ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P2.09
https://doi.org/10.26493/2590-9770.1332.594
(Also available at http://adam-journal.eu)
Open problems from NCS 2018
Jonathan Leech
Westmont College Santa Barbara, CA, USA
João Pita Costa ∗
Jožef Stefan Institute, Ljubljana, Slovenia
Received 14 July 2019, accepted 19 September 2019, published online 31 December 2019
Abstract
All participants in the workshop NCS 2018 were invited to submit a list of open prob-
lems, typically problems relevant to the subject matter of their talk. These problems follow,
grouped according the particular individual presenting them. These individuals appear with
their problem sets in alphabetical order, the sole exception being that the Honoree of the
workshop appears first. Some of the presenters give a bit of background to accompany their
problems. In other cases, such as myself, the presenter assumes that enough background
was given in their talk as published herein. Thus the reader is invited to refer back to the
relevant article. It is hoped that in the months to follow some of these problems will be
solved, or at least, considerable light will be shed on them.
Keywords: Skew lattice, ring, primitive skew lattice, dual congruence distributivity, orthodox semi-
group, sheaf representation, left regular band, quasilattice, paralattice, covering lattice, skew Bool-
ean algebra, congruence lattice, antilattice, skew Heyting algebra, coset structure, comodernistic
lattice.
Math. Subj. Class.: 06A75, 06B20, 06B75
1 Jonathan Leech
Problem 1.1. The following problem considers skew lattices in the context of rings.
(1) Given a finite left (right) regular band B, must the skew lattice generated from B in
the semigroup ring Z(B) be finite also?
(2) If B is the free left (right) regular band on generators, x1, x2, . . . , xn, what must its
generated skew lattice in Z(B) look like. . . in detail?
∗The author thanks the support of the Croatian Science Foundation’s funding of the project EVOSOFT, with
the reference UIP-2014-09-7945.
E-mail addresses: leech@westmont.edu (Jonathan Leech), joao.pitacosta@ijs.si (João Pita Costa)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 2 (2019) #P2.09
The Z-coefficient case has implications for the case of an arbitrary commutative ring
K with identity being used as the coefficient ring. Of course, if the generated skew lattice
in (2) is finite for all n, then the answer to (1) is YES. In this case, for any finite left regular
band in any ring, the generated skew lattice is finite.
Problem 1.2. Find necessary and sufficient conditions for a skew lattice S to be embed-
ded into a ring. That is for S to have an isomorphic copy consisting of idempotents in a
ring with the operations being multiplication and either the quadratic or cubic join. Pre-
conditions include: being distributive (and thus categorical); being cancellative (and thus
symmetric). Clearly skew Boolean algebras and their duals can be embedded into a ring.
More generally, strongly distributive skew lattices [satisfying x∧(y∨z) = (x∧y)∨(x∧z)
and (x∨y)∧z = (x∧z)∨ (y∧z)] and dual strongly distributive lattices can be embedded
into a ring. Are there other sufficient conditions?
Problem 1.3. In particular, find necessary and sufficient conditions for a primitive skew
lattice to be embedded into a ring. A primitive skew lattice is both distributive and cancella-
tive, Perhaps all primitive skew lattices can be so embedded. If not, then a third general
requirement for a skew lattice to be embedded into a ring is that all of its primitive subal-
gebras can be so embedded. More generally, find necessary and sufficient conditions for
skew chains A > B > C or A > B > C > D or longer, to be so embedded.
Problem 1.4. What can be said about the subvariety of skew lattices generated from all
primitive skew lattices? It must be: distributive (and thus categorical), cancellative (and
thus symmetric) and strictly categorical. What else? Again, what are necessary and suffi-
cient conditions for a skew lattice to be in this subvariety?
Problem 1.5. What can one say about the subvariety of skew lattices generated jointly
from the varieties of normal and conormal skew lattices?
2 Robert J. Bignall
Problem 2.1. Find a Mal’cev condition (see [21]) for dual congruence distributivity at a
constant.
Problem 2.2. Characterise the class of dual binary discriminator varieties that are dually
congruence distributive at their discriminating constant.
Background
Let K be a class of algebras of some given similarity type and let 0 be a constant term.
