise, S 1 is the union of the D-class A and all lower D-classes in the generalized Boolean lattice S/D, while S 2 consists of all D-classes B that meet A and its lower D-classes at {0} in S/D. In fact S 1 and S 2 are ideals of S where by an ideal of a skew lattice S is meant any subset I such that e∨ f ∈ I for all e, f ∈ I, and both e∧ g, g∧ e ∈ I for all e ∈ I and all g ∈ S. S 1 corresponds to the principal ideal in S/D determined by the element A of S/D while S 2 corresponds to the ideal in S/D consisting of all elements of S/D that meet A at 0. If S = E( R), how does this play out in the full context of the host ring, R? We begin by passing from skew lattice ideals I in E( R) to their induced ring ideals QI in R. Lemma 6.5.2 Let R be idempotent-closed and let I be an ideal of E( R). Then the least ideal QI of R containing I consists of all sums ∑ eixifi where both ei, fi ∈ I and xi ∈ R. (As with Theorem 6.4.3, all elements in QI also have the form ∑ eixiei where ei ∈ I.) Proof. Since −( xy) = (− x) y = x(− y), the set of all such sums is at least a subring Rʹ of R. Clearly I ⊆ Rʹ ⊆ QI. On the other hand, QI consists of sums of the form ∑ xieiyi where ei ∈ I and xi, yi ∈ R. But each such sum lies in Rʹ since its terms must. Indeed, given e ∈ I, f = e + xe − exe satisfies fe = f and ef = e, forcing f to be an idempotent in I, thus ensuring xe = f – e + exe ∈ Rʹ. Similarly ey and thus xey also lie in Rʹ. £ Theorem 6.5.3 Let ring R be idempotent-closed and dominated, and let I and J be ideals of E( R) such that each e ∈ E( R) is uniquely f + g for some f ∈ I and g ∈ J. Then: i) R = QI + QJ and QIQJ = { qq′ ⎢ q ∈ QI & q′ ∈ QJ} = {0} = QJQI. ii) σ: QI ⊕ QJ →R given by σ( q, q′) = q + q′ is a ring homomorphism onto R, that restricts to isomorphisms, σI : QI ⊕ {0} ≅ QI and σJ :{0} ⊕ QJ ≅ QJ. iii) In general, QI ∩ QJ ⊆ ann( R) with QI ∩ QJ possibly exceeding {0}; iv) σ is an isomorphism when QI ∩ QJ = {0}, and in particular when ann( R) = {0}. 243 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof. The given decomposition of E( R) implies first that I ∩ J = {0} and thus IJ = JI = {0}. Lemma 6.5.2 now gives QIQJ = {0} = QJQI. Since E( R) = I + J and R is both idempotent-closed and dominated, R consists of sums of elements of the form ( e+f) x( e+f) where e ∈ I and f ∈ J. But ( e+f) x( e+f) = exe + exf + fxe + fxf = exe + fxf ∈ QI + QJ. Indeed exf ∈ QI ∩ QJ and so exf = e( exf) = 0 since QIQJ = {0}. Likewise fxe = 0 and (i) is seen. (ii) is a consequence of (i), as is the inclusion of (iii). That QI ∩ QJ can exceed {0} is seen in the next example. Since ker( σ) = {( x, – x)⎪ x ∈ QI ∩ QJ}, (iv) follows. £ ⎧⎡ 0 a b p ⎤ ⎫ ⎪⎢ ⎥ ⎪ ⎪⎪ Example 6.5.1 R is the matrix ring 0 u 0 c ⎨⎢ ⎥ a, b, c, d, u, v, p ∈ F⎬ where F is any 0 0 v d ⎪⎢ ⎥ ⎪ ⎪⎢ 0 0 0 0 ⎩⎣ ⎦⎥ ⎭⎪ field. E( R) is multiplicatively closed with four D-classes described as follows: ⎧⎡ 0 m n mp + nq ⎤⎫ ⎧⎡ ⎤⎫ ⎧⎡ ⎤⎫ ⎪⎢ ⎥ 0 a 0 ac 0 0 b bd ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 1 0 p ⎪⎢ ⎥ ⎪⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎨ > 0 1 0 c ⎨⎢ ⎥⎬ & 0 0 0 0 ⎨⎢ ⎥⎬ > 04×4 { } ⎢ ⎬ 0 0 1 q ⎥ ⎪ 0 0 0 0 ⎪⎢ ⎥ 0 0 1 d ⎪⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪ 0 0 0 0 ⎩⎣ ⎦⎭⎪ ⎩⎪⎢ 0 0 0 0 ⎣ ⎦⎥⎭⎪ ⎩⎪⎢ 0 0 0 0 ⎣ ⎦⎥⎭⎪ where a, b, c, . . . , q vary freely over F. If I and J are the primitive skew lattice ideals determined by the middle left and middle right D-classes, then QI, QJ and ann( R) = ann( QI) are represented respectively by ⎡ 0 a 0 v ⎤ ⎡ 0 0 b v ⎤ ⎡ 0 0 0 v ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 u 0 c ⎢ ⎥ 0 0 0 0 0 0 0 0 , ⎢ ⎥, ⎢ ⎥ . £ 0 0 0 0 ⎢ ⎥ 0 0 u d ⎢ ⎥ 0 0 0 0 ⎢ ⎥ ⎢ 0 0 0 0 ⎣ ⎦⎥ ⎢ 0 0 0 0 ⎣ ⎦⎥ ⎢ 0 0 0 0 ⎣ ⎦⎥ Theorem 6.5.3 can be applied repeatedly. Doing so leads to a modification of direct sum decompositions of rings, which we describe this in full generality, independent of any special assumption on R and E( R). We begin with a ring R and a set of subrings of R, { Qi⎪ i ∈ I}. Let ⊕ ∑ Q i∈ I be the direct sum of the Q i i, the subring of the direct product ∏ Qi consisting of all I- tuples with only finitely many non-0 components. Define σ: ⊕ ∑ Q i∈ I → R by σ(⟨ x i i⎪ i ∈ I⟩) = ∑ xi. σ preserves addition. It preserves multiplication precisely when ∑ xi∑ yi = ∑ xiyi holds, which it does if xiyj = 0 for all i ≠ j. If in addition the image ∑ Q = R, then each Q i i must be an ideal of R. When R is a sum of ideals Qi where QiQj ={0} for all i ≠ j, we say that R is the orthosum of the Q ⊕ –1 ⊕ i. With σ as above, σ([ ∑ ann( Q ] = ann( R) and [ ann( R)] = ∑ . From i ) i∈ I σ ann( Qi) i∈ I this we have: 244 VI: Skew Lattices in Rings Proposition 6.5.4 If R is an orthosum of ideals Q ⊕ i and σ: ∑ Q i∈ I → i R is the sum epimor- phism, then σ: ⊕ ∑ ⊕ Q i∈ I → ∑ is isomorphic with i R is an isomorphism if and only if ann( Qi) i∈ I ann( R) under the restricted map. An isomorphism ⊕ τ: ∑ i∈ I ( Q ) i ann( Qi ) → R/ ann( R) is defined by τ(⟨ xi + ann( Qi)⎪ i ∈ I⟩) = ∑ xi + ann( R) . Proof. Since ann( ⊕ ∑ ⊕ ⊕ Q i∈ I ) ∑ , if σ is an isomorphism of ∑ with i = ann( Qi) i∈ I Q i∈ I i R, then σ restricts to an isomorphism, ⊕ ∑ ann( Qi) i∈ I ≅ ann( R). Conversely, σ is at least surjective. Suppose σ(⟨ xi⎪ i ∈ I⟩) = 0 in R. If some xi ≠ 0, then for some finite set of non-0 elements from distinct Qi we have x 1 + x 2 + … + xn = 0. Since 0 is in ann( R), each xi lies in the annihilator of it respective ideal. Thus if σ restricts to an isomorphism, ⊕ ∑ ann( Qi) i∈ I ≅ ann( R), then ⟨ xi⎪ i ∈ I⟩ = ⟨0⎪ i ∈ I⟩, making σ an isomorphism. The final isomorphism is the one induced from the combined epimorphism ⊕ ∑ Q i∈ I i σ ⎯ → ⎯ R π ⎯ → ⎯ R/ ann( R) where π is the canonical map. £ Regarding the idempotents, we also have: Proposition 6.5.5 If R is the orthosum of ideals Q ⊕ i and σ : ∑ Q i∈ I → R is the sum i epimorphism, then the restriction σ ⊕ E: ∑ E( Q i∈ I → E( R) is a bijection of sets that is an i ) isomorphism of skew Boolean algebras, whenever E( R) is multiplicative. ⊕ ∑ E( Q i∈ I i ) is the subset of ⊕ ∑ ⊕ Q i∈ I where all x ∑ ) and is multiplicative iff each E( Q i i ∈E( Qi). It equals E( Q i∈ I i i) is.) Proof. Since any sum of mutually orthogonal idempotents is also idempotent, σ E is at least well-defined. Let e ∈ E( R) equal x 2 2 1 +...+ xr, with each xi ∈ Qi. Then e = e 2 = x 1 +...+ xr and thus for each i ≤ r, x 2 4 2 2 i = xi + ai where ai ∈ ann( R). Again one has xi = xi so that each xi ∈ E( Qi) and σ E is at least surjective. Let e ∈ E( R) be represented as both e 1 +...+ er and f 1 +...+ fr where ei, fi ∈ E( Qi). (By letting some values be 0 we may assume a common indexing.) But then ei = fi + ai where ai ∈ ann( R) for each index i. Squaring both sides gives ei = fi. Thus σE is a bijection. Finally, since σ is a ring homomorphism, the final assertions are clear. £ Given a ring R, E( R) satisfies the descending chain condition (the DCC) if any sequence e 1 ≥ e 2 ≥ e 3 ≥ . . . in E( R) eventually stabilizes: en = en+1 = . . . . The ascending chain condition on E( R) (the ACC) is defined in dual fashion. The latter implies the former since a descending chain e 1 ≥ e 2 ≥ e 3 ≥ . . . in E( R) induces a corresponding ascending chain of idempotents e 1 − e 2 ≤ e 1 – e 3 ≤ . . . with both stabilizing, if they do, simultaneously. 245 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond In what follows by a rectangular ring we mean an idempotent-dominated ring R for which E( R) is a 0-rectangular band. Theorem 6.5.6 Given an idempotent-closed and dominated ring R for which E( R) satisfies the descending chain condition: i) E( R) = ⊕ ∑ P i∈ I where the P i i are the primitive bands given by the union the minimal nonzero D- classes of E( R) with {0}. ii) R is an orthosum ideals ∑ Q i∈ I where each ideal Q i i is a rectangular subring for which E( Qi) = Pi. iii) R/ ann( R) ⊕ ≅ ∑ Q where annihilators of all quotient rings vanish. i ann( Qi ) i∈ I iv) In particular, R ⊕ ≅ ∑ Q i∈ I when ann( R) vanishes. i v) As skew Boolean algebras, E( R) ≅ E[ R/a nn(R)] and Pi ≅ E[ Qi/ann( Qi)]. Proof. (iii) through (v) follow from (i) and (ii) and Results 6.4.6 and 6.5.3 - 6.5.5. The DCC on E( R) plus the normality of E( R) guarantee that each idempotent e > 0 is a unique sum of primitive idempotents, e = p 1 + ··· + pn, where each pi covers 0 in (E( R), ≤) and p 1, ···, pn are mutually orthogonal, coming from different primitive subalgebras of E( R). Assertion (i) follows from this. Next let x ∈ Γ(R) be given. By Theorem 6.4.3. x = exe = ( p 1 + ··· + pn) x( p 1 + ··· + pn) = ∑ = ∑ i, j pi xp j i pixpi for the appropriate primitive idempotents, where pixpj = 0 for i ≠ j thanks to Theorem 6.5.3. Thus x = exe = ∑ where p i pi xpi ixpi ∈ Qi and ( pixpi)( pjxpj) = 0 for i ≠ j. Since every element in R is a finite sum of elements in Γ( R), (ii) follows. £ Corollary 6.5.7 The conclusions of Theorem 6.5.6 hold if we assume the ascending chain condition on E( R). In this case the number of summands Qi is finite, equaling the number of atoms in E( R)/D. Conversely, when only finitely many summands Qi exist, E( R) satisfies the ascending chain condition. Proof. The DCC must hold on E( R) also. The ACC also prevents E( R) from having an infinite number of 0-minimal D-classes and thus R from having an infinite number of ortho-summands Qi. The converse is clear. £ Corollary 6.5.8 If R satisfies the conditions of Theorem 6.5.6, then K R = ∑K Qi. Upon choosing a nonzero idempotent ei ∈ Qi for each i, A = ∑ is an idempotent-covered i eiQiei abelian ring such that R = A ⊕ K R as additive groups and R/K R ≅ A as rings. £ When the DCC holds on E( R), the question of E( R) being multiplicative can be reduced as follows. To begin, let M( R) denote the set of primitive idempotents of E( R) and let M0( R) 246 VI: Skew Lattices in Rings denote M( R) ∪ {0}. If E( R) satisfies this chain condition, then for any e > 0 in E( R) an m ∈ M( R) exists such that e ≥ m. A result of Dolžan [8] for a case when R is abelian can be extended: Theorem 6.5.9 If E( R) satisfies the descending chain condition, then E( R) is multiplicative if and only if M0( R) is multiplicative. Proof. Let M0( R) be multiplicative and let S consist of all possible finite sums ∑ ei of elements from distinct D-classes in M0( R). Since all products ef from distinct D-classes in M0( R) equal 0, S is also a set of idempotents that is closed under multiplication. Given e > 0 in E( R), let m 1 ∈ M( R) be such that e ≥ m 1. If e = m 1, we stop. Otherwise we have e > e − m 1 ≥ m 2 in M( R) with m 2 orthogonal to m 1 in E( R), since m 1 D m 2 implies m 2 = m 2( e – m 1) m 2 = m 2 − m 2 = 0. If e – m 1 = m 2, then e = m 1 + m 2 with m 1⊥ m 2 in E( R). Otherwise, e − m 1 – m 2 ≥ some m 3 in M( R). The DCC insures this process eventually halts to give e = m 1 + · · · + mn with the mi mutually orthogonal and thus E( R) = S. The converse is trivial. £ Although we do not use this, it can be proved that if an idempotent-closed and dominated ring R satisfies the DCC [ACC] on (left, right) ideals, then it must satisfy the DCC [ACC] on idempotents. Rectangular rings Thus to within isomorphism, the rings of the last section are direct sums of rectangular rings ⊕ ∑ ⊕ ⊕ R i∈ I or quotient rings ( ∑ ) /I for some ideal I ⊆ ann( ∑ ). We study these i R i∈ I i R i∈ I i “atomic” rings Ri with the goal of describing them in terms of rectangular bands S and rings A with identity 1 for which E( A) = {0, 1}. Our main concern is not the precise structure of the latter “subatomic” ring A, but rather their role in the larger “atomic” picture. We begin with a special case that is suggestive of what occurs generally. Given a ring A such that E( A) = {0, 1} and a rectangular band S, we form a ring A[ S]. Under addition A[ S] is the free A-module on generating set S. Thus it consists of formal sums ∑ ass with as ∈ A and as ≠ 0 for only finitely many s. Addition is given by: ∑ ass + ∑ bss = ∑( as + bs) s; multiplication is given by distributivity subject to the constraints: ( as)( bt) = ( ab)( st) and 0 s = 0 = ∑0 s. If s ∈ S is identified with 1 s ∈ A[ S], then S is a multiplicative band inside E( A[ S]). But is it a maximal rectangular band in A[ S]? In what follows, at times we use just finite expressions a 1 s 1 +...+ ansn with ai in A and si in S, assuming that aj = 0 for all ajsj terms not showing. 247 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Lemma 6.5.10 Given A and S as above and s ∈ S, then in A[ S] L S = {∑ aisi ⎪∑ ai = 1 in A & ai ≠ 0 ⇒ si L s in S} and R S = {∑ bjtj⎪∑ bj = 1 in A & bj ≠ 0 ⇒ tj R s in S} are the sets of idempotents respectively L- related or R- related to s in A[ S] . Moreover, M S = L S R S = {∑( aibj)( sitj)⎪∑ ai = 1 = ∑ bj in A with aibj ≠ 0 ⇒ si L s R tj in S} is the maximal rectangular band in A[ S] containing s and hence all of S. Proof. Indeed, given x = ∑ aisi where ∑ ai = 1 in A and si L s in S if ai ≠ 0, one easily sees that xs = x, sx = s and thus x 2 = xsx = x. Conversely, if ∑ aisi is an idempotent that is L-related to s, then s = s(∑ aisi) s = ∑ ai( ssis) = ∑ ais = (∑ ai) s so that ∑ ai = 1. Moreover, ∑ aisi = (∑ aisi) s = ∑ ai( sis). Since all sis are L-related to s in S, by the uniqueness of the representation, all si with nonzero coefficients in ∑ aisi are L-related to s in S. Thus L S and likewise R S are indeed as described. Finally, for any rectangular band M, given e ∈ M one has R e = eM, L e = Me in M so that M = MeM = L e R e. Thus we need only show that under multiplication M S is a rectangular band. This follows from the easily verified identity in A[ S]: (∑ aisi)(∑ bjtj)(∑ ckuk)(∑ dmvm) = (∑ aisi)(∑ dmvm) given ∑ ai = ∑ bj = 1 = ∑ ck = ∑ dm in A and si, tj uk and vm are in S. £ One can use the above “Inflation Lemma” to show that for nontrivial A and S, M S will properly include S except in three cases: &2[ S] for | S| = 2 (two cases) and &2[ S] where S is the 4-element rectangular band of Example 3 below. On the other hand, if S has an L- or R-class with ≥ 3 distinct elements a, b, c, then a new element in the inflated class is given by a − b + c. Or if α ∈ A \ {0, 1}, then given a ≠ b in an L- or R-class of S, α a + (1 − α) b is a new element in the inflated class. Now, that the status of S and M S in A[ S] is decided, we turn to all of E( A[ S]). Theorem 6.5.11 A[ S] is idempotent-dominated and E( A[ S]) is the 0-rectangular band M S ∪ {0} and K A[ S] = {(∑ aisi⎪∑ ai = 0 in A}. 248 VI: Skew Lattices in Rings Proof. Since A[ S] = ∑ s∈ SAs and As = sA[ S] s, the first assertion holds. We show that any f = f 2 ≠ 0 in A[ S] lies in M S. Let f = ∑ aisi. Given s in S, with i and j representing the same index values we have fsf = (∑ aisi) s(∑ ajsj) = ∑ aiajsissj = ∑ aiajsisj = (∑ aisi)(∑ ajsj) = ff = f and sfs = s(∑ aisi) s = ∑ aissis = ∑ ais = (∑ ai) s. Since ( sfs)2 = sfsfs = sfs, ∑ ai ∈ E(A). E( A) = {0, 1} by assumption. If ∑ ai = 0, so that sfs = 0, then also f = fsfsf = 0, contradicting f ≠ 0. This leaves ∑ ai = 1, so that f = fsf = (∑ aisi) s(∑ aisi) = (∑ aisis)(∑ aissi) ∈L S RS = M S. Next, since for all a, b ∈ A and all s, t ∈ S, ( as)(∑ aisi)( bt) = [ a(∑ ai) b]( st), the condition ∑ ai = 0 is necessary for ∑ aisi to be in K A[ S]. Since A[ S] is additively generated from all a• s terms, it is also sufficient. £ In the general rectangular case, letting M( R) denote E( R) \ {0}, we have: Theorem 6.5.12 If ring R is a rectangular ring, then i) Γ( R) = ∪ e ∈M( R) eRe with ( eRe)( fRf) = efRef for all e, f ∈ M( R). ii) Given e, f ∈ M( R), the map χ: x → fxf defines a ring isomorphism of eRe with fRf. Thus every element y in fRf is uniquely expressed as fxf for some x in eRe. iii) Given y = fxf ∈ fRf and z = gx′g ∈ gRg, yz = ( fg)( xx′)( fg) ∈ fgRfg. iv) R = ∑ e ∈M( R) eRe with all summands being isomorphic subrings. Proof. Lemma 6.4.7 gives the first equality in (i). The second equality in (i) follows from ( eRe)( fRf) = efeRefRfef ⊆ efRef = ( efRe) f ⊆ ( eRe)( fRf). To see (ii), note that χ is at least an additive homomorphism from eRe to fRf. Let x, y ∈ eRe be given. Then f( xy) f = f( xey) f = f( xefey) f = ( fxef)( feyf) = ( fxf)( fyf) so that χ is a homomorphism of rings. Indeed it is an isomorphism with inverse isomorphism given by y → eye from fRf back to eRe. (iii) follows from yz = ( fxf)( gx′g) = ( fexef)( gex′eg) = fexex′eg = fgexx′efg = fgxx′fg. The final assertion is now clear. £ 249 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond ⎡ 0 0 0 ⎤ ⎡ 0 a ab ⎤ Example 6.5.1 continued for n =3, i = j = k = 1, e = ⎢ 0 1 0 ⎥ and f = ⎢ 0 1 b ⎥ . ⎣⎢ 0 0 0 ⎦⎥ ⎣⎢ 0 0 0 ⎦⎥ ⎧⎡ 0 0 0 ⎤ ⎪ ⎫⎪ ⎧⎡ 0 ad abb ⎤ ⎪ ⎫⎪ Here eRe = ⎢ ⎥ ⎨ ⎢ ⎥ 0 d 0 ⎢ ⎥ d ∈ F ⎬ , fRf = ⎨ 0 d db ⎢ ⎥ d ∈ F ⎬ and the isomorphism x → fxf ⎢ 0 0 0 ⎩⎪⎣ ⎦⎥ ⎭⎪ ⎢ 0 0 0 ⎩⎪⎣ ⎦⎥ ⎭⎪ ⎡ 0 0 0 ⎤ ⎡ 0 ad abb ⎤ of Theorem 6.5.12 (ii), sends ⎢ ⎢ ⎥ 0 d 0 ⎥ in eRe to 0 d db ⎢ ⎥ in fRf. £ ⎣⎢ 0 0 0 ⎦⎥ ⎢ 0 0 0 ⎣ ⎦⎥ How close is this class of A[ S]-rings to the class of all rectangular rings? We begin our answer with the following canonical co-representation: Proposition 6.5.13 Let R be a rectangular ring. If A = fRf for a fixed f ∈ M = M( R), then A is a ring with identity 1 = f such that E( A) = {0, f}; moreover the map β: A[ M] → R defined by β(∑ aiei) = ∑ eiaiei in R is a homomorphism onto R that is bijective between the copy {1 e⎪ e ∈ M} of M in A[ M] and M in R, and also between subrings Ae in A[ M] and their images eAe in R. Proof. By Theorem 6.5.12 (ii) and (iv), R = ∑ e ∈ M eRe with all summands being isomorphic to A. Thus β is an additive epimorphism that is bijective where stated. That it preserves multiplication follows from Theorem 6.5.12 (iii) and distribution. £ Thus, all rectangular rings essentially arise as rings of the form A[ S] as above and certain homomorphic images of these rings. The entire situation is put more precisely as follows. Theorem 6.5.14 Given a ring A with identity such that E( A) = {0, 1} and a rectangular band S, then A[ S] is a rectangular ring. Moreover given any ideal K of A[ S] such that K ⊆ K A[ S], the quotient ring A[ S]/ K is also rectangular. Conversely, to within isomorphism every rectangular ring is obtained in this fashion. Proof. The first assertion is Theorem 6.5.11. The second comes from Theorem 6.4.14. To see the converse note that for the map β above, if β(∑ aiei) = ∑ eiaiei = 0 in R, then in fRf = A, ∑ ai = ∑ faif = f(∑ ai) f = f 0 f = 0. Thus ker(β) ⊆ K A[ M] by Theorem 6.5.11. The converse now follows by Theorem 6.4.14. £ Example 6.5.2 Let A = &2 and let S be the rectangular band determined by the array: a R b L L e. g. a∧ d = b and a∧ d = c. c R d Setting s = a+b+c+d, the sixteen elements of &2[ S] are arrayed in the following diagram. x 0 a b c d a + b a + c a + d . s − x s b + c + d a + c + d a + b + d a + b + c c + d b + d b + c 250 VI: Skew Lattices in Rings Again E(& 2[ S]) = {0} ∪ S, K = {0, s, a+b, c+d, a+c, b+d, a+d, b+c} so that A[ S]/K ≅ & 2 and ann(&2[ S]) = {0, s}. &2[ S] /ann(Z2[ S]) has 8 elements that are parameterized by the x-row entries. &2[ S] is weakly commutative with K equaling the nil radical N. £ 6.6 Idempotent-closed rings of matrices In this section F is again a field and n is a positive natural number. We characterize those idempotent-closed subrings of M n( F) that are maximal subject to certain constrains. This is done modulo maximal idempotent-closed matrix subrings R for which E( R) ⊆ {0, I}. While we are unaware of any general characterization of the latter, if F is the field of complex numbers 0, a result of Livshits et al [2003] implies that if A is an algebra in M n(0) such that E( A) ⊆ {0, I,}, then either A = N or A = 0 I + N for some nil algebra N. A maximal such algebra would be simultaneously similar to the algebra of all upper-triangular matrices with constant diagonals, since any subring of nilpotent matrices in M n(0) is triangularizable. (See Example 6.7.2 below. See also Okninski [1997] or Radjavi and Rosenthal [2000].) The abelian case We begin with maximal idempotent-covered abelian subrings of M n( F). Such subrings necessarily have and identity E and the “maximal” constraint insures that E = I. The following pair of extreme examples are suggestive of the general case. Example 6.6.1. R is the abelian subring of all diagonal matrices in M n( F) and E( R) is the set of all diagonal matrices with only 0 or 1 entries. R is a maximal idempotent-closed ring in M n( F). For suppose that ring R′ in M n( F) properly contains R. Let A ∈ R′ be a non-diagonal matrix and let i ≠ j be such that E i AE j ≠ 0. (E i is the matrix with the i-th diagonal entry 1, and 0 elsewhere.) Then (E i + E i AE j)2 = E i + E i AE j ∈ E( R), but (E i + E i AE j)E j = E i AE j is nilpotent. Thus R′ is not idempotent-closed. ⎡ d 0  0 ⎤ 1 ⎢ ⎥ 0 d  0 2 ⎢ ⎥ ⎢     ⎥ ⎢ 0 0  d ⎥ n ⎣ ⎦ 251 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Example 6.6.2. Let R consist of all upper triangular matrices in M n( F) with constant diagonals. R is trivially idempotent-closed since E( R) = {0, I}. ⎡ d a  a ⎤ 1, 2 1, n ⎢ ⎥ 0 d ⎢  a 2, n ⎥ ⎢     ⎥ ⎢ ⎥ 0 0 ⎣  d ⎦ For any field, R is maximal subject to E( R) = {0, I}. In fact, R is a maximal idempotent-closed subring of M n( F). For its verification, see the discussion in Example 9 in Cvetko-Vah and Leech [2011]. That a subring has only idempotents 0 and I is unremarkable. But that a maximal idempotent-closed subring of M n( F) can be thus is interesting. £ Theorem 6.6.1. For each n ≥ 1, M n( F) has a maximal idempotent-closed subring R that is an algebra over F and for which E( R) = {0, I}. £ Conversely, one may ask: are all maximal idempotent-closed subalgebras R of M n( F) for which E( R) = {0, I} simultaneously similar to such an example, as happens when F = 0? In any case, a general way of constructing maximal idempotent-closed subrings of M n( F) with identity I (and hence abelian) follows from the next result. Proposition 6.6.2. An idempotent-closed ring R in M n( F) containing I is similar simultaneously to a ring in block form (1) below where for each index i all blocks Di form a subring Ri in M n( i)( F) with E( Ri) = {0 n( i), In( i)}. R is maximally idempotent-closed in M n( F) if and only if each Ri is maximally idempotent-closed in the matrix ring M n( i)( F). ⎡ D ⎤ 1 0  0 ⎢ ⎥ (1) U = 0 D ⎢ 2  0 ⎥ ⎢     ⎥ ⎢ ⎥ 0 0  D ⎣ n ⎦ Proof. By Proposition 6.4.1, E( R) is a Boolean algebra. Let E1, ... , E k be the atoms of E( R). Then E1 +... + E k = I and we can choose a basis for Fn such that in this basis each E i is similar to the diagonal matrix that has In( i) (of the appropriate dimension) on the i-th diagonal block entry and 0s elsewhere. All elements of E( R) are sums of atoms and thus diagonal with the block form (1) where each Di = In( i) or 0 n( i). Given 252 VI: Skew Lattices in Rings ⎡ V  ⎤ 11 V 12 V 1 n ⎢ ⎥ V ⎢  V V = 21 V 22 2 n ⎥ ⎢     ⎥ ⎢ ⎥ V  V ⎢ 1 n V 2 n nn ⎣ ⎦⎥ in R. If V ij ≠ 0 for some i ≠ j, then (E i + E i VE j) ∈ E( R) but (E i + E i VE j)E j = E i VE j ∉ E( R) and R is not idempotent-closed. Thus under this basis R is indeed similar to a ring in the stated block from with each block ring Ri ≅ {E i VE⎪V ∈R}. The atomic nature of the E i insures that each E( Ri) is as stated. Given the diagonal block design, the final assertion is clear. £ R is isomorphically a direct sum ⊕ iRi. Internally, R = E1 R E1 ⊕ ... ⊕ E kR E k. What is more, E( R) = E(E1 R E1) ⊕ ··· ⊕ E(E kR E k) = {E1, 0}⊕ ··· ⊕{E k, 0} in that each idempotent in R decomposes uniquely as a sum of idempotents in each E iR E i. The general case: the idempotents To pass to idempotent-closed subrings of M n( F) in general, we first look at bands and skew lattices in matrix rings. As it turns out: all bands in M n( F) are simultaneously similar to a band of upper-triangular matrices. Consequently, each skew lattice in M n( F) is simultaneously similar to a skew lattice of upper-triangular matrices. The result for bands was proved for algebraically closed fields by Radjavi [1997]. The arbitrary case for bands follows from Okninski’s results [1997]. For maximal normal bands, i.e. maximal skew Boolean algebras, results of Cvetko-Vah ([2005b] and [2007]) are relevant. We summarize her results for convenience. 253 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Theorem 6.6.3. In any matrix ring M n( F) , the following are true: 1. Given a normal skew lattice S in M n( F) , all matrices in S can be simultaneously triangularized to form an isomorphic skew lattice of matrices S Δ of the following fixed block form where 0ʹ and 0ʺ are fixed square 0-blocks, each Di is a 0 or I square block of fixed dimensions, AiDi =Ai, DiBi = Bi and C= ∑ AiBi: ⎡ 0′ A ⎤ 1  Ak C ⎢ ⎥ 0 D ⎢ 1  0 B 1 ⎥ (2) U = ⎢      ⎥ ⎢ 0 0  D ⎥ k Bk ⎢ ⎥ ⎣ 0 0  0 0′ ⎦ 2. This triangularization can be chosen so that for all such U, the following diagonal matrix EU also lies in SΔ with U D EU, with the EU collectively form a lattice section S0 of SΔ . ⎡ 0′ 0  0 0 ⎤ ⎢ 0 D ⎥ 1  0 0 E ⎢ ⎥ U =      ⎢ ⎥ 0 0  D ⎢ k 0 ⎥ ⎢ ⎥ ⎣ 0 0  0 0′ ⎦ 3. When S is a maximal normal skew lattice, and the elements of SΔ in block form look like ⎡ 0 A C ⎤ ⎡ 0 0 0 ⎤ U = ⎢ 0 D B ⎥ with E ⎢ ⎥ U = 0 D 0 ⎣⎢ 0 0 0 ⎦⎥ ⎣⎢ 0 0 0 ⎦⎥ where D is any possible 0-1 diagonal matrix of a fixed size and A, B and C are all possible submatrices of appropriate dimensions such that AD = A, DB = B and AB = C. The lattice section S0 of all EU forms a Boolean algebra. (The pattern allows for the possibility that either 0ʹ or 0ʺ vanishes or both. In the first case the 0ʹ- row and 0ʹ-column disappear so that the main diagonal begins with D. Similar remarks hold when 0ʺ or both vanish. The maximal case of “both” is the idempotent part of Example 6.6.1.) The general case: the idempotent-closed subrings If R is an idempotent-closed ring in M n( F) and E is in the maximal D-class of E( R) then E R E is a maximal abelian subring of R with the identity E and B = E(E R E) is a Boolean algebra. 254 VI: Skew Lattices in Rings Let E1, ..., E k be the atoms of B. Then for all i ≤ k, Ri = E iR E i is a ring with identity E i and no non-trivial idempotents, and A ≅ R 1 ⊕ ··· ⊕ Rk is an idempotent-covered ring with identity E and E(A) ≅ E1E(R)E1 ⊕ ··· ⊕ E kE(R)E k as above. We are now ready to state the first of the two main results in this section. Theorem 6.6.4. If R is a maximal idempotent-closed and idempotent-dominated ring in M n( F) , then R is simultaneously similar to the ring of all matrices of block form ⎡ 0′ A  ⎤ 1 Ak C ⎢ ⎥ 0 D  ⎢ 1 0 B 1 ⎥ (3) ⎢      ⎥ ⎢ ⎥ 0 0  D ⎢ k Bk ⎥ ⎢ 0 0 ⎣  0 0′ ⎦⎥ where each Di comes from a maximal idempotent-closed matrix subring Ri of M n( i)( F) for which E( R) = { In( i), 0 n( i)} . Idempotent matrices of this form have only 0 n( i) s or In( i) s on the diagonal, and satisfy AiDi = Ai, DiBi = Bi and C = ∑A i B i. (Remarks similar to those given for Theorem 6.6.1 apply when 0ʹ or 0ʺ vanish. When both vanish one has a maximal idempotent covered subring with central idempotents.) Proof. We first choose a basis B for Fn such that relative to B all idempotents in R have matrix form (1). Assuming that all elements in R are matrix-represented relative to B, the assumption that R is idempotent-dominated implies the leftmost column of blocks and the bottom row of blocks of these representations consist only of zero blocks, as in (3). What can arise in the central block of all R-matrices, the say m × m blocks that exclude the two outermost rows of blocks and the two outermost columns of blocks relative to (1). ⎡ D ! ? ⎤ 1 ⎢ ⎥ ⎢ " # " ⎥ D ∈M n ⎢ i ( i)( F) ? ! D ⎥ k ⎣ ⎦ Given i ∈{1, . . . , k} let Ei be the matrix in M n( i)( F) with D = and 0 elsewhere. By i In( i) Theorem 6.6.1 we may assume that all Ei lie in E( R). If E = E 1 + · · · + Ek, then A = ERE is an idempotent-closed subring of R with identity E as are each EiREi where E( EiREi) = { Ei, 0}. Note that for A-matrices, all blocks in the first row and last column are also 0-blocks. Thus A is isomorphic to an idempotent-closed ring in Mm(F) with identity Im under the map sending each matrix P in A to its central m × m block. A is further block diagonalized with each set of Di- blocks forming a ring Ri in M n( i)( F) that is isomorphic to the subring EiREi. Since ERE = E 1 RE 1 255 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond ⊕ ... ⊕ EkREk by the remark after Proposition 6..7.2, given any matrix in A, all nondiagonal blocks in its central submatrix are 0-blocks. Indeed, since P and EPE share the same central submatrix, every matrix P in R has this pattern in its central submatrix. Hence, given our assumptions on E( R), every matrix in R is at least of the block form (3). ⎡0 A C⎤ ⎡0 A′ C′⎤ ⎡0 AD′ AB′⎤ Using simplified block-of-blocks format, ⎢0 D B⎥ ⎢ B′⎥ = ⎢ ⎥ ⎢ ⎥ 0 D′ ⎢ ⎥ 0 DD′ DB′ ⎢ ⎥ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ ⎡ 2 0 A C ⎤ ⎡0 A C⎤ ⎧ AD = A, DB = B, ⎪ and ⎢ 0 D B ⎥ = ⎢0 D B⎥ iff It follows that any such ring ⎢ ⎢ ⎥ ⎨ 0 0 0 ⎣ ⎦⎥ 0 0 0 ⎣ ⎦ AB = C and D 2 = D. ⎩⎪ of matrices is idempotent-closed if and only if its central ring of D-matrices is idempotent-closed. Thus the status of a subring R as being idempotent-closed in M n(F) remains unchanged if its design is extended by first allowing arbitrary Ai, Bi and C blocks and then enlarging each ring Ri of Di-blocks to a maximal idempotent-closed subring of M n( i)( F) that includes 0 n( i) and In( i). If such a subring extension is idempotent-covered, then the Ai, Bi, C blocks and the Ri rings already had this maximal status as stated in the theorem. It remains to see that the ring R of all matrices of form (3) is idempotent-dominated. We again use this simplified block form. To begin, the identities ⎡ 0 A C ⎤ ⎡0 0 0⎤ ⎡0 A 0⎤ ⎡0 0 C⎤ ⎢ 0 D B ⎥ = ⎢ ⎥ + ⎢ ⎥ + ⎢ ⎥ , ⎢ 0 D B ⎢ ⎥ 0 0 0 ⎢ ⎥ 0 0 0 ⎢ ⎥ 0 0 0 ⎣ ⎦⎥ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ ⎡0 0 0⎤ ⎡0 0 0⎤ ⎡0 0 0⎤ ⎡0 A 0⎤ ⎡0 A 0⎤ ⎡0 0 0⎤ ⎢0 D B⎥ = ⎢ ⎥ ⎢ ⎥ and ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 I 0 ⎢ ⎥ 0 D B ⎢ ⎥ 0 0 0 ⎢ ⎥ 0 0 0 ⎢ ⎥ 0 I 0 ⎢ ⎥ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ ⎡0 0 C⎤ reduce this verification to showing any ⎢0 0 0 ⎥ lies in Q ⎢ ⎥ R. Such a matrix is a sum of matrices 0 0 0 ⎣ ⎦ of this type having only a single nonzero entry; but any such summand easily factors into a ⎡0 A 0⎤ ⎡0 0 0⎤ product of the form ⎢0 0 0⎥ ⎢ ⎥ and thus lies in Q ⎢ ⎥ 0 0 B ⎢ ⎥ R. £ 0 0 0 ⎣ ⎦ 0 0 0 ⎣ ⎦ A is the maximal abelian subring of R and the subring of all matrices in R for which all Di- blocks are 0-blocks is the nilpotent ideal K R encountered in Section 4. R = A ⊕ K as additive groups and as rings R/K ≅ A making A the maximal abelian image of R. (See Section 6.4.) Again in our simplified block notation: 256 VI: Skew Lattices in Rings ⎡ 0 0 0 ⎤ ⎡ 0 A C ⎤ ⎡ 0 A AB ⎤ A is all ⎢ ⎥ ⎢ ⎥ 0 D 0 ⎢ ⎥ ⎢ ⎥ ; K R is all 0 0 B ⎢ ⎥ ; max D-class in E( R) is all 0 I B . ⎢ ⎥ 0 0 0 ⎣ ⎦ 0 0 0 ⎣⎢ ⎦⎥ 0 0 0 ⎣ ⎦ Also, in the situation above, in the decomposition of R into an orthosum of rectangular ideals, the particular ideals involved are the k distinct rectangular subrings obtained by letting all Di, Ai and Bi blocks be 0-matrices except for a particular index j where the Dj, Aj and Bj blocks are subject to just the constraints of Theorem 6.6.2 and the C block is generated by the AjBj outcomes. When F = 0 we have the following crisp result: Corollary 6.6.5. If R is a maximal idempotent-closed and idempotent-dominated ring in M n(0) , thenR is simultaneously similar to the ring of all matrices with the block form (3) above, where the Di lie in the subring of all upper triangular matrices in M n( i)(0) with constant diagonals. £ Can such a ring be extended to a larger subring of Mn(F), which although no longer idempotent-dominated, has no new idempotents and thus is still idempotent-closed? (A maximal idempotent-dominated and idempotent-closed subring R of M n( F) cannot be contained in a properly larger idempotent-closed subring R′ of M n( F) unless R′ has no new idempotents.) Suppose that 0′ in the upper left corner of the above block design is a p×p 0-matrix and that 0′′ in the lower right is a q×q 0-matrix. Let T ′ be a subring of M p( F) that is maximal with respect to only having 0 as an idempotent. Likewise, let T′′ be a subring of M q( F) that is maximal with respect to only having 0 as an idempotent. Such “fringe” subrings are trivially idempotent-closed. Consider the following design ⎡ G A  ⎤ 1 Ak C ⎢ ⎥ 0 D  ⎢ 1 0 B 1 ⎥ (4) ⎢      ⎥ ⎢ ⎥ 0 0  D ⎢ k Bk ⎥ ⎢ 0 0 ⎣  0 H ⎦⎥ where the As, Bs, Cs and Ds are as in Theorem 6.6.4 above but G ∈ T′ and H ∈ T′′. When G and H are 0 we are in the previous context. The collection Rʹ of all matrices with this design forms a subring of M n( F) . But in this larger ring no new idempotents are created since 257 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond ⎡ 2 G A  ⎤ ⎡ ⎤ 1 Ak C  ⎢ ⎥ G A 1 Ak C ⎢ ⎥ 0 D ⎢   1 0 B 1 ⎥ 0 D ⎢ 1 0 B 1 ⎥ ⎢      ⎥ = ⎢ ⎥ ⎢ ⎥      ⎢ ⎥ 0 0  D ⎢ 0 0  D k Bk ⎥ ⎢ k Bk ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 ⎣  0 H ⎦ 0 0 ⎣  0 H ⎦ precisely when first G2 = G = 0 and H2 = H = 0 and the As, Bs, Cs and Ds behave as described in Theorem 6.6.4. We have proved the following theorem. Theorem 6.6.6. Let R be the set of all matrices of a Type (4) design in M n( F) where: 1) The Ai, Bj and C blocks can be any matrix of the prescribed dimensions. 2) The Di-blocks are subject to the constraints of Theorem 6.6.4. 3) G and H belong to subrings Tʹ and Tʺ of M p( F) and M q( F) respectively, that are maximal with respect to having no nonzero idempotent. Then R is a maximal idempotent-closed subring of M n( F) . Its Type (3) matrix subring is a maximal idempotent-closed and dominated subring of M n( F) . The converse ( a maximal idempotent-closed subring of M n( F) containing a maximal idempotent-closed and dominated subring of M n( F) , is simultaneously similar to a ring of the above type) is also true. Its proof is given in Cvetko-Vah and Leech [2011]. Historical remarks The results in Section 6.1 are from Leech’s initial paper on skew lattices [1989]. Those in Sections 6.2 and 6.3 are from Cvetko-Vah and Leech’s 2008 paper, some of which generalized results in two earlier papers of Cvetko-Vah ([2004] and ([2005a]. All results in Sections 6.4 and 6.5 are from the 2012 paper of Cvetko-Vah and Leech. Section 6.6 is based on earlier results of Cvetko-Vah ([2005b] and [2007]).except for those mentioned at the end which are from their 2011 paper. 258 VI: Skew Lattices in Rings References K. Cvetko-Vah, Pure skew lattices in rings, Semigroup Forum, 68 (2004), 268–279. Pure ∇ -bands in rings, Semigroup Forum, 71 (2005a), 93–101. Skew lattices in matrix rings, Algebra Universalis 53 (2005b), 471–479. On the structure of semigroups of idempotent matrices, Linear Algebra Appl. 426 (2007), 204–213. K. Cvetko-Vah and J. Leech, Associativity of the ∇ -operation on bands in rings, Semigroup Forum 76 (2008), 32–50. On maximal idempotent-closed subrings of M n( F), International Journal of Algebra and Computation 21 (2011), 1097–1110. Rings whose idempotents are multiplicatively closed, Communications in Algebra 40 (2012), 3288–3307. D. Dolžan, Multiplicative sets of idempotents in a finite ring, Journal of Algebra 304 (2006), 271– 277. J. Leech, Skew lattices in rings, Algebra Universalis 26 (1989), 48–72. Small skew lattces in rings, Semigroup Forum 70 (2005), 307-311. L. Livshits, G. MacDonald, B. Mathes, J. Okninski and H. Radjavi, Matrix semigroups with commutable rank, Semigroup Forum, 67 (2003), 288–316. J. Okninski, Triangularizable semigroups of matrices, Linear Algebra Appl., 262 (1997) 111–118. H. Radjavi, Invariant subspaces and spectral conditions on operator semigroups, Linear Operators, Banach Center Publications, Vol. 38 (Polish Academy of Sciences, Warsaw, 1997), 287– 296. H. Radjavi and P. Rosenthal, Simultaneous Triangularizations, Universitext (Springer, New York, 2000). 259 VII: FURTHER TOPICS IN SKEW BOOLEAN ALGEBRAS In Chapter IV the implicit perspective of a skew Boolean algebra is that of a skew lattice with added structure, namely a constant 0 and a difference operation \, which with ∧ and ∨ satisfy certain identities. There is an alternative approach: to consider algebras with a reduced signature (∧, \, 0) or even (\, 0) that are subject to a set of identities, and view skew Boolean algebras as instances where these simpler algebras acquire added structure. These reduced algebras can, however, be of independent interest, as was the case for the iBCK algebras encountered below. This chapter begins by looking at algebras of signature (\, 0) that satisfy “subtractive” identities such as x\0 = x and x\ x = 0 = 0\ x. Indeed Section 7.1 considers seven such identities. The first six (indeed the first four) characterize implicative BCS-algebras (or just iBCS-algebras). The significance of these algebras lies in the fact that (\, 0)- reducts of skew Boolean algebras are iBCS-algebras. The seventh identity, when joined to the rest, characterizes implicative BCK-algebras (or just iBCK-algebras). A skew Boolean algebra is a generalized Boolean algebra if and only if its (\, 0)- reduct is an iBCK-algebra. A skew Boolean algebra is simultaneously both a strongly distributive skew lattice and an iBCS-algebra, having both types of algebras as reducts. The Signature Bisection Theorem (Theorem 7.1.1) tells how a strongly distributive skew lattice and an iBCS-algebra, if defined on a common set, must interact to form a skew Boolean algebra having the initial pair of algebras as reducts. We next define an alternative iBCK-difference / on skew Boolean ∩-algebras by x/ y = x \ x∩ y, in which case the reduct (S; /, 0) is an iBCK-algebra. Theorem 7.1.2 characterizes skew Boolean ∩-algebras as algebras of signature (∨, ∧, /, 0). The section continues by looking at the role of discriminator terms and discriminator varieties. Skew Boolean algebras form a binary discriminator variety (Theorem 7.1.4) and all binary discriminator algebras have an iBCS-algebra reduct; what is more, all iBCS-algebras also have left-normal-band-with-0 reducts. Thus much, but not all, of the structure of a left-handed skew Boolean algebra is encoded in its iBCS reduct (S; /, 0). The main result (Theorem 7.1.9) states: If V is a binary discriminator variety with constant term 0 and additive term x + y, then every algebra A of V has a left-handed skew Boolean algebra term reduct AS. (Here x + y is a binary term satisfying x + 0 = x = 0 + x.) Clearly a right-handed version also holds. The section concludes by looking at ternary discriminator varieties. Theorems 7.1.10 and 7.1.11 reveal a close connection between skew Boolean ∩-algebras and pointed ternary discriminator varieties. Most of Section 7.1 is based on the work of Robert Bignall and his student Matthew Spinks, with some input from Jonathan Leech. 261 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond 7.1 Differences, discriminators and connections with other algebras Besides a strongly distributive skew lattice reduct (S; ∨, ∧), a skew Boolean algebra also has a complementary reduct, (S; \, 0). To understand the behavior of the latter, consider the following identities: ( a) x \ x ≈ 0. ( b) x \ ( y \ x) ≈ x. ( c) ( x \ y) \ z ≈ ( x \ z) \ y. ( d) ( x \ y) \ z ≈ ( x \ z) \ ( y \ z). ( e) x \ 0 ≈ x ( f) 0 \ x ≈ 0, ( g) x \ ( x \ y) ≈ y \ ( y \ x). An algebra (S; \, 0) of type (2, 0) satisfying ( a) – ( d) is called an implicative BCS-algebra ( iBCS- algebra for short), in which case it also satisfies both ( e) and ( f) making it a 0-subtractive algebra. Indeed ( e) follows by putting y = x in ( b) and then using ( a), while ( f) follows by setting x = 0 in ( b) and then using ( e). If in addition (S; \, 0) satisfies the "commutative" identity ( g), it is called an implicative BCK-algebra ( iBCK-algebra for short). iBCK-algebras were introduced in Lyndon [1951]. They have been studied by various authors including Abbott [1967], Cornish [1982], Iseki and Tanaka [1978] and Kalman [1960]. iBCS-algebras were introduced and studied by Bignall and Spinks in [2003] and [2007]. For skew Boolean algebras we have the following Signature Bisection Theorem. Theorem 7.1.1. An algebra (S; ∨, ∧, \, 0) of type 〈2, 2, 2, 0〉 forms a skew Boolean algebra if and only if: i) (S; ∨, ∧) is a strongly distributive skew lattice. ii) (S; \, 0) is an iBCS-algebra. iii) The identity e \ ( e \ f) ≈ e∧ f∧ e holds. (S; ∨, ∧, \, 0) is a generalized Boolean algebra if and only if (ii) can be strengthened to: iiʹ) (S; \, 0) is an iBCK-algebra. Proof Given skew Boolean algebra, (S; ∨, ∧, \, 0), (i) holds and it is easily seen that the reduct (S; \, 0) is an iBCS algebra and that e \ ( e \ f) ≈ e∧ f∧ e. Just check the situation for 0-primitive algebras. Thus (i) - (iii) follow. The converse will be proved after Theorem 1.11 below. Given (i)-(iii), (S; \, 0) is an iBCK-algebra if and only if identity (g) holds, which in this context is equivalent to e∧ f∧ e ≈ f∧ e∧ f holding. But the latter is equivalent to e∧ f ≈ f∧ e, making (S; ∨, ∧) a lattice and (S; ∨, ∧, \, 0) a generalized Boolean algebra. £ To distinguish an iBCK operation from the more general iBCS operation, we use the symbol / when referring to the former. Given an iBCK algebra (S; /, 0), set x∩ y = x/( x/ y). Then: 262 VII: Further Topics in Skew Boolean Algebras x∩ x = x/( x/ x) = x/0 = x; x∩0 = x/( x/0) = x/ x = 0; and x∩ y = y∩ x by ( g). With a bit more work one sees that (S; ∩, 0) is a meet semilattice with a minimum 0 such that each principal ideal ⎡ x⎤ = { y⎪ y ≤ x} forms a Boolean lattice, with x / y being the complement of x∩ y in ⎡ x⎤. For any pair u, v ∈ ⎡ x⎤, u ∨ v is given as x / [( x/ u) ∩ ( x/ v)]. (See the references above.) The algebra (S; ∩, /, 0) is sometimes called a Boolean semilattice. This is relevant to skew Boolean ∩-algebras. Given such an algebra (S: ∨, ∧, \, ∩, 0), define the iBCK difference e / f on S by e / f = e \ e∩ f. For all e and f both