ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P2.06
https://doi.org/10.26493/2590-9770.1333.152
(Also available at http://adam-journal.eu)
Regular antilattices
Karin Cvetko-Vah ∗
Department of Mathematics, Faculty of Mathematics and Physics,
University of Ljubljana, Jadranska 21, SI-1000 Ljubljana, Slovenia
Michael Kinyon †
Department of Mathematics, University of Denver, Denver, CO 80208, USA
Jonathan Leech
Department of Mathematics, Westmont College, Santa Barbara, CA 93108, USA
Tomaž Pisanski ‡
Faculty of Mathematics, Natural Sciences and Information Technologies,
University of Primorska, Glagoljaška 8, SI-6000 Koper, Slovenia, and
Andrej Marušič Institute, University of Primorska,
Muzejski trg 2, SI-6000 Koper, Slovenia, and
Faculty of Mathematics and Physics, University of Ljubljana,
Jadranska 21, SI-1000 Ljubljana, Slovenia, and
Institute of Mathematics, Physics and Mechanics (IMFM),
Jadranska 19, SI-1000 Ljubljana, Slovenia
Received 18 November 2019, accepted 21 December 2019, published online 30 December 2019
Abstract
Antilattices (S;∨,∧) for which the Green’s equivalences L(∨), R(∨), L(∧) and R(∧)
are all congruences of the entire antilattice are studied and enumerated.
Keywords: Noncommutative lattice, antilattice, Green’s equivalences, lattice of subvarieties, enumer-
ation, partition, composition.
Math. Subj. Class.: 06B75, 05A15, 05A17, 03G10, 11P99
∗Corresponding author.
†Michael Kinyon was supported by the Simons Foundation Collaboration Grant 359872.
‡Work of Tomaž Pisanski is supported in part by the ARRS grants P1-0294, J1-7051, N1-0032, and J1-9187.
E-mail addresses: karin.cvetko@fmf.uni-lj.si (Karin Cvetko-Vah), michael.kinyon@du.edu (Michael
Kinyon), leech@westmont.edu (Jonathan Leech), pisanski@upr.si (Tomaž Pisanski)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 2 (2019) #P2.06
1 Introduction
In the study of noncommutative lattices, lattices still play an important role. They are
the commutative cases of the algebras being considered and indeed play an important role
in the general theory of that larger class of algebras. (As with rings, “noncommutative”
is understood inclusively to mean not necessarily commutative). But also, typically, a
second subclass of algebras exists that plays counterpoint to the subclass of lattices. It has
become common to refer to their members as “antilattices.” Typically they resist any kind
of nontrivial commutative behavior. That is, an instance of xy = yx for a relevant binary
operation can occur only when x = y. Antilattices, however, are not without their special
charm. Indeed, they have been studied in connection with magic squares and finite planes.
(See [8].)
In this paper we study the class of regular antilattices for which the Green’s equiva-
lences are congruences. Precise definitions occur in Section 2 where basic concepts such
as bands, quasilattices and the condition of regularity are described, along with some rele-
vant preliminary results.
Regular antilattices themselves are the focus of Section 3. The main results are a very
precise decomposition given in Theorem 3.3 and its several consequences. A closer look at
the lattice of subvarieties (see Figure 1) occurs in the final fourth section.
The reader seeking further information on bands is referred to the presentations given
in Clifford and Preston [4], in Grillet [5] and in Howie [6]. For further background on
skew lattices and quasilattices, see [7] and [9]. The basic facts of universal algebras, and in
particular varieties, may be found in the second chapter of [2].
2 Preliminary concepts and results
A band is a semigroup (S; ·) for which all elements are idempotent, that is, xx = x holds.
A band is rectangular if it satisfies the identity xyz = xz, or equivalently, xyx = x. (As
often occurs, if just a single binary operation is involved, its appearance is suppressed in
equations.) A semilattice is a commutative band (xy = yx). Clearly, rectangular semilat-
tices form the class of trivial 1-point bands. Indeed both classes are structural opposites
that play important roles in the general structure of bands. To see how and to set the stage
for further preliminaries requires the use of Green’s relations, defined first for bands.
