On Strongly Symmetric Skew Lattices
Karin Cvetko-Vah
Abstract. Skew lattices are a non-commutative generalization of lattices. In the
past 20 years several varieties of skew lattices have been introduced. In the present
paper we study the variety of strongly symmetric skew lattices.
1. Introduction
Skew lattices are the most successful non-commutative generalization of
lattices to date. The study of non-commutative variations of lattices originates
in Jordan’s 1949 paper [7]. A comparative study of different notions of noncommuative lattices was carried out by Laslo and Leech in [8]. The current
study of skew lattices began with the 1989 paper of Leech [9], where the
fundumental structural theorems were proved, and has been very lively in the
past 20 years.
A skew lattice S is an algebra (S; ∧, ∨) of type (2, 2) in which both operations
∧ and ∨ are associative and they satisfy the absorption laws x ∧ (x ∨ y) = x =
(y∨x)∧x and their duals. Both ∧ and ∨ are idempotent by the usual argument:
x∧x = x∧(x∨(x∧x)) = x and its dual. As a result we have the basic dualities
of skew lattices: x ∧ y = y iff x ∨ y = x and x ∧ y = x iff x ∨ y = y.
Like other associative structures, many instances of commutations occur
under either operation. In general, however, elements commuting under ∨ need
not commute under ∧ and vice-versa. A skew lattice such that x ∧ y = y ∧ x
if and only if x ∨ y = y ∨ x for all x, y is called symmetric. Symmetric skew
lattices form a variety that is characterized by the following identities given
by Spinks. (See [15] Theorem SSL-6.)
x ∨ y ∨ (x ∧ y) = (y ∧ x) ∨ y ∨ x
and
x ∧ y ∧ (x ∨ y) = (y ∨ x) ∧ y ∧ x.
The importance of symmetry was clear in the early study of skew lattices.
Not only was it a key requirement for some central results, but the two most
significant classes of examples, skew lattices of idempotents in rings (see, e.g.,
[9]) and skew Boolean algebras (see [1], [2] and [10]) consisted of symmetric
skew lattices. While skew lattices in rings provided accessible examples of skew
lattices, skew Boolean algebras introduced connections with areas of interest in
2000 Mathematics Subject Classification: Primary: 06A11; Secondary: 06F05.
Key words and phrases: skew lattice, symmetric, strongly symmetric, cancellative,
variety.
2
K. Cvetko-Vah
Algebra univers.
general algebra. From the perspective of general algebra, a prime motivation
for studying skew Boolean algebras is their structural role in studying discriminator varieties and their generalizations. For example, results in [1] imply that
every algebra A in a pointed discriminator variety is term equivalent to a skew
Boolean intersection algebra hA; ∧, ∨, \, ∩, 0i whose congruences coincide with
those of A. (A skew Boolean intersection algebra is a skew lattice with zero
hA; ∧, ∨, 0i structurally enriched with a relative complementation operation \
and a semilattice meet ∩ given by the natural partial order.)
Symmetry clearly affects the type of subalgebras on two generators that a
skew lattice can have. It happens that infinitely many nonisomorphic, symmetric skew lattices on two generators exist. Indeed there are infinitely many
infinite such cases. (See [11] Theorems 4.10 and 4.12.) This contrasts sharply
with the situation for skew lattices in rings where there are three cases of order
2, four cases of order 4, three cases of order 8 and finally one case of order
16. The same thing holds for skew lattices on two generators occurring within
skew Boolean algebras, since the latter can be embedded in rings. (See [3].)
A skew lattice is called strongly symmetric if it satisfies the identities
x ∧ (y ∨ x) = (x ∧ y) ∨ x and x ∨ (y ∧ x) = (x ∨ y) ∧ x. The variety of strongly
symmetric skew lattices was introduced by Spinks in a 1998 unpublished monograph [15], who initially termed them “quasi-absorptive” skew lattices because
of their connection to certain skewed absorption identities holding for left and
right-handed skew lattices. He also provided a 65-step automated proof via
OTTER 3.0.4 showing that all such skew lattices are symmetric.