differences agree, that is e / f = e \ f, if and only if e and f commute. Indeed one has e \ e∧ f∧ e = e \ e∩ f if and only if e∧ f∧ e = e∩ f. But then e∧ f = e∧ f∧ e∧ f = ( e∩ f) ∧ f = e∩ f and likewise, f∧ e = e∩ f. The converse is clear. In particular, e / f = e \ f, if e ≥ f. Both ∩ and the skew Boolean difference \ can be recovered from the iBCK difference / by e∩ f = e / ( e / f) and e \ f = e / ( e∧ f∧ e). Thus skew Boolean ∩-algebras can be viewed as algebras with three binary operations: ∨, ∧, and /, plus a constant, 0. What identities involving {∨, ∧, /, 0} characterize such algebras? Our remarks above, together with an examination of what occurs in the primitive case, yield a Second Signature Bisection Theorem due to Bignall and Leech [1995]. Theorem 7.1.2. Every skew Boolean ∩- algebra (S: ∨, ∧, \, ∩, 0) is term equivalent to an algebra, (S; ∨, ∧, /, 0) of type 〈2, 2, 2, 0〉 where: i) (S; ∨, ∧, 0) is a symmetric, normal skew lattice with 0. ii) (S; /, 0) is an iBCK-algebra. iii) The induced meet, e∩ f = e / ( e / f), of (S; /, 0) satisfies the identities of Lemma 4.3.2: e ∩ ( e∧ f∧ e) = e∧ f∧ e and e ∧ ( e∩ f) = e∩ f = ( e∩ f) ∧ e. Proof. Given a skew Boolean ∩-algebra, (i) is clear. The reduct (S; /, 0), where e / f = e \ e∩ f, satisfies the conditions for an iBCK algebra on primitive skew Boolean algebras and hence on all skew Boolean ∩-algebras so that (ii) follows. Finally, the iBCK meet e / ( e / f) reduces to the natural intersection ∩. Indeed since both e/ f, e∩ f ≤ e, we get e / ( e / f) = e \ ( e / f) = e \ ( e \ e∩ f) = e∩ f, and (iii) follows. Conversely, given (i), (S; ∨, ∧, 0) is a symmetric, normal skew lattice with zero 0. By (iii) it shares a common natural partial order ≤ with the /-induced meet ∩. This forces both algebras to share a common natural meet ∩. Each common principal ideal ⎡ e⎤ = { y⎪ y ≤ e} is a Boolean lattice by (ii), which forces (S/D; ∨, ∧) to be a generalized Boolean lattice. It follows that (S; ∨, ∧, ∩, 0) is an implicit skew Boolean ∩-algebra with iBCK difference /. £ 263 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond (Dual) binary discriminator varieties and iBCS algebras Let A be a set with distinguished element 0. The binary 0-discriminator and the dual binary 0-discriminator on A are defined respectively by ⎧ ⎧ x \ y = x if y = 0 ⎨ and x 0 if y = 0 ∧ y = ⎨ . 0 otherwise ⎩ x otherwise ⎩ Clearly x \ y and x∧ y are left-handed skew Boolean operations on A as a primitive skew Boolean algebra. When no ambiguity exists about the constant 0 we simply use the term ( dual) binary discriminator. The details of the following lemma are easily verified. Lemma 7.1.3. Given a set A with constant 0, the functions x \ y and x∧ y satisfy identities: B1. a ∧ a = a. B2. a ∧ ( b ∧ c) = ( a ∧ b) ∧ c. B3. a ∧ ( b ∧ c) = a ∧ ( c ∧ b). B4. a ∧ 0 = 0 ∧ a = 0. B5. a ∧ b = a \ ( a \ b). B6. a \ a = 0. B7. ( a \ b) \ c = ( a \ c) \ b. B8. ( a \ b) \ c = ( a \ c) \ ( b \ c). B9. a \ ( b \ a) = a. B10. a \ 0 = a. B11. 0 \ a = a. B12. ( a \ b) \ b = a \ b. £ Identities B1 to B4 characterize a left normal band with zero. Identities B6 through B9 reprise identities ( a) – ( d) at the onset of this section, with any algebra (A; \, 0) satisfying them being an iBCS-algebra. For such algebras B10 – B12 also hold. B10 and B11 are just ( e) and ( f) above, while ( a \ b) \ b =B8 ( a \ b) \ ( b \ b) =B6 ( a \ b) \ 0 =B10 a \ b. A variety V with a constant term 0 is a [dual] binary discriminator variety if a binary term x \ y [ x ∧ y] exits such that V is generated by a class K of algebras on which that binary term induces the [dual] binary 0-discriminator. Binary discriminator varieties are widespread. Examples include Stone algebras, pseudo- complemented semilattices, implicative BCK-algebras and, as we will shortly show, any ternary discriminator variety with a constant term. Any binary discriminator variety is also a dual binary discriminator variety, due to B5, but not conversely. Indeed the variety of left normal bands with zero is a dual binary 0- discriminator variety, but it cannot be a binary 0-discriminator variety because any term t( x, y) in the language of left normal bands with zero in which both variables appear explicitly must satisfy the implication: if y = 0 then t( x, y) = 0. Hence a binary term x \ y satisfying the identity x \ 0 ≈ x cannot be defined in the band. 264 VII: Further Topics in Skew Boolean Algebras These definitions generalize the concepts of pointed ternary discriminator and pointed ternary discriminator variety. Indeed from a ternary discriminator d( x, y, z) and 0 one defines the binary discriminator by setting x \ y = d(0, y, x). Thus pointed ternary discriminator varieties are binary discriminator varieties and all the later are dual binary discriminator varieties. These relationships cannot be reversed in general. Allowing for a bit of repetition we have: Theorem 7.1.4. Skew Boolean algebras form a binary discriminator variety with \ being a binary 0-discriminator on primitive skew Boolean algebras. Thus if (S: ∨, ∧, \, 0) is a skew Boolean algebra, then the reduct (S: \, 0) is an iBCS algebra. £ It is well known that the variety of left normal bands with zero is generated by the three- element band ({0, 1, 2}; ∧, 0), where ∧ is the dual binary discriminator on the base set {0, 1, 2}. Since any algebra of the form (A; ∧, 0) is a left normal band when ∧ is the dual binary discriminator on A, the class of left normal bands with zero is called the generic dual binary discriminator variety. Strongly distributive skew lattices with zero provide another example of a dual binary discriminator variety. Before giving a semigroup characterization of iBCS-algebras we will need some further properties of iBCS-algebras. We follow Matthew Spinks’ Monash University dissertation [2002]. Lemma 7.1.5. Upon setting x∧ y = x \ ( x \ y), an iBCS- algebra also satisfies identities: B13. ( a \ b) \ ( c \ a) ≈ a \ b. B14. a \ ( b \ ( c \ a)) ≈ a \ b. B15. a \ ( a \ ( a \ b)) ≈ a \ b. B16. ( a ∧ b) \ c = ( a \ c) \ ( a \ b). B17. ( a \ c) \ b = ( a \ c) \ ( a \ b). B18. ( a \ c) ∧ ( b \ c) = ( a \ c) \ ( a \ b). B19. a ∧ ( b \ c) = ( a \ c) \ ( a \ b). Proof. B13. ( a \ b) \ ( c \ a) =B7 [ a \ ( c \ a)] \ b =B9 a \ b. B14. a \ ( b \ ( c \ a)) =B9 [ a \ ( c \ a)] \ [ b \ ( c \ a)] =B8 ( a \ b) \ ( c \ a) =B13 a \ b. B15. a \ b =B9 ( a \ b) \ [ a \ ( a \ b)] =B8 { a \ [ a \ ( a \ b)]} \ { b \ [ a \ ( a \ b)]} =B13 { a \ [ a \ ( a \ b)]} \ ( b \ a) =B7 [ a \ ( b \ a)] \ [ a \ ( a \ b)] =B9 a \ ( a \ ( a \ b)). B16. ( a ∧ b) \ c =B5 [ a \ ( a \ b)] \ c =B7 ( a \ c) \ ( a \ b). B17. ( a \ c) ∧ b =B5 ( a \ c) \ [( a \ c) \ b] =B7 ( a \ c) \ [( a \ b) \ c] =B8 [ a \ ( a \ b)] \ c =B5 ( a ∧ b) \ c =B16 ( a \ c) \ ( a \ b). B18. ( a \ c) ∧ ( b \ c) =B5 ( a \ c) \ [( a \ c) \ ( b \ c)] =B8 ( a \ c) \ [( a \ b) \ c] =B7 ( a \ c) \ [(( a \ c) \ b] =B5 ( a \ c) ∧ b =B17 ( a \ c) \ ( a \ b). B19. a ∧ ( b \ c) =B5 a \ [( a \ ( b \ c)] =B9 [ a \ ( c \ a)] \ [ a \ ( b \ c)] =B7 ( a \ [ a \ ( b \ c)]) \ ( c \ a) 265 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond =B13 ( a \ [( a \ ( b \ c)]) \ ( c \ [ a \ ( b \ c)]) =B8 ( a \ c) \ ( a \ ( b \ c)) =B7 ( a \ [ a \ ( b \ c)]) \ c =B8 ( a \ c) \ ([( a \ ( b \ c)] \ c) =B7 ( a \ c) \ [( a \ c) \ ( b \ c)] =B5 ( a \ c) ∧ ( b \ c) =B18 ( a \ c) \ ( a \ b). £ This leads us to: Theorem 7.1.6. Given an iBCS-algebra (A; \, 0), upon setting x∧ y = x \ ( x \ y) the derived algebra (A; ∧, 0) is a left normal band with zero. Proof. B1 follows from a ∧ a = a \ ( a \ a) = a \ 0 = a. B4 follows from a ∧ 0 = a \ ( a \ 0) = a \ a = 0 and 0 ∧ a = 0 \ (0 \ a) = 0 provided we know that 0 \ a = 0 holds in general. But the latter follows from 0 \ a = ( a \ a) \ a = ( a \ a) \ ( a \ a) = 0 \ 0 = 0. To verify B2 and B3 we again follow Spinks [2002]. To begin, observe that ( a ∧ b) ∧ c =B5 ( a ∧ b) \ [( a ∧ b) \ c] =B5, 16 [ a \ ( a \ b)] \ [( a \ c) \ ( a \ b)] =B8 [ a \ ( a \ c)] \ ( a \ b) =B7 [ a \ ( a \ b)] \ ( a \ c) where the latter expression must also equal ( a ∧ c) ∧ b. Hence establishing B2 will also establish B3. But ( a ∧ b) ∧ c =again [ a \ ( a \ c)] \ ( a \ b) =B8 [ a \ ( a \ b)] \ [( a \ c) \ ( a \ b)] =B7 { a \ [( a \ c) \ ( a \ b)]} \ ( a \ b) =B19 { a \ [ a ∧ ( b \ c)]} \ ( a \ b) =B5 [ a \ ( a \ ( a \ ( b \ c)))] \ ( a \ b) =B15 [ a \ ( b \ c)] \ ( a \ b) =B19 a ∧ [ b \ (( b \ c))] =B5 a ∧ ( b ∧ c). £ Theorem 7.1.6 holds trivially for any iBCS-algebra arising as a reduct of an algebra in a binary discriminator variety, for one begins with a generating class K of binary discriminator algebras that implicitly satisfy B1-B5, where B5 defines ∧. From these algebras B1-B5 are passed to all algebras in the variety. Although we do not show this here, the variety of iBCS-algebras is in fact a binary discriminator variety (Bignall and Spinks [2007]). Theorem 7.1.6 is thus a corollary to this fact. This theorem, however, takes us only halfway to characterizing iBCS-algebras in terms of left normal bands with zero. Indeed: Theorem 7.1.7. If (S; ∧, 0) is the derived left normal band with zero of an iBCS-algebra (S; \, 0), then for each a ∈ S the set ⎡ a⎤ = { b ∈S⎮ b ≤ a} is both a subalgebra of S and a Boolean lattice under the natural partial ordering of S. Conversely, given a left normal band with zero (S; ∧, 0) such that for each a ∈ S the set ⎡ a⎤ is a Boolean lattice under ≥, a derived iBCS-structure (S; \, 0) is given by letting a \ b be the relative complement of a∧ b in ⎡ a⎤ for all a, b in S. Finally, both derivations (S; \, 0) → (S; ∧, 0) and (S; ∧, 0) → (S; \, 0) are reciprocal. In particular, a \ b is the complement of a∧ b in ⎡ a⎤. 