D : xD y iff both xyx = x and yxy = y;
L : xL y iff both xy = x and yx = y;
R : xR y iff both xy = y and yx = x.
For bands, L and R commute under the usual composition of relations, with the common
outcome being D, i.e., L ◦ R = R ◦ L = R ∨ L = D. Here R ∨ L denotes the join
of the two relations. Moreover, we have the following fundamental result of Clifford and
McLean [3, 10]:
Theorem 2.1. Given a band (S; ·), the relation D is a congruence for which S/D is the
maximal semilattice image and each D-class of S is a maximal rectangular subalgebra of
S. In brief, every band is a semilattice of rectangular bands.
So what do rectangular bands look like? First there are two basic cases. A left-zero
band is a band (L; ·) with the trivial composition: xy = x. A right-zero band is a band
(R; ·) with the trivial composition: xy = y. In other words, we either have just a single
K. Cvetko-Vah et al.: Regular antilattices 3
L-class or just a single R-class. Finally, there are products of both types, L × R, and up
to isomorphism, that is it. Thus a rectangular band may be pictured as a rectangular grid
consisting of rows that areR-classes and columns that are L-classes.
• • • • • •
• • • • • •
• • • • • •
• • • • • •
The product xy of elements x and y is the unique element in the row of x and the
column of y. Given a rectangular band (S; ·) and x in S, if L denotes the L-class of x and
R denotes the R-class of S, and ϕ : L × R → S is defined by ϕ(u, v) = uv ∈ S, then ϕ
is an isomorphism of rectangular bands. Rectangular bands are precisely the bands that are
anti-commutative in that xy = yx iff x = y.
While bands have a very simple local structure – their rectangular D-classes – it is
not immediately clear how elements from different D-classes combine under the binary
operation.
A band is regular if the relations L and R are both congruences. Semilattices and
rectangular bands are both regular. In the semilattice case L and R reduce to the identity
relation, so that regularity is trivial. One might expect all bands to be regular, but that is not
so. In the rectangular case there is more: L and R commute under composition, not only
with each other, but with every congruence θ:
L ◦ θ = θ ◦ L = θ ∨ L and R ◦ θ = θ ◦ R = θ ∨R.
A double band is an algebra (S;∨,∧) for which both reducts (S;∨) and (S;∧) are
bands. A lattice is thus a double band where both (S;∨) and (S;∧) are semilattices that
jointly satisfy the standard absorption identities for a lattice: x∧(x∨y) = x = x∨(x∧y).
A very general class of noncommutative lattices is as follows. A quasilattice is a double
band that satisfies the following (modified) absorption identities:
x ∧ (y ∨ x ∨ y) ∧ x = x = x ∨ (y ∧ x ∧ y) ∨ x.
Note that if commutativity is assumed, both identities reduce to the absorption identities
for a lattice.
A skew lattice is a noncommutative lattice that satisfies the dual absorption identities:
x ∧ (x ∨ y) = x = (x ∨ y) ∧ x,
x ∨ (x ∧ y) = x = (x ∧ y) ∨ x.
A skew lattice is a quasilattice, but not conversely. In a quasilattice, both operations
share common D-classes that also form subalgebras, although on these classes both opera-
tions need not agree! Clearly, for a quasilattice (S;∨,∧), D is a congruence. Indeed, S/D
is the maximal lattice image of S. This leads us to:
Definition 2.2. An antilattice is a double band (S;∨,∧) for which both reducts, (S;∨) and
(S;∧), are rectangular bands, i.e., satisfy the identity xyz = xz or equivalently xyx = x.
An antilattice is trivially a quasilattice. Conversely, each D-class of a quasilattice is a
subalgebra that is an antilattice.