Given an algebra A ∈ K and Θ ∈ ConA, denote the congruence class {a ∈ A | a Θ 0}
by [0]Θ, where 0 = 0A. A is said to be congruence distributive at 0, or congruence
0-distributive if for all Θ,Ψ,Φ ∈ ConA
[0](Θ ∨Ψ) ∧ Φ = [0](Θ ∧ Φ) ∨ (Ψ ∧ Φ).
A is said to be dually congruence distributive at 0, or dually congruence 0-distributive if
for all Θ,Ψ,Φ ∈ ConA
[0](Θ ∧Ψ) ∨ Φ = [0](Θ ∨ Φ) ∧ (Ψ ∨ Φ).
J. Leech and J. Pita Costa: Open problems on skew lattices 3
The class K is congruence 0-distributive (resp. dually congruence 0-distributive) if every
A ∈ K is congruence 0-distributive (resp. dually congruence 0-distributive); see [2]. These
properties are described as being ideal-theoretic, because [0]Θ is an ideal of A with respect
to the constant 0.
Dual congruence 0-distributivity is a stronger condition than congruence 0-distributi-
vity. For example, meet semilattices with 0, or more generally normal bands with 0, are
congruence 0-distributive but not dually congruence 0-distributive. On the other hand,
skew Boolean algebras are dually congruence 0-distributive, and hence are also congru-
ence 0-distributive. Skew Boolean algebras, or more generally algebras in skew Boolean
discriminator varieties, enjoy a number of other strong ideal-theoretic properties, despite
the fact that their congruence lattices in general satisfy no lattice identities.
I. Chajda [2] has given a Mal’cev condition for the class of congruence 0-distributive
varieties, modelled on Bjarni Jónsson’s well-know Mal’cev condition for the class of con-
gruence distributive varieties [11]. However, it would appear that no-one has as yet solved
the harder problem of finding a Mal’cev condition for the class of dually congruence 0-
distributive varieties (or dually congruence 0-distributive SP classes; that is classes of al-
gebras that are closed under the taking of products, subalgebras and isomorphisms.) The
congruence lattices of algebras in dual binary discriminator varieties do not in general have
any special properties. However, the ideals of such algebras play a central role. Another
significant ideal theoretic property is congruence permutability at 0. A class K of algebras
with a constant 0 is congruence permutable at 0 if for any A ∈ K and any Θ,Φ ∈ ConA,
[0]Θ ◦ Φ = [0]Φ ◦Θ.
K is arithmetic at 0 if it is both congruence distributive and congruence permutable at 0, in
which case it is also dually congruence distributive at 0.
The class of binary discriminator varieties has an entirely ideal-theoretic characterisa-
tion as that class of pointed varieties that are arithmetic at their constant term, have equa-
tionally definable principal ideals, and are generated by a sub-class of ideal simple mem-
bers. A worthwhile but as yet incomplete research program would be to characterise each
pointed discriminator variety sub-class of the class of dual binary discriminator varieties in
terms of the ideal theoretic properties of its members.
3 Des FitzGerald
Problem 3.1. Let A be an algebra and S its monoid of endomorphisms. It is known that if
S has commuting idempotents then A has the properties:
(RI) the intersection of two retracts of A is also a retract,
(UR) to each retract of A corresponds a unique idempotent with that range,
and their duals. Is the converse true? This was posed as Problem 7.4 of [10] where a proof
of the forwards implication may be found.
Problem 3.2. My paper [5] necessarily included (in Section 2) a development of groupoids
over a band of objects. It seemed likely that this topic had been considered already, but I
have since found nothing directly comparable in the literature. Hence the problem: describe
the class of orthodox semigroups which are obtained as groupoids over a band of objects
in the manner shown there.
4 Art Discrete Appl. Math. 2 (2019) #P2.09
Background for 3.1
First note that every homomorphism (written as a right mapping) in a variety has a factori-
sation f = hj with h surjective and j injective. If there are h : A → R, j : R → A such
that jh = 1R, then h is called a retraction (and in fact is surjective) and j is a section (and
is injective). R is a retract of A and the congruence ρh := h ◦ h−1 is a coretract of A. Of
course R ∼= A/ρh, so the distinction between retracts and coretracts is solely whether we
consider R as a subalgebra or a quotient algebra. Put f = hj; then jh = 1R is equivalent
to f2 = f (this is easy folklore).