266 VII: Further Topics in Skew Boolean Algebras Proof. First, note that for left normal bands ≤ is described by b ≤ a if a ∧ b = b and ⎡ a⎤ is the set { a ∧ s⎮s ∈S}. Let b, c lie in ⎡ a⎤. By Lemma 4.4.9, b \ c = ( a ∧ b) \ ( a ∧ c) = a ∧ ( b \ c) ∈ ⎡ a⎤ also. In addition 0 = a ∧ 0 ∈ ⎡ a⎤ so that ⎡ a⎤ is seen to be an iBCS-subalgebra of S. Since ∧ is commutative on (⎡ a⎤; \, 0), the latter satisfies the iBCK identities (i) – (iv) above and in particular (ii) which implies that ∧ is commutative on ⎡ a⎤. Since (⎡ a⎤; \, 0) is an iBCK-algebra with maximal element a, it forms a Boolean lattice (⎡ a⎤; ∨, ∧, \, a, 0) with ∧ and \ as already given, and x ∨ y defined as a \ [( a \ x) ∧ ( a \ y)] for all x, y ≤ a. Conversely, let (S; ∧, 0) be a left normal band with zero such that for each a ∈ S the set ⎡ a⎤ is a Boolean lattice under ≤. Let a \ b denote the usual relative complement a \ a∧ b in ⎡ a⎤. The clearly B6 holds. B7 reduces to ( a \ a∧ b) \ [( a \ a∧ b) ∧ c] ≈ ( a \ a∧ c) \ [( a \ a∧ c) ∧ b]. This holds in the Boolean case of ⎡ a⎤ where both sides reduce to [ a \ ( a∧ b ∨ a∧ c)] ∨ ( a∧ b∧ c). In similar fashion B8 and B9 are seen to hold. That both processes are reciprocal follows from B14 which implies a \ ( a ∧ b) = a \ b and the Boolean identity a \ ( a \ a∧ b) = a∧ b. £ Observe that this theorem implies the converse of the main statement of Theorem 7.1.1 in the case of left-handed skew Boolean algebras. That is, we have: Theorem 7.1.1L. An algebra (S; ∨, ∧, \, 0) of type 〈2, 2, 2, 0〉 forms a left-handed skew Boolean algebra if and only if: i) (S; ∨, ∧) is a left-handed strongly distributive skew lattice. ii) (S; \, 0) is an iBCS-algebra. iii) The identity e \ ( e \ f) ≈ e∧ f holds. Proof. (⇒) is clear. For (⇐), (ii) and (iii) along with the previous theorem imply that for all a in S, ⎡ a⎤ = { b ∈S⎮ b ≤ a} is a Boolean lattice on which \ is the relative complement. (S; ∨, ∧, 0) is thus at least a skew Boolean algebra reduct. The \ of its skew Boolean structure agrees with the given \ within all ⎡ a⎤. But then they agree in general since a\ b = a\( a∧ b) holds for the algebra. £ Given a skew lattice (S; ∨, ∧), set x∧L y = x∧ y∧ x and x∨R y = y∨x∨y. Then (S; ∧L, ∨R) is a left-handed skew lattice. (See Section 3.4.) Observe that: i) x∧L y = y∧L x iff x∧ y = y∧ x, in which case all four expressions are equal. ii) x∨R y = y∨R x iff x∨ y = y∨ x, in which case all four expressions are equal. iii) Thus (S; ∧L, ∨R) is symmetric iff (S; ∧, ∨) is symmetric. iv) (S; ∧L, ∨R) and (S; ∧, ∨) share the same natural partial order ≥ and natural quasi-order ≻. v) Thus (S; ∧L, ∨R) is normal iff (S; ∧, ∨) is normal vi) In particular, (S; ∧L, ∨R) forms skew Boolean algebra iff (S; ∧, ∨) does. vii) In which case, (S; ∧L, ∨R, 0) and (S; ∧, ∨, 0) share a common difference \. 267 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Proof of The Bisection Theorem (7.1.1) completed. Given (S; ∨, ∧, \, 0) satisfying (i) – (iii) of Theorem 7.1.1, consider first the left-handed algebra (S; ∧L, ∨R, \, 0) where x∧L y = x∧ y∧ x and x∨R y = y∨ x∨ y. (S; ∨L, ∧L, \, 0) satisfies (i) – (iii) in 7.1.1L making it a left-handed skew Boolean algebra. Thus (S; ∨, ∧, \, 0), with the identical partial order, is a skew Boolean algebra. £ A stronger version of the above theorem exists; it is Theorem 3.3.21 in Spinks’ 2002 dissertation. Its proof, closely modeled after the one above, is left to the reader. Theorem 7.1.8. (Spinks [2002]) An algebra ( S; ∨, ∧, \, 0) of type <2, 2, 2, 0> is a skew Boolean algebra if and only if the following conditions hold: i) ( S; ∨, ∧) is a symmetric skew lattice; ii) ( S; \, 0) is an iBCS-algebra; iii) x ∧ y ∧ x ≈ x \ ( x \ y) holds. In particular, under these conditions, \ is the skew Boolean algebra difference. £ When a binary discriminator variety is also additive A binary operation + on an algebra A with a constant element 0 is additive if for all a in A, a + 0 = a = 0 + a. An algebra A with a constant element 0 is additive if an additive operation can be polynomial defined on A. A variety with a constant 0 is said to be additive if it a binary term x + y can be polynomial defined satisfying the identities x + 0 ≈ x ≈ 0 + x. Theorem 7.1.9. If V is a binary discriminator variety with constant term 0 and additive term x + y, then every algebra A of V has a left-handed skew Boolean algebra term reduct AS. Proof: By Theorem 7.1.6 the binary term x ∧ y = x \ ( x \ y) induces a left normal band operation on every member of V. Let x + y be the additive term of V and define x ∨ y to be the term y + ( x \ y). We need to show that for any A ∈ V, AS = (A; ∨, ∧, \, 0) is a left-handed skew Boolean algebra. It is sufficient to show that the left-handed skew Boolean algebra identities hold on any member A of V on which x \ y induces the binary discriminator. (That is, the primitive case.) This is done directly by straightforward case-splitting arguments. We consider the associativity of ∨. Let A be a member of V for which \ is the discriminator. Given a, b, c ∈ A, a ∨ ( b ∨ c) = a ∨ ( c + ( b \ c)) = ( c + ( b \ c)) + ( a \ ( c + ( b \ c))) and similarly ( a ∨ b) ∨ c = c + (( b + ( a \ b)) \ c). Denoting the two expressions on the right above in succession by (L) and (R), we consider four cases that together cover all possibilities. 268 VII: Further Topics in Skew Boolean Algebras a = 0: L = ( c + ( b \ c)) + (0 \ ( c + ( b \ c)) = ( c + ( b \ c)) + 0 = c + ( b \ c). R = c + (( b + (0 \ b)) \ c) = c + (( b + 0) \ c) = c + ( b \ c) = L. b = 0. L = ( c + 0) + (( a \ ( c + 0)) = c + ( a \ c). R = c + (0 + ( a \ 0) \ c) = c + ((0 + a) \ c) = c + ( a \ c) = L c = 0: In a similar fashion, L = b + ( a \ b) = R. a, b, c ≠ 0. Thus a \ b = b \ c = a \ c = 0 so that L = ( c + 0) + ( a \ ( c + 0)) = c + ( a \ c) = c + 0 = c, while R = c + (( b + 0) \ c) = c + ( b \ c) = c + 0 = c. Thus a ∨ ( b ∨ c) = ( a ∨ b) ∨ c in all possible cases. The absorption identities are similarly checked for \ being the binary discriminator. We consider only a ∧ ( a ∨ b) = a. Here there are three cases. a = 0: 0 ∧ (0 ∨ b) = 0 ∧ ( b + 0\ b) = 0 ∧ ( b + 0) = 0 ∧ b = 0. b = 0: a ∧ ( a ∨ 0) = a ∧ (0 + a\0) = a ∧ (0 + a) = a ∧ a = a. a, b ≠ 0: a ∧ ( a ∨ b) = a ∧ ( b + a\ b) = a ∧ (b + 0) = a ∧ b = a. Thus (S; ∨, ∧, 0) is at least a normal skew lattice with a zero. To show symmetry we verify that a∧ b = b∧ a implies a∨ b = b∨ a. (The converse holds for all normal skew lattices.) Since S is primitive, two main (nonexclusive) cases of a∧ b = b∧ a occur. Either a or b is 0, say a = 0. Here 0 ∨ b = b + (0 \ b) = b while b ∨ 0 = 0 + ( b \ 0) = 0 + b = b. The other case is a equals b. Here commutativity of ∨ is trivial. Thus (S; ∨, ∧) is symmetric, normal skew lattice. The rest follows from Theorem 7.1.6. £ Example 7.1.1. Let SA denote the variety of Stone algebras. A member of SA has the form (A; ∩, ∪, *, 0,1), where (A; ∩, ∪, 0,1) is a bounded distributive lattice and * is a relative pseudo-complementation operation; thus a* is the largest element of A such that a ∩ a* = 0. Stone algebras form a subvariety of the variety of pseudo-complemented distributive lattices. They are distinguished from other pseudo-complemented distributive lattices by the identity x* ≈ x***. Since they are generated by the two and three-element chains, it is not difficult to see that Stone algebras form a binary discriminator variety with binary discriminator term x \ y = x ∩ y*. For a, b ∈ A let a ∨ b be b ∪ ( a ∩ b*) and a ∧ b be a ∩ ( a ∩ b*)*. Then by theorem above, the derived algebra (A; ∨, ∧, \, 0) is a left-handed skew Boolean algebra. £ 269 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond Ternary discriminators and ternary discriminator varieties A ternary discriminator on a set S is a function d: S3 → S defined on a given set S by ⎧ x if x ≠ y d( x, y, z) = ⎨ . z otherwise ⎩ An algebra A is a ternary discriminator algebra if d can be polynomial-defined on its underlying set A. Ternary discriminator algebras are simple. Indeed given elements a ≠ b in such an algebra A and let θ be a congruence on A for which a θ b. Then for all c ∈ A, a = d( a, b, c) θ d( a, a, c) = c. Thus the only non-identity congruence on A is the universal congruence. Murskii [1975] proved that in a certain sense almost all finite algebras are ternary discriminator algebras. If K is a class of algebras of common type having a common ternary discriminator term, the variety V generated from K is a ternary discriminator variety. Burris and Sankappanavar [1981] described such a variety as “the most successful generalization of Boolean algebras to date, successful because we obtain Boolean product representations.” Examples of these varieties include Boolean algebras, n-dimensional cylindric algebras, p-rings and skew Boolean ∩- algebras. The remaining results in this section are from Bignall and Leech [1995]. Theorem 7.1.10. In the variety of skew Boolean ∩- algebra s the polynomial term d( x, y, z) = ( x / y) ∨ [ z \ ( ( x / y) ∨ ( y / x) )] is a ternary discriminator on any primitive algebra. Thus skew Boolean ∩- algebras form a ternary discriminator variety. As such they are both congruence distributive and congruence permutable. (The congruence distributive property was already observed in Theorem 4.4.3.) Proof. If x = y, then clearly d( x, y, z) = z on any algebra, primitive or otherwise. If x ≠ y on some primitive algebra P, then x∩ y = 0, so that x / y = x and y / x = y on P. Thus the displayed polynomial reduces to x ∨ [ z \ ( x ∨ y)]. If x ≠ 0, the latter reduces to x ∨ 0 = x. If x = 0, then y ≠ 0 so that x ∨ [ z \ ( x ∨ y)] reduces to z \ y = 0 = x on P. The congruence distributive property has already been seen in the case of ∩-algebras. That it and the congruence permutable property hold on all discriminator varieties follows from results of Bulman-Flemming, Keimel and Werner. (See Theorem IV.9.4 in Burris and Sankppanavar [1981].) £ A pointed ternary discriminator variety is a ternary discriminator variety with a constant term 0. PD0 denotes the pointed ternary discriminator variety generated by the class of all ternary discriminator algebras (A; d, 0) with 0 as a nullary operation, i.e., a constant. 