Skew lattices in rings and skew Boolean algebras are strongly symmetric.
In fact both types of algebras are distributive in that they satisfy the identities
x∧(y∨z)∧x = (x∧y∧x)∨(x∧z∧x) and x∨(y∧z)∨x = (x∨y∨x)∧(x∨z∨x). In
Corollary 4.4 we show that distributive, symmetric skew lattices are strongly
symmetric. What is more, Corollary 5.6 states that a skew lattice S is strongly
symmetric if and only if all of its 2-generator subalgebras are isomorphic to
one of the 11 cases mentioned above. We also prove a Dedekind-Birkhoff
type result, Corollary 5.9, characterizing strongly symmetric skew lattices by
certain forbidden subalgebras.
A reader that is unfamiliar with the subject is referred to [12] for a comprehensive survey of the theory of skew lattices.
2. Preliminaries
2.1. Green’s relations. On a skew lattice S the Green’s equivalence relations
are defined by
aRb ⇔ (a ∧ b = b & b ∧ a = a) ⇔ (a ∨ b = a & b ∨ a = b),
aLb ⇔ (a ∧ b = a & b ∧ a = b) ⇔ (a ∨ b = b & b ∨ a = a),
aDb ⇔ (a ∧ b ∧ a = a & b ∧ a ∧ b = b) ⇔ (a ∨ b ∨ a = a & b ∨ a ∨ b = b).
Vol. 00, XX
On Strongly Symmetric Skew Lattices
3
An R-, L- and D-class containing a ∈ S shall be denoted by Ra , La and Da ,
respectively.
A skew lattice S is rectangular if x ∧ y ∧ x = x, or dually y ∨ x ∨ y = y,
holds on S. A skew lattice S is rectangular if and only if it consists of a single
D-class.
2.2. Leech’s decomposition theorems. Leech’s first decomposition theorem for skew lattices [9] states that on any skew lattice S the Green’s relation
D is a congruence, each D-class is a rectangular band and S/D is the maximal
lattice image of S. (See [9] for details.) A skew lattice is called a skew chain
if its maximal lattice image is linearly ordered.
On a skew lattice S the natural preorder is defined by
a b ⇔ a ∧ b ∧ a = a ⇔ b ∨ a ∨ b = b.
Note that the natural preorder on S is also derived from the partial order in
the lattice S/D.
The natural partial order can be defined on S by
x ≤ y ⇔ x ∧ y = y ∧ x = x ⇔ x ∨ y = y ∨ x = y..
The partial order plays an important role in S, providing many details
missing from the broad outline of S given by the First Decomposition Theorem.
Consider incomparable D-classes A and B, their join class J and their meet
class M in the D-class lattice, S/D.
J A
}} AAA
}
AA
}
AA
}}
}}
B
AA
AA
}}
AA
}
AA
}}
A }}}
M
Theorem 2.1. Given the situation above with a ∈ A, b ∈ B, j ∈ J and
m ∈ M:
i) If m ≤both a, b, then a ∧ b = m = b ∧ a, and conversely.
ii) If both a, b ≤ j, then a ∨ b = j = b ∨ a, and conversely.
Proof. (i) Since a ∧ b ∈ M , a ∧ b = (a ∧ b) ∧ m ∧ (a ∧ b) = a ∧ (b ∧ m ∧ a) ∧ b = m
since m ≤ both a and b. Conversely, given a∧b = m = b∧a, a∧m = a∧a∧b =
a ∧ b = m and likewise m ∧ a = m. Thus m ≤ a and similarly m ≤ b. (ii) is
shown in similar fashion.
A skew lattice is right-handed if it satisfies the identities, x ∧ y ∧ x = y ∧ x
and x ∨ y ∨ x = x ∨ y. Hence x ∧ y = y and x ∨ y = x hold on each D-class.
Left-handed skew lattices are defined by the dual identities.
4
K. Cvetko-Vah
Algebra univers.
Leech’s Second Decomposition Theorem for skew lattices [9] states that on
every skew lattice S the Green’s relations R and L are congruences and S
is isomorphic to the fibre product of a left-handed and a right-handed skew
lattice over a common maximal lattice image, that is to S/R ×S/D S/L.
2.3. The coset structure. A skew lattice consisting of only two D-classes
is called primitive. The structure of primitive skew lattices was thoroughly
studied in [11]. Let P be a primitive skew lattice with equivalence classes A
and B and assume A > B in P/D. (See [9] Theorem 1.15.)
For b ∈ B the set A ∧ b ∧ A = {a ∧ b ∧ a0 ; a, a0 ∈ A} is a coset of A in B.
Dually, a coset of B in A is any subset of the form B ∨ a ∨ B for some a ∈ A.
All cosets of A in B and all cosets of B in A have equal power [11]. Therefore
if P is finite then the power of each coset divides the powers |A| and |B|.
The class B is partitioned by the cosets of A, and the class A is partitioned
by the cosets of B. Given a ∈ A, in each coset Bj of A in B there is exactly