Now (RI) and (UR) are as stated, but to unpick my little concealment, we could rephrase
(UR) as: to each section j : R → A corresponds a unique right inverse h : A → R. Then
the duals are
(RI*) the join of two coretracts of A is a coretract of A;
(UR*) each retraction h : A→ R has a unique section (left inverse).
I know no reference for my claim that CI: S = EndA has commuting idempotents
implies all of these; here’s my proof.
Proof. Let two retracts of A be determined (as Af and Ag) by f, g ∈ S with f = f2,
g = g2, and fg = gf . Then (fg)2 = fg and Afg is a retract (and ρfg is a coretract).
Clearly
Afg = Agf ⊆ Af,Ag
and so Afg ⊆ Af ∩ Ag. But if x ∈ Af ∩ Ag then x = xf = xg = xfg, so x ∈ Afg.
This proves Af ∩ Ag = Afg, a retract, and establishes (RI). Similarly, any two coretracts
of A are of the forms ρf and ρg , with f, g idempotents of EndA. Then ρf ∨ ρg ⊆ ρfg
(xf = x′f ⇒ xfg = x′fg, etc.) But if xfg = x′fg, we have
x ρf xf ρg x
′f ρf x
′ and x ρg xg ρf x′g ρg x′,
and thus (x, x′) ∈ ρf ∨ ρg whence ρfg ⊆ ρf ∨ ρg . Thus ρf ∨ ρg = ρfg , a coretract; so
(RI*) holds.
Now for (UR). Suppose there are f = f2, g = g2 such that the retract R = Af = Ag.
Then for any x ∈ A, there are y, z ∈ A such that xf = yg, xg = zf . Then
xfg = yg = xf
= xgf = zf = xg,
and so f = g and we see (UR). To (UR*): Suppose there is h : A → R and sections
j, k : R → A such that jh = kh = 1R. Then hj, hk are idempotents in EndA and we
have in turn hjhk = hkhj, hk = hj, and k = j.
4 Sam van Gool
Problem 4.1. What is the relationship, if any, between the duality for sheaf representa-
tions of distributive-lattice-ordered algebras of Gehrke and van Gool [6] and the duality for
strongly distributive skew lattices of Bauer, Cvetko Vah, Gehrke, van Gool, and Kudryavt-
seva [1]? Sheaves play prominent but apparently distinct roles in these two dualities; any
connection between the two could be an interesting direction for further research.
J. Leech and J. Pita Costa: Open problems on skew lattices 5
5 Michael Kinyon
5.1 Left regular bands and left-handed skew lattices as ordered sets
When I’ve given talks about skew lattices to nonexperts, one of the things that typically
bothers members of the audience is that unlike lattices, skew lattices do not have a nice
order theoretic definition.
There was one attempt to give such a characterization by Gerhardts [7], who studied
a variety of noncommutative lattices he called Fastverbanden. They have an unintuitive
definition, but it is not difficult to show that they are precisely what we now call left-
handed binormal (normal and conormal) skew lattices. Gerhardts attempted to characterize
Fastverbanden entirely in terms of their natural partial order and their natural preorder.
Unfortunately, [7] has at least one mistake in it. However, the definitions can be modified
to get his idea to work. For simplicity, I will just focus on bands here instead of skew
lattices.
The basic structure (X,,≤) consists of a set X with a preorder  and a partial order
≤ which refines  (that is, ≤ ⊆ ). Let L denote the equivalence relation associated to .
We assume the following:
If x L y and x, y ≤ z, then x = y. (∗)
For a, b ∈ X , an element c ∈ X is said to be a lower bound of the pair (a, b) if c ≤ a and
c  b. We say that c is an infimum of (a, b) if
(i) c is a lower bound of (a, b), and
(ii) if x  a, b, then x  c.
It is easy to prove that a pair (a, b) can have at most one infimum. (The definition of
infimum is where I differ from Gerhardts.)
A left normal band (X,≤,) (in the order theoretic sense) is a structure as above such
that every ordered pair (a, b) of elements has an infimum, which we denote by a ∧ b. We
then can prove the expected theorem: left normal bands in the order theoretic sense are
precisely the same as left normal bands in the algebraic sense.