4 Art Discrete Appl. Math. 2 (2019) #P2.06
If the antilattice is a skew lattice, it is also called a rectangular skew lattice. As an
antilattice, it is characterized by x ∧ y = y ∨ x. D-classes of skew lattices are always
rectangular skew lattices.
Similar to bands, a version of the Clifford-McLean Theorem holds:
Theorem 2.3. Given a quasilattice (S;∨,∧), the relation D is the same for both, (S;∨)
and (S;∧). For this common congruence, the quotient algebra S/D is the maximal lattice
image and each D-class of S is a maximal sub-antilattice of S. In brief, every quasilattice
is a lattice of antilattices. (Compare Corollary 3 of [7]; see also [9].)
Antilattices have been studied, not only due to their connection to quasilattices, but also
in connection with magic squares and finite planes. (See [8].)
Like quasilattices and semigroups, by definition antilattices do not have prescribed con-
stants, thus making the empty set a viable subalgebra. In so doing, this allows for the
existence of a complete lattice of subalgebras for any given antilattice.
3 Regular antilattices
Given an antilattice (S;∨,∧), both reducts (S;∨) and (S;∧) are regular in that L(∨) and
R(∨) are congruences on (S;∨), and likewise L(∧) and R(∧) are congruences on (S;∧).
The antilattice is regular if all four equivalences are congruences for the whole algebra. In
general, a quasilattice (S;∨,∧) is regular if L(∨), R(∨), L(∧) and R(∧) are congruences
of (S;∨,∧). Skew lattices are regular, but in general, quasilattices need not be regular.
Theorem 3.1. Regular antilattices form a subvariety of the variety of antilattices.
Proof. We show that antilattices for which L(∨) is a congruence form a subvariety. To
begin, in an antilattice S, xL(∨) u ∨ x holds for all u, x ∈ S, and conversely, if xL(∨) x′,
then trivially, x′ = x′ ∨ x. Since L(∨) is already a congruence on the reduct (S;∨), for
L(∨) to be a congruence on (S;∧), precisely the following identities need to hold:
(y ∧ x) ∨ [y ∧ (u ∨ x)] = y ∧ x & [y ∧ (u ∨ x)] ∨ (y ∧ x) = y ∧ (u ∨ x)
and
(x ∧ y) ∨ [(u ∨ x) ∧ y] = x ∧ y & [(u ∨ x) ∧ y] ∨ (y ∧ x) = (u ∨ x) ∧ y.
Thus this class of antilattices indeed forms a subvariety. Similar remarks verify the same
claim forR(∨), L(∧) andR(∧). The theorem now follows.
SinceL andR commute under composition with all congruences on a rectangular band,
L(∨) ◦ L(∧) = L(∧) ◦ L(∨); L(∨) ◦ R(∧) = R(∧) ◦ L(∨); R(∨) ◦ L(∧) = L(∧) ◦ R(∨);
and R(∨) ◦ R(∧) = R(∧) ◦ R(∨) hold for regular antilattices. All four outcomes are thus
congruences on the antilattice, and indeed form the join congruences of the respective pairs
of congruences.
An antilattice (S;∨,∧) is flat if the reduct (S;∨) is either a left-0 semigroup (x ∨ y =
x) or a right-0 semigroup (x ∨ y = y), and likewise the reduct (S;∧) is either a left-0
semigroup (x ∧ y = x) or a right-0 semigroup (x ∧ y = y). That is, for each operation,
either D = L or D = R. Clearly there are 4 distinct classes of flat antilattices:
• the class ALL of all antilattices where x ∨ y = x = x ∧ y.
K. Cvetko-Vah et al.: Regular antilattices 5
• the class ALR of all antilattices where x ∨ y = x but x ∧ y = y.
• the class ARL of all antilattices where x ∨ y = y but x ∧ y = x.
• the class ARR of all antilattices where x ∨ y = y = x ∧ y.
Clearly each class is a subvariety of regular antilattices. What is more:
Lemma 3.2. Flat antilattices S and T of the same class are isomorphic if and only if they
have the same cardinality. When the latter is the case, an isomorphism is given by any
bijection between S and T .