one element b ∈ B such that b < a. Dually, given b ∈ B, in each coset Ai of
B in A there is exactly one element a ∈ A such that b < a. Given cosets Ai
in A and Bj in B there is a natural bijection of cosets φji : Ai → Bj , where
φji (x) = y iff x ≥ y, i.e. x ∧ y = y ∧ x = y. Moreover, both operations ∧ and
∨ are determined by the coset bijections.
In the right-handed case, the description of cosets can be simplified:
A ∧ b ∧ A = b ∧ A and B ∨ a ∨ B = B ∨ a.
Indeed, for instance a ∧ b ∧ a0 = (a ∧ b) ∧ (b ∧ a0 ) = b ∧ a0 .
2.4. Skew lattice configurations. A skew lattice configuration is a (potential) subset of a skew lattice with all D-relationships indicated (usually by –)
and partial ordering relationships indicated (usually by \, |, / or similar symbols). Put otherwise, it is a partial Hasse diagram. However, unlike in the case
of lattices, the skew lattice operations are not uniquely defined by the configuration. The configuration therefore possesses some but not all the information
about a skew lattice. For example, given the configuration
1 N=NN
== NNN
== NNN
== NNN
NN
c
a>
b
p
>>
ppp p
>>
ppp
>>
> ppppp
p
0
(1)
we have two possible ways how to define a skew lattice structure on
S = {0, a, b, c, 1} in order to correspond to the configuration (1). Namely,
we can define a right-handed structure by setting b ∧ c = c, c ∧ b = b, b ∨ c = b
Vol. 00, XX
On Strongly Symmetric Skew Lattices
5
and c ∨ b = c, or a left-handed structure by setting b ∧ c = b, c ∧ b = c, b ∨ c = c
and c ∨ b = b.
If we decide for the right-handed case then the operations ∧ and ∨ are given
by the Cayley tables below:
∧
0
a
b
c
1
0 a b
0 0 0
0 a 0
0 0 b
0 0 b
0 a b
c
0
0
c
c
c
1
0
a
b
c
1
∨
0
a
b
c
1
0 a b c 1
0 a b c 1
a a 1 1 1
b 1 b b 1
c 1 c c 1
1 1 1 1 1
3. Basic properties of strongly symmetric skew lattices
A skew lattice is called left strongly symmetric if it satisfies the identity
(x ∧ y) ∨ x = x ∧ (y ∨ x)
(2)
and is called right strongly symmetric if it satisfies the identity
(x ∨ y) ∧ x = x ∨ (y ∧ x).
(3)
A skew lattice is strongly symmetric if it is both left and right strongly symmetric.
The following lemma is an easy observation.
Lemma 3.1. Any right-handed skew lattice is right strongly symmetric. Dually, any left-handed skew lattice is left strongly symmetric.
If a skew lattice S satisfies an identity, then S/R and S/L satisfy the same
identity. Conversely, since S is embedded in S/R × S/L, any identity satisfied
by both S/R and S/L is also satisfied by S, cf. [5]. In particular we have:
Lemma 3.2. A skew lattice S is strongly symmetric if and only if its left
factor S/R and its right factor S/L are both strongly symmetric.
Given any skew lattice S the left factor S/R is a left-handed skew lattice
and the right factor S/L is a right-handed skew lattice.
Proposition 3.3. A skew lattice S is strongly symmetric if and only if its left
factor S/R is right strongly symmetric and its right factor S/L is left strongly
symmetric.
Next, we prove that the two strongly symmetric identities can be joined
together into one, which simplifies to one of the defining identities (2), (3)
either in the left-handed case or in the right-handed case.
Theorem 3.4. A skew lattice S is strongly symmetric if and only if it satisfies
the identity
(y ∨ x ∨ y) ∧ x ∧ (y ∨ x ∨ y) = (y ∧ x ∧ y) ∨ x ∨ (y ∧ x ∧ y).
(4)
6
Algebra univers.
K. Cvetko-Vah
Proof. Observe that (4) simplifies to (2) in the right-handed case and to (3) in
the left-handed case. S satisfies (4) if and only if S/R and S/L both satisfy (4),
which is further equivalent to S/R satisfying (2) and S/L satisfying (3). A skew lattice S is cancellative if x ∧ y = x ∧ z together with x ∨ y = x ∨ z
imply y = z, and x ∧ y = z ∧ y together with x ∨ y = z ∨ y imply x = z.
Cancellation is equivalent to distributivity in the commutative case.
The Dedekind-Birkhoff Theorem (also known as the “M3 –N5 Theorem”)
states that a lattice is distributive if and only if it contains no sublattice
isomorphic to either of the lattices M3 , N5 below:
M3
1?
 ???