It is easy to get the corresponding result for left-handed binormal skew lattices, and it
is also not hard to generalize from the one-sided case to normal bands by using both the L-
andR-preorders.
Left normality essentially comes from property (∗) above.
Problem 5.1. Generalize the above characterization to left regular bands. What property
should replace (∗)?
5.2 Quasilattices and paralattices
Recall that a double band (X,∨,∧) is a
• quasilattice if the natural preorders dualize each other,
• paralattice if the natural partial orders dualize each other.
Both classes of noncommutative lattices form varieties. Quasilattices are axiomatized by
x ∧ (y ∨ x ∨ y) ∧ x = x = x ∨ (y ∧ x ∧ y) ∨ x,
6 Art Discrete Appl. Math. 2 (2019) #P2.09
while paralattices are axiomatized by
(x ∨ y ∨ x) ∧ x ∧ (x ∨ y ∨ x) = x = (x ∧ y ∧ x) ∨ x ∨ (x ∧ y ∧ x).
(These are not the four identities axiomatizing paralattices in [12] but it is easy to see these
two are equivalent to those.)
Let us say that a double band (X,∨,∧) is a -lattice if it satisfies the following
implications:
x ≤∧ y ⇒ y ∨ x
x ≤∨ y ⇒ y ∧ x
(Part of the problem here is to find a suitable name for this class of noncommutative lat-
tices.) It is clear that -lattices include both quasilattices and paralattices as special
cases. It is also straightforward to show that -lattices form a variety axiomatized by
x ∧ (x ∨ y ∨ x) ∧ x = x = x ∨ (x ∧ y ∧ x) ∨ x.
Problem 5.2 (Suggested by J. Leech). Describe the join variety of the varieties of paralat-
tices and quasilattices.
6 Jurij Kovič
Problem 6.1. Define the concept of (and develop a theory for) a covering of a (skew)
lattice. Use the analogy with covering graphs (voltages).
Background
The concept of a covering graph. Definitions of graphs, covering graphs and voltages are
in detail explained in [15, pp. 96–97]. Covering graphs were first introduced in [9] for
directed graphs, but they may be applied also to undirected graphs, as in [16]. Lifting
graph automorphisms from the underlying graph is explained in [14].
7 Ganna Kudryavtseva
Some problems on skew Boolean algebras are proposed below. For a concise background
on skew Boolean algebras, please refer to [13].
Problem 7.1. Study objects obtained by relaxations of the definition of an SBA S. What
if normality is dropped and one requires only S is symmetric, has 0 and S/D is a Boolean
lattice? What if symmetricity is dropped and one requires only S is normal, has 0 and
S/D is a Boolean lattice? These, in each case, are generalizations of SBAs. Study their
structure. Do they form a variety (in each case)? How are they related to SBAs?
Problem 7.2. Let S be a LSBIA (left handed skew Boolean intersection algebra). Then
S∗ is a (locally compact) Boolean space so that there is its dual GBA, B. Is it possible to
extend the assignment S 7→ B to a functor? What are adjoints, if exist? What can be said
about the relationship between S/D and B?
Problem 7.3. There is a forgetful functor from Boolean inverse semigroups to LSBAs
(indeed, domain and range maps of an étale groupoids are étale maps). Does it have a left
adjoint? Then how is it constructed? A similar question can be asked also about Boolean
inverse ∧-semigroups and LSBIAs.
J. Leech and J. Pita Costa: Open problems on skew lattices 7
8 Tomaž Pisanski
Problem 8.1. Given a finite lattice K. Find a test that will tell whether K is isomorphic to
a congruence lattice Con(N) of some antilattice.
Problem 8.2. Any automorphism α of N induces an automorphism α̂ of Con(N). Hence,
there is an obvious group homomorphism h : AutN → AutCon(N) mapping α to α̂.
Classify those antilattices N for which h is an isomorphism.
Problem 8.3. Let us call a lattice L transitive if for each pair of maximal chains a and b of
L there exists an automorphism of L that maps one to the other one, i.e. it maps a into b.
We call an antilattice N transitive, if its congruence lattice Con(N) is transitive. Classify
transitive antilattices.
Background
For an antilatticeN , a congruence θ is an equivalence relation onN that is compatible with
both ∧ and ∨. Congruence relations form a lattice, Con(N).