Theorem 3.3 (Decomposition Theorem). Every nonempty regular antilattice (S;∨,∧) fac-
tors into the direct product SLL×SLR×SRL×SRR of its four maximal flat images, one
from each class above, with the respective factors being unique up to isomorphism.
Proof. The factorization is obtained by first factoring with respect to say ∨:
S ∼= S/R(∨) × S/L(∨)
to get two factors for which the ∨-operation is flat. Then similarly factor both factors
further with respect to the relevantR(∧) and L(∧) congruences to get four flat factors:
S ∼= SLL × SLR × SRL × SRR
where: SLL = S/(R(∨) ∨R(∧)); SLR = S/(R(∨) ∨L(∧)); SRL = S/(L(∨) ∨R(∧)) and
SRR = S/(L(∨) ∨ L(∧)).
Further factorization can take place. But first, given a positive integer n, let nLL, nLR,
nRL and nRR denote the relevant flat antilattices on the set {1, 2, 3, . . . , n}. This leads us
to the following finite version of the Decomposition Theorem:
Theorem 3.4. Let (S;∨,∧) be a nonempty finite regular antilattice with the above factor-
ization SLL × SLR × SRL × SRR. If nLL = |SLL|, nLR = |SLR|, etc., then
S ∼= nLL × nLR × nRL × nRR.
Clearly these four parameters characterize (S;∨,∧). It is also clear that factorization
can continue on each of the four factors. For instance say |SLL| = 180 = 4× 5× 9. Then
we have SLL ∼= (2LL)2 × (3LL)2 × 5LL. Up to isomorphism, the only 1-point algebra is
the trivial algebra 1 = {0}.
Corollary 3.5. A regular antilattice (S;∨,∧) is directly irreducible iff |S| is either 1 or
a prime. Every finite regular antilattice of order > 1 thus factors into a direct product of
finitely many flat antilattices of prime order.
Corollary 3.6. A regular antilattice (S;∨,∧) is subdirectly irreducible iff either |S| = 1
or |S| = 2. Every (finite) regular antilattice of order > 1 is thus isomorphic to a subdirect
product of (finitely) many flat antilattices of order 2.
A sub-(pseudo)variety of regular antilattices is positive, if it is not the sub-variety {∅}.
Corollary 3.7. The lattice of all positive subvarieties of regular antilattices is a Boolean
algebra with 16 elements and 4 atoms: ALL, ALR, ARL and ARR (see Figure 1).
6 Art Discrete Appl. Math. 2 (2019) #P2.06
Corollary 3.8. The lattice of all positive sub-pseudovarieties of finite regular antilattices is
a Boolean algebra with 16 elements and the four atoms as above, but with their respective
classes now restricted to finite algebras: fALL, fALR, fARL and fARR.
We will take a closer look at the positive subvarieties involved in the fourth section.
What can one say about the congruence lattice of a regular antilattice? To begin ob-
serve that the four classes of flat antilattces are mutually term equivalent with each other
and with the class of all left-zero semigroups and also the class of all right-zero semi-
groups. In all these special cases the congruence lattice is precisely the full lattice Π(S) of
all equivalences of the underlying set S. Following the situation for rectangular bands in
general, we have:
Theorem 3.9. Let a nonempty regular antilattice (S;∨,∧) be factored into the direct prod-
uct of its four maximal flat images: SLL×SLR×SRL×SRR. Then the congruence lattice
of S is given by Π(SLL) × Π(SLR) × Π(SRL) × Π(SRR). That is, if the elements if S
are expressed as 4-tuples (x, y, z, w) given by the factorization, then each congruence θ on
S can be represented as a 4-tuple (θLL, θLR, θRL, θRR) of congruences on each factor in
that:
(x, y, z, w) θ (x′, y′, z′, w′) iff x θLL x′, y θLR y′, z θRL z′&w θRR w′.