270 VII: Further Topics in Skew Boolean Algebras Theorem 7.1.11. PD0 is term equivalent to the variety of all right-handed skew Boolean ∩- algebras. Given (A; d, 0) in PD0, right-handed skew Boolean ∩- operations are given by x ∨ y = d( x, 0, y), x ∧ y = d( x, d( x, 0, y), y) and x ⁄ y = d( x, y, 0). Conversely, given a right-handed skew Boolean ∩- algebra (S; ∨, ∧, /, 0), an algebra (S; d, 0) in PD0 is given by d( x, y, z) = x / y ∨ ( x ∧ z) ∨ ( y ∨ z)/ y. Finally both processes, { d, 0} → {∨, ∧, /, 0} and {∨, ∧, /, 0} → { d, 0}, are reciprocal. Proof. If (A; d, 0) is a pointed ternary discriminator algebra, then ⎧ ⎧ ⎧ d( x, 0, y) = x if x ≠ 0 ⎨ ; d( x, d( x, 0, y), y) = y if x ≠ 0 ⎨ ; d( x, y, 0) = 0 if x = y ⎨ . y if x ⎩ = 0 0 if x ⎩ = 0 x if x ⎩ ≠ y But these identities describe respectively ∨, ∧ and / on the primitive right-handed Boolean ∩- algebra with upper D-class A \ {0} and zero-class {0}. Hence all algebras in PD0 induce right- handed skew Boolean ∩-algebras. Conversely, given (S; ∨, ∧, /, 0) and d( x, y, z) as stated, then ⎧ d( x, x, z) = ( x 0 if y ≠ 0 ∧ z) ∨ ( x ∨ z)/ x = z and d(0, y, z) = ( y ∨ z)/ y = . ⎨ z if y ⎩ = 0 When x is neither y nor 0, then d( x, y, z) = x ∨ ( x ∧ z) ∨ ( y ∨ z)/ y = x. Thus in all cases d( x, y, z) is indeed the ternary discriminator on S and (S; d, 0) is a pointed discriminator algebra. Thus all algebras (S; d, 0) induced from right-handed skew Boolean ∩-algebras lie in PD0. That the operations are reciprocal is easily checked at the “entry level” of pointed discriminator algebras and primitive right-handed Boolean ∩-algebras. £ Corollary 7.1.12. Any skew Boolean ∩- algebra A has a right-handed skew Boolean ∩- algebra reduct AR with the property that its congruences and its ∩, / and \ operations coincide with those of A. The new skew lattice operations ∨R and ∧R are defined in terms of the old by x ∨R y = x∨ y∨ x and x ∧R y = y∧ x∧ y. Proof. Indeed one has x ∨R y = d( x, 0, y) = ( x / 0) ∨ [ y \ ( ( x / 0) ∨ (0 / x) )] = x ∨ ( y \ x) = x ∨ y ∨ x with the last identity holding first on all primitive ∩-algebras and hence on all ∩-algebras. Also, x ∧R y = d( x, d( x, 0, y), y) = d( x, x ∨ y ∨ x, y) = y \ [( x ∨ y ∨ x) / x] = y∧ x∧ y with the last identity holding again first on all primitive ∩-algebras and thus on all ∩-algebras. That the standard differences agree is clear. Intersections also agree since the natural partial ordering is unchanged: x∧ y = y∧ x = x iff x∧ y∧ x = y∧ x∧ y = x. Thus BCK differences are also 271 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond unchanged. By Theorem 4.4.13 the congruence structure is unchanged since both algebras share the same maximal lattice image. £ In view of Corollary 7.1.12 and McKenzie [1975] Theorem 1.3, we also have: Corollary 7.1.13. Any algebra A in a pointed ternary discriminator variety has a skew Boolean ∩- algebra polynomial reduct whose congruences coincide with those of A. For more on skew Boolean algebras and discriminator varieties, see the paper by Karin Cvetko-Vah and Antonino Salibra in the following references. Also of interest is the paper by Murskii which, among other things, shows that almost all finite algebras are discriminator algebras (with “almost all” under stood in a certain sense). References J. C. Abbott, Semi-Boolean algebras, Matematicki Vesnik 4 (1967), 177– 198 R. J. Bignall, Quasiprimal varieties and components of universal algebras, Flinders University of South Australia Dissertation,1976. R. J. Bignall and J. Leech, Skew Boolean algebras and discriminator varieties, A. Universalis 33 (1995), 387 – 398. R. J. Bignall, and M. Spinks, Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae Mathematicae Japonicae, 58 (2003), 629 – 638. Corrigendum: Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae Mathematicae Japonicae, 66 (2007), 387–390. S. Burris, S. and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York, 1981. W. H. Cornish, On Iseki’s BCK-algebras, Algebraic Structures and Applications, Lecture Notes in Pure and Applied Mathematics, Vol. 74 (1982), Marcel Dekker, 101 – 122. K. Cvetko-Vah and A. Salibra, The connection of skew Boolean algebras and discriminator varieties to Church algebras, Algebra Universalis, 73 (2015), 369-390. K. Iseki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japonica 23 (1978), 1 – 26. 272 VII: Further Topics in Skew Boolean Algebras J. A. Kalman, Equational completeness and families of sets closed under subtraction, Nederl. Akad. Wetensch. Proc. Ser. A, 63 (1960), 402–405. R. C. Lyndon, Identities in two-valued calculi, Transactions Amer. Math. Soc., 71 (1951), 457–465. R. McKenzie, On spectra and the negative solution of the decision problem for algebras having a finite nontrivial model, Journal of Symbolic Logic, 40 (1975), 186 – 195. V. L. Murskii, The existence of a finite basis, and some other properties, for “almost all” finite algebras (Russian), Problemy Kilbernet., 50 (1975). M. Spinks, Contributions to the Theory of Pre-BCK Algebras, Monash University Dissertation, 2002. 273 Bibliography A. Bauer and K. Cvetko-Vah [2013] Stone duality for skew Boolean intersection algebras, Houston Journal of Mathematics 39, 73 – 109. A. Bauer, K. Cvetko-Vah, M. Gerkhe, G. Kudryatseva and S. van Gool [2013] A noncommutative Priestly duality, Topology and Applications 160, 1423 – 1438. J. Berendsen, D. Jansen, J. Schmaltz and F. Vaandrager [2010] The axiomatization of overriding and update, J. of Applied Logic, 8, 141–150. R. J. Bignall [1976] Quasiprimal varieties and components of universal algebras, Dissertation, The Flinders University of South Australia. [1991] Non-commutative multiple-valued logic, Proc. of the 21st International Symposium on Multiple-valued Logic, IEEE Computer Society Press, 49 – 54. R. J. Bignall and J. E. Leech [1995] Skew Boolean algebras and discriminator varieties, Algebra Universalis 33, 387 – 398. R. J. Bignall and M. Spinks [1996] Propositional skew Boolean logic, Proc. 26th International Symposium on Multiple- valued Logic, IEEE Computer Soc. Press, 43 – 48. [1997] Multiple-valued logic as a programming language, Proc. 27th International Symposium on Multiple-valued Logic, IEEE Computer Soc. Press, 227 – 232. [1998] Multiple-valued logics for theorem-proving in first order logic with equality, Proc. 28th Int. Symposium on Multiple-valued Logic, IEEE Computer Soc. Press, 102-107. [2003] Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae Mathematicae Japonicae, 58, 629 – 638. 275 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond [2007] Corrigendum: Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae Mathematicae Japonicae, 66 (2007), 387–390. J. Cirulis [2111] Nearlattices with an overriding operation, Order 28, 33 – 51. W. H. Cornish [1980] Boolean skew algebras, Acta Math. Hungary 36, 281-291. [1982] On Iseki’s BCK-algebras, in Algebraic Structures and Applications, Lecture Notes in Pure and Applied Mathematics, Vol. 74, Marcel Dekker, 101 – 122. K. Cvetko-Vah [2004] Pure skew lattices in rings, Semigroup Forum 68, 268-279. [2005a] Pure ∇- bands, Semigroup Forum, 71, 93-101. [2005b] Skew lattices of matrices in rings, Algebra Universalis 53, 471-479. [2005c] Skew Lattices in Rings, Dissertation, University of Ljubljana, 2005. [2006] A new proof of Spinks’ Theorem, Semigroup Forum 76, 267 – 272. [2007a] Internal decompositions of skew lattices, Communications in Algebra 35, 243 -247. [2007b] On the Structure of Semigroups of Idempotents in Rings of Matrices, Linear Algebra and its Applications 426, 204-213. [2011] On strongly symmetric skew lattices, Algebra Universalis 66, 99-113. [2017] On skew Heyting algebras, Ars Mathematica Contemporanea 12, 37-50. K. Cvetko-Vah and D. Carfi [2011] Skew lattice structures on the financial events plane, Applied Sciences 13, 9 – 20. K. Cvetko-Vah, M. Kinyon, J. E. Leech and M. Spinks [2011] Cancellation in skew lattices, Order 28, 9 – 32. 276 Bibliography K. Cvetko-Vah and J. E. Leech [2008] Associativity of the s- operation on bands in rings, Semigroup Forum 76, 32 – 50. [2011] On maximal idempotent-closed subrings of M n( F), International Journal of Algebra and Computation 21, 1097–1110. [2012] Rings whose idempotents are multiplicatively closed, Communications in Algebra 40, 3288–3307. K. Cvetko-Vah, J. E. Leech and M. Spinks, [2013] Skew Lattices and Binary Operations on Functions, J. Applied Logic, 11, 253-265. K. Cvetko-Vah and J. Pita Costa, [2010] On the coset laws for skew lattices in rings, Novi Sad J. Math. 83, 11– 25. [2011] On coset laws for skew lattices, Semigroup Forum 83, 395 – 411. [2016] Flat coset decompositions of skew lattices, Semigroup Forum 92, 361– 376. K. Cvetko-Vah and A. Salibra, [2015] The connection of skew Boolean algebras and discriminator varieties to Church algebras, Algebra Universalis, 73, 369-390. P. Fillmore, G. MacDonald, M. Radjabalipour and H. Radjavi [1994] Towards a classification of maximal unicellular bands, Semigroup Forum 49, 195 – 215. [1999] Principal ideal bands, Semigroup Forum 59, 362 – 373. K. Iseki and S. Tanaka [1978] An introduction to the theory of BCK-algebras, Math. Japonica 23, 1 – 26. N. Kimura [1958] The structure of idempotent semigroups, I, Pacific J. of Math. 8, 257– 275. 277 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond M. Kinyon and J. E. Leech [2013] Categorical skew lattices, Order 30, 763-777. M. Kinyon, J. E. Leech and J. Pita Costa, [2015] Distributivity in skew lattices, Semigroup Forum 91, 378 - 400. G. Kudryavtseva [2012] A refinement of Stone duality to skew Boolean algebras, Algebra Universalis 67, 397– 416. [2013] A dualizing object approach to noncommutative Stone duality, J. Australian. Math. Soc . 95, 383-403. [2016] Free skew Boolean intersection algebras and set partitions, Order 31, 1-22. G. Kudryavtseva and M. V. Lawson [2016] Boolean sets, skew Boolean algebras and a non-commutative Stone duality, Algebra Universalis 75, 1 – 19. G. Kudryavtseva and J. E. Leech [2016] Free skew Boolean algebras, International J. of Algebra and Computation 26, 1323-1348. Laslo, G. [1997] The classification of noncommutative lattices according to absorption laws, Proc. of the 1997 Annual Meeting of the Romanian Soc. of Math. Sciences, Tome 1, 77 – 84. [1999] Contributii la dezvoltarea teoriei laticilor necomutative, Dissertation, Babes-Bolyai University, 1999. [2006] On postulates for lattices, preprint, 2006. Laslo, G. and I. Cozac [1998] On noncommutative generalizations of lattices, Mathematica 41, 186 – 196. Laslo, G. and J. E. Leech [2002] Green’s equivalences on noncommutative lattices, Acta Sci. Math. (Szeged) 68, 501–533. 278 Bibliography Leech, J. E. [1986] Towards a theory of noncommutative lattices, Semigroup Forum 34, 1-4. [1989] Skew lattices in rings, Algebra Universalis 26, 48-72. [1990] Skew Boolean algebras, Algebra Universalis 27, 48-72. [1992] Normal skew lattices, Semigroup Forum 44, 1 – 8. [1993] The geometric structure of skew lattices, Trans. of the Amer. Math. Soc. 335, 823 – 42. [1996] Recent developments in the theory of skew lattices, Semigroup Forum 52, 7-24. [2005a] Small skew lattices in rings, Semigroup Forum, (2005), [2005b] Magic squares, finite planes and simple quasilattices, Ars Combinatoria 77, 75 –96. J. Leech and M. Spinks [2008] Skew Boolean algebras derived from generalized Boolean algebras, Algebra Universalis 58, 287–302. [2017] Varieties of skew Boolean algebras with intersections, J. Australian Math. Soc. 102, 290-306. F. Pastijn [1983] Idempotent distributive semirings, II, Semigroup Forum 26, 151-166. Pastijn, F. and A. Romanowska [1982] Idempotent distributive semirings, I, Acta Sci. Math. 44, 239-253. J. Pita Costa [2011] On the coset structure of a skew lattice, Demonstratio Mathematica 44, 673 - 692. [2012] Coset laws for categorical skew lattices, Algebra Universalis 68, 75–89. [2012] On the coset structure of skew lattices, University of Ljubljana Dissertation, 2012. [2014] On the coset category of a skew lattice, Demonstratio Mathematica 47, 539-554. 