9 Joao Pita Costa
Problem 9.1. Can we build on the coset structure of a skew distributive lattice to charac-
terize skew Heyting algebras through their coset structure. And what can be said about the
coset structure of skew Boolean algebras? Will it give us some insights on the respective
dual spaces?
Problem 9.2. Can we always describe coset law-like identities that can determine a variety
of skew lattices? Or are there varieties of skew lattices that do not permit such characteri-
zation?
Problem 9.3. The index theorems presented for some varieties of skew lattices in [3, 17]
and [19] show a combinatorial perspective on these algebras as a direct consequence of
their coset structure. Can we always obtain such results? What can they bring to the study
of skew lattices?
For a concise background on the coset structure of several varieties of skew lattices and
how they relate to each other, and to their index theorems, please refer to [18].
10 Andreja Tepavčević
Problem 10.1. Representation of algebraic lattices by a weak congruence lattice. All weak
congruences form an algebraic lattice under inclusion. The diagonal relation (equality) has
a particular role within weak congruence lattice. A 30-years unsolved problem [22] is as
follows: if we take an algebraic lattice L and pick an element a, under which conditions
there is an algebra such that its weak congruence lattice is isomorphic to L and the diagonal
element corresponds to the fixed element a under this isomorphism. This is not possible
for every element in a lattice, since this element should be codistributive and should satisfy
some particular conditions. For further background please read [4] and [23].
Problem 10.2. Congruence lattice characterization. Characterize congruence lattices (and
later weak congruence lattices) of skew lattices. It is well-known [8] that the congruence
lattice of a lattice is distributive since there is the corresponding Mal’cev term. Is there a
similar result for skew lattices?
8 Art Discrete Appl. Math. 2 (2019) #P2.09
Background
Congruences are compatible equivalence relations. Weak congruences are compatible rela-
tions satisfying the same as congruences except they are not reflexive. Weak congruences
are congruences on subalgebras of an algebra A (equivalently - symmetric, transitive and
compatible relations of A).
11 Russ Woodroofe
Problem 11.1. What is the correct universal algebra generalization of order congruence
lattices? More broadly: find examples of comodernistic lattices!
Background
An element m of a lattice L is left-modular if for every pair x ≤ y of elements in L, we
can write x ∨m ∧ y without parentheses. That is, m is left-modular if
∀x ≤ y, (x ∨m) ∧ y = x ∧ (m ∧ y) .
Left-modularity is a lattice-theoretic generalization of the Dedekind modular identity of
group theory.
A lattice L is comodernistic if every nontrivial interval [a, b] of the lattice has a coatom
that is left-modular in the interval. That is, for every a < b in L, there is an m covered by
b, so that m satisfies the left-modular relation with respect to every pair a ≤ x ≤ y ≤ b.
Jay Schweig and myself introduced comodernistic lattices in [20].
Comodernistic lattices generalize dual semimodular lattices, which can be defined as
those lattices where every coatom of every interval is left-modular. Comodernistic lattices
also generalize supersolvable lattices, which are those graded lattices which have a maxi-
mal chain consisting of left-modular elements. There are additional examples:
Example 11.2. The subgroup lattice of any finite solvable group is comodernistic. This
follows from the fact that normal subgroups are left-modular, by the Dedekind modular
identity.
Example 11.3. Similarly, the subalgebra lattice of any finite Lie algebra is comodernistic.
Example 11.4. The order congruence lattice O(P ) of a finite poset P consists of all the
level sets of all the order preserving maps with domain P , ordered by refinement. Order
congruence lattices are comodernistic, as can be seen in two steps:
(1) Let µ be the partition with a maximal element in a singleton block, and all other
elements together in a big block. Then µ is left-modular.
(2) Intervals in O(P ) are direct sums of order congruence lattices of quotients of sub-
posets of P . Existence of modularity in coatoms of quotients follows from (1). Ex-
istence of left-modular coatoms is preserved by direct sum.
Back to the Problem
I’m interested in finding comodernistic lattices (or dually, modernistic lattices) in other
settings. Perhaps there is a wider class of congruence lattices from universal algebra or
similar which yields comodernistic lattices?
J. Leech and J. Pita Costa: Open problems on skew lattices 9
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