Conversely, in this manner every such 4-tuple of congruences defines a congruence on the
full antilattice S.
In similar fashion:
Theorem 3.10. Given a nonempty regular antilattice S with factorization SLL × SLR ×
SRL × SRR, if a = |SLL|, b = |SLR|, c = |SRL| and d = |SRR|, then the number of
subalgebras of S is:
1 + (2a − 1)(2b − 1)(2c − 1)(2d − 1).
One can ask: given a positive integer n ≥ 1, up to isomorphism, how many nonisomor-
phic regular antilattices are there of size n? By Theorem 3.4 it is the number ρ(n) of 4-fold
positive factorizations abcd of n, where the order of the factors a, b, c, d is important. Here
a is the size of the LL-factor, b is the size of the LR-factor, etc.
To begin, thanks to Corollary 3.5, given the prime power factorization n = 2e23e3
5e5 · · · pepkk :
ρ(n) = ρ(2e2)ρ(3e3)ρ(5e5) · · · ρ(pepkk ).
Thus things can be reduced to calculating ρ(pe) for any prime power pe .
From a combinatorial perspective, this is equivalent to asking in how many distinct
ways can e identical balls be distributed into 4 labeled boxes. This question has a simple
answer:
ρ(pe) =
(
e+ 3
3
)
.
By putting all these together we obtain the following closed formula for ρ(n):
Theorem 3.11. Let n have the following prime power factorization n = 2e23e35e5 · · · pepkk :
Then
ρ(n) =
(
e2 + 3
3
)(
e3 + 3
3
)(
e5 + 3
3
)
· · ·
(
epk + 3
3
)
.
K. Cvetko-Vah et al.: Regular antilattices 7
See also Table 1.
One can ask a more general question: In how many distinct ways can e identical balls
be distributed into k labeled boxes? This question has analogous answer, namely
(
e+k−1
k−1
)
;
see, for instance [1]. We will use some of these in the next section. Note that such an
ordered partition of an integer n into k possibly empty parts is sometimes called a compo-
sition; see [1].
4 Semi-flat antilattices and other subvarieties
An antilattice (S;∨,∧) is semi-flat if either (S;∨) or (S;∧) is flat. Flat antilattices are
trivially semi-flat. The class of all semi-flat antilattices consists of four distinct subclasses
that are not necessarily disjoint:
• AL#, the class of all antilattices (S;∨,∧) s.t. (S;∨) is a left-0 band;
• AR#, the class of all antilattices (S;∨,∧) s.t. (S;∨) is a right-0 band;
• A#L, the class of all antilattices (S;∨,∧) s.t. (S;∧) is a left-0 band;
• A#R, the class of all antilattices (S;∨,∧) s.t. (S;∧) is a right-0 band.
Theorem 4.1. These four classes are subvarieties of the variety of regular antilattices.
Proof. First observe that each is at least a subvariety in the variety of all antilattices. We
show this for AL#, the other cases being similar. The identity characterizing AL# in the
variety of antilattices is clearly x∨ y = x. Thus AL# is indeed a subvariety of antilattices.
To see that all semi-flat antilattices are regular, again we need only consider, say, AL#. So
let (S;∨,∧) be an antilattice for which (S;∨) is a left zero-band. ThusL(∨) is the universal
equivalence ∇ on S, and thus trivially a congruence on (S;∧) while R(∨) is the identity
equivalence and thus again trivially a congruence on (S;∧). Next consider L(∧) andR(∧).
Being congruences on (S;∧), they are at least equivalences on S. But all equivalences on
S are congruences on the left zero-band (S;∨), and thus L(∧) and R(∧) are congruences
on (S;∨,∧).
Consider next the following diagram.
ALL ALR
ARL ARR
AL#
AR#
A#L A#R
The four flat varieties occupy the middle rectangle. If two distinct flat varieties are
adjacent on this rectangle, their join variety is the semi-flat variety labeling the line between
them. But if they are diagonal opposites, we have the following:
• ALL ∨ARR = the subvariety of antilattices for which x ∨ y = x ∧ y.