279 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond H. Radjavi [1985] On the reduction and triangularization of semigroups of operators, J. Operator Theory 13, 65 - 71. H. Radjavi and P. Rosenthal [2000] Simultaneous Triangularization, Springer, New York. M. Spinks [1998] Automated deduction in non-commutative lattice theory, Tech. Report 3/98, Monash University, Gippsland, June 1998 [2000] On middle distributivity for skew lattices, Semigroup Forum 61, 341-345. [2002] Contributions to the Theory of Pre-BCK Algebras, Dissertation, Monash University, Gippsland. M. Spinks and R. Veroff [2006] Axiomatizing the skew Boolean propositional calculus, J. Automated Reasoning 37, 3–20. H. Yutani [1977] On a system of axioms of a commutative BCK algebra, Math. Sem. Notes 5, 255 – 256. 280 Addendum 2020 This survey has focused on developments in skew lattices research up through much of 2017. The research does not end there. To pick up the trail, we begin with a conference held in Slovenia in May of 2018. Its official title was Noncommutative Structures 2018: a Workshop in Honor of Jonathan Leech. I gave the opening address, which was later published as the following article: J. Leech, My journey into noncommutative lattices and their theory, The Art of Discrete and Applied Mathematics, 2 (2019) #P2.01. This paper provides the interested seeker with a fairly thorough overview of much that transpired in the first thirty years of the renewed study of noncommutative lattices. In the same issue of this online journal were contributions by other workshop participants: K. Cvetko-Vah, M. Kinyon, J. Leech & T. Pisanski, Regular antilattices . The Art of Discrete and Applied Mathematics, 2 (2019) #P2.06. D. G. FitzGeraltd: Groupoids on a skew lattice of objects. The Art of Discrete and Applied Mathematics, 2 (2019) #P2.03. D. Ellerman: A graph-theoretic method to define any Boolean operation on partitions. The Art of Discrete and Applied Mathematics, 2 (2019) #P2.02. J. Jovanović & A. Tepavčević: Ω-lattices from skew lattices. The Art of Discrete and Applied Mathematics, 2 (2019) #P2.04. A. Bucciarelli & A. Salibra: On noncommutative generalisations of Boolean algebras. The Art of Discrete and Applied Mathematics, 2 (2019) #P2.07. R. J. Bignall & M. Spinks: Dual binary discriminator varieties. The Art of Discrete and Applied Mathematics, 2 (2019) #P2.08. J. Pita Costa & J. Leech: On the coset structure of distributive skew lattices, The Art of Discrete and Applied Mathematics, 2 (2019) #P2.05 Open problems from NCS 2018. The Art of Discrete and Applied Mathematics, 2 (2019) #P2.09 281 Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond More recent skew lattice research has come from Karin CvetkoVah and friends: K. CvetkoVah: Noncommutative frames, Journal of Algebra and Its Applications 18 (2019), 1950011. K. CvetkoVah, J. Hemelaer & J. Leech: Noncommutative frames revisited, Ars Mathematica Contemporanea 19 (2020),147-154. K. CvetkoVah, Jens Hemelaer & Lieven Le Bruyn: Duality for noncommutative frames, to appear in Topology and its Applications. K. CvetkoVah, J. Hemelaer & L. Le Bruyn: What is a noncommutative topos?, Journal of Algebra and Its Applications, 18 (2019), 1950011 (13 pages) K. Cvetko-Vah & C. Verwimp: Skew lattices and set-theoretic solutions of the Yang-Baxter equation, Journal of Algebra, 542 (2020). 65 – 92. K. Cvetko-Vah, M. Sadrzadeh, D. Kartsaklis & B. Blundell, Non-commutative logic for compositional distributional semantics, Logic, Language, Information, and Computation, Springer, Berlin, Lecture Notes in Computer Science, 10388 (2017), 110–124. Until 2017 most of the research on skew lattices has, to my knowledge, occurred in Australia, North America and Europe. But that is changing and with the change comes research that pushes the theory in new directions. From China: Y. Zhi, X. Zhou & Q. Li: Residuated skew lattices, Information sciences Sci. 460/461 (2018), 190–201. Residuated rough sets induced by ideals in skew lattices, Journal of Intelligent and Fuzzy Systems, 33 (2017), 5959-5972. Y. Zhi, & X. Zhou, Roughness in substructures of skew lattices: Journal of Intelligent and Fuzzy Systems 36 (2019), 5959-5972; 282 Addendum 2020 From East Africa B. A. Alaba, M. Alamneh* & Y. M. Gubena: Skew semi-Heyting algebras, International Journal of Computing Science and Applied Mathematics, 4 (2018), 10-14. *From India From India Y. L. Anasuya & P. R. Krishna Kishore: Characterization for a certain class of non-symmetric skew lattices, Southeast Asia Bulletin of Mathematics 35 (2011), 625 – 629. P. R. Krishna Kishore: Prime Filters in a Skew Lattice, Southeast Asia Bulletin of Mathematics, 36 (2012), 255 – 263. B. Assaye, M. Alamneh, L. Narayan Mishra and Y. Mebrat: Dual skew Heyting almost distributive lattices, Applied Mathematics and Nonlinear Sciences, 4 (2019), online. B. R. Bijila & P. R. Kumar: On the atom-based graph of a rectangular skew lattice, Malaya Journal of Mathematik, 8 (2020), 1022-1025 From Iran R. Koohnavard & A. Borumand Saeid (Skew) filters in residuated skew lattices, Scientific Annals of Computer Science, 28 (2018), 115–140. (Skew) filters in residuated skew lattices II, Honam Mathematical Journal 40 (2018), 401–431. On pseudo residuated skew lattices, Boletín de la Sociedad Matemática Mexicana, 26 (2020), 775–794. . Filter theory on pseudo residuated skew lattices, Afrika Matematika 31 (2020), 1251-1271. On residuated skew lattices, Analele Universitii “Ovidius” Constanta, 21 (2019), on line. Onward! 283 Acknowledgement: I would like to express my appreciation to all who have done research on skew lattices. In particular I am grateful to those I have worked directly with: Robert J. Bignall, Karin Cvetko-Vah, Michael Kinyon, Ganna Kudryavtseva, Gratiela Laslo, Tomaž Pisanski, Joao Pita Costa, and Matthew Spinks. I would also like to express my gratitude to the late Charles Wells, who in 1983 provided the venue at Case Western Reserve University where I began my research on skew lattices and skew Boolean algebras. Charles was very supportive of my initial research. In addition, many thanks to those involved with bringing this survey of skew lattice theory to publication: the editorial team (Tomaž Pisanski, Karin Cvetko Vah and Michael Kinyon), the technical support, Nino Bašić, the University of Primorska and its Rector Klavdija Kutnar and the University of Primorska Press Editor-In-Chief Jonatan Vinkler To ALL, thank you! 284 Jonathan E. Leech NONCOMMUTATIVE LATTICES: Skew Lattices, Skew Boolean Algebras and Beyond The extended study of non-commutative lattices 2 3 5 7 11 13 17 19 23 2was b9 3 egun in 1949 b 1y Ernst P 37 4 ascual Jordan, a theoretical and mathematical physicist and co-worker of Max Born and Werner Karl Heisenberg. Jordan introduced noncommutative lattices as algebraic structures potentially suitable to encompass the logic of the quantum world. The modern theory of noncommutative lattices began forty years later with Jonathan Leech’s 1989 paper “Skew lattices in rings.” Recently, noncommutative generalizations of 1 89 97 101 103 107 109 113 127 131 137 139 1 lattices and related structures hav 49 151 e seen an upsurge in interest, with new ideas and applications emerging, from quasilattices to skew Heyting algebras. Much of this 157 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 2 activit 57 26 y is deriv 3 269 27 ed in some wa 1 277 281 283 y from the initiation of 293 307 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 Jonathan Leech’s program of research in this area. The present book consists of seven chapters, mainly covering skew lattices, quasilattices and paralattices, skew lattices of idempotents in rings and skew Boolean algebras. As such, it is the first research monograph covering major results due to this renewed study of noncommutative lattices. It will serve as a valuable graduate textbook on the subject, as well as a handy reference to researchers of noncommutative algebras. Založba Univerze na Primorskem Titov trg 4, SI-6000 Koper www.hippocampus.si zalozba@upr. si ISBN 978-961-293-027-1 Document Outline Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: Založba Univerze na Primorskem/University of Primorska Press About the Author Colophone Table of Contents I: PRELIMINARIES 1.1 Lattices 1.2 Bands 1.3 Noncommutative lattices – initial observations References II: SKEW LATTICES 2.1 Fundamental results 2.2 Instances of commutative behavior 2.3 Normal skew lattices 2.4 Primitive skew lattices and skew lattice structure 2.5 Partial skew lattices and coset projections 2.6 Decompositions of normal, symmetric skew lattices References III: QUASILATTICES, PARALATTICES & THEIR CONGRUENCES 3.1 Congruences on quasilattices 3.2 Antilattices that are simple as algebras 3.3 Regular quasilattices 3.4 Paralattices and refined quasilattices 3.5 The effects of the distributive identities 3.6 Deriving simple antilattices from magic squares References IV: SKEW BOOLEAN ALGEBRAS 4.1 Skew Boolean algebras 4.2 Finiteness, orthosums and free algebras 4.3 Connections with strongly distributive skew lattices 4.4 Skew Boolean algebras with intersections 4.5 Omega algebras and skew Boolean covers References V: FURTHER TOPICS IN SKEW LATTICES 5.1 Symmetric skew lattices 5.2 Distributive identities in the symmetric case 5.3 Cancellation in skew lattices 5.4 Categorical skew lattices 5.5 Distributive skew lattices 5.6 Midpoints and distributive skew chains 5.7 Counting theorems and cancellative skew lattices References VI: SKEW LATTICES IN RINGS 6.1 Quadratic skew lattices in rings 6.2 ∇-bands and cubic skew lattices 6.3 The question of s-closure 6.4 Idempotent-closed rings 6.5 Decomposing E(R) and R 6.6 Idempotent-closed rings of matrices References VII: FURTHER TOPICS IN SKEW BOOLEAN ALGEBRAS 7.1 Differences, discriminators and connections with other algebras References Bibliography Addendum 2020 Acknowledgement