• ALR ∨ARL = the subvariety of antilattices for which x ∨ y = y ∧ x.
8 Art Discrete Appl. Math. 2 (2019) #P2.06
These are the antilattice subvarieties that are, respectively, skew∗ lattices or skew lattices.
Next are the four double joins. Consider ALL ∨ ALR ∨ ARL. It consists of regular
antilattices for which the ARR-factor is trivial. Since ∨ and ∧ are idempotent, this reduces
to no nontrivial ARR-subalgebra occurring in the given antilattice. More briefly, no copy
of 2RR occurs as a subalgebra. This is guaranteed by the identity x ∨ (x ∧ y) = x (that
is equivalent to the implication: uR(∧) v ⇒ uL(∨) v) along with its ∨ − ∧ dual. This
subvariety is, of course, the Boolean complement ACRR of ARR. The three other double
joins are treated similarly to obtain:
ALL ∨ALR ∨ARR = ACRL,
ALL ∨ARL ∨ARR = ACLR,
ALR ∨ARL ∨ARR = ACLL.
Finally, above these four lies the full variety of all regular antilattices and just below
the four flat cases lies the variety of trivial 1-point algebras. The resulting lattice of all
subvarieties of regular antilattices is, of course, isomorphic to the lattice of all subsets of
any 4-element set, which brings us back to Corollary 3.7.
RA
ACRR A
C
RL A
C
LR A
C
LL
SA
AL# A#R A#L AR#
S∗A
ALL ALR ARL ARR
1
Figure 1: The Hasse diagram of the Boolean lattice of all positive subvarieties of antilat-
tices.
The Hasse diagram of this Boolean lattice is explained in Table 1.
Equipped with all necessary tools we may now perform enumeration of the pseudo-
variety of finite regular antilattices and their sub-pseudo-varieties; see Table 2. These se-
quences can be found in OEIS [11].
K. Cvetko-Vah et al.: Regular antilattices 9
Table 1: The 16 positive subvarieties of regular antilattices.
Symbol Subvariety ρ(pe)
RA regular antilattices
(
e+3
3
)
ACRR complement of ARR
(
e+2
2
)
ACRL complement of ARL
(
e+2
2
)
ACLR complement of ALR
(
e+2
2
)
ACLL complement of ALL
(
e+2
2
)
SA skew antilattices
(
e+1
1
)
= e+ 1
AL# L∗ semi-flat
(
e+1
1
)
= e+ 1
A#R
∗R semi-flat
(
e+1
1
)
= e+ 1
A#L
∗L semi-flat
(
e+1
1
)
= e+ 1
AR# R∗ semi-flat
(
e+1
1
)
= e+ 1
S∗A skew∗ antilattices
(
e+1
1
)
= e+ 1
ALL LL-flat
(
e+0
0
)
= 1
ALR LR-flat
(
e+0
0
)
= 1
ARL RL-flat
(
e+0
0
)
= 1
ARR RR-flat
(
e+0
0
)
= 1
1 trivial antilattice 1 if e = 0
Table 2: Enumeration of small regular antilattices and their subvarieties.
ACRR,A
C
RL, SA,AL#,A#R, ALL,ALR,
n RA ACLR,A
C
LL A#L,AR#,S
∗A ARL,ARR 1
OEIS A007426 A007425 A000005 A000012
1 1 1 1 1 1
2 4 3 2 1 0
3 4 3 2 1 0
4 10 6 3 1 0
5 4 3 2 1 0
6 16 9 4 1 0
7 4 3 2 1 0
8 20 10 4 1 0
9 10 6 3 1 0
10 16 9 4 1 0
11 4 3 2 1 0
12 40 18 6 1 0
13 4 3 2 1 0
14 16 9 4 1 0
15 16 9 4 1 0
16 35 15 5 1 0
10 Art Discrete Appl. Math. 2 (2019) #P2.06
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