??

?

•?
•
•
??

??

?? 

0
N5
•/
//
//
//
//
//
//
/
1?
??
??
??
•
0




•
In [6] analogous results were proved for symmetric and for cancellative skew
lattices, ie. the two varieties were characterized in terms of “forbidden” subalgebras.
Theorem 3.5 ([6]). A skew lattice is symmetric if and only if it contains no
L,0
R,1
R,0
copy of NSR,0
or NSL,1
and NSL,0
are given
7 , NS7 , NS7
7 , where NS7
7
by the configuration:
j
j1 TTT
TTTTjjjjjj 2 OOOOO
ooo
o
j
T
OOO
o
j TTTT
o
OOO
jjjj
TTTT
ooo
j
o
j
j
OOO
o
TTTT
o
jj
j
O
j
j
T
ooo
j
a1 UUU a2 OO
b1 jjj b2
o
UUUU
o
OOO
oo jjjj
UUUU OO
oojojjjjj
UUUUOOO
o
o
j
o j
UUUOUOO
UUOUO ojojojojojjj
0
Vol. 00, XX
On Strongly Symmetric Skew Lattices
7
are given by the configuration:
and NSL,1
and NSR,1
7
7
iinin 1 PUPUPUPUPUUU
iinininn
PPPUUUUU
i
i
i
n
PPP UUUU
iiinnnn
i
i
i
PPP UUUU
ii nnn
i
i
UUU
P
i
i
n
i
a1 PP
a U
b1
b2
i
i
i
PPP 2 UUUUUU
nnn
iii
n
i
UUUU
PPP
n
i
i
UUUU iiiii
PPP
nnn
PPP
nnn
iUiUiUUUU
i
n
i
n
P
i
n
U
i
m1 i
m2
and
are right-handed, while NSL,0
and NSR,1
Moreover, algebras NSR,0
7
7
7
L,1
NS7 are their left-handed duals. Cayley tables for NS7R,0 are given by:
∨ 0
0
0
am am
bm bm
jm jm
an
an
am
jm
jm
bn
bn
jm
bm
jm
jn
jn
jm
jm
jm
and
∧
0
am
bm
jm
0
0
0
0
0
an
0
an
0
an
bn
0
0
bn
bn
jn
0
an
bn
jn
Theorem 3.6 ([6]). A skew lattice S is cancellative if and only if it contains
L,0
R,1
L,1
R
L
no copy of M3 , N5 , NSR,0
7 , NS7 , NS7 , NS7 , NC5 or NC5 , where
L
R
NC5 [NC5 ] is the right- [left-] handed skew lattice given by the configuration
(1).
In the remainder of this section we give direct proofs showing that the
variety of strongly symmetric skew lattices is included in the variety of symmetric skew lattices, and includes the variety of cancellative skew lattices. The
first result was already proved by Spinks in [15] while the second result was
observed by the authors of [6].
Theorem 3.7. Any strongly symmetric S skew lattice is symmetric.
Proof. If S is not symmetric then by Theorem 3.5 S contains a subalgebra
L,0
R,1
isomorphic to one of the algebras NSR,0
and NSL,1
7 , NS7 , NS7
7 . Assume
R,0
that S contains a copy of NS7 as a subalgebra. Then (a1 ∧ b2 ) ∨ a1 =
0 ∨ a1 = a1 while a1 ∧ (b2 ∨ a1 ) = a1 ∧ j2 = a2 . Therefore NS7R,0 is not strongly
symmetric and hence neither is S.
Theorem 3.8. Any cancellative skew lattice S is strongly symmetric.
Proof. If S is not strongly symmetric, then by Leech’s Second Decomposition
Theorem at least one of the factors S/R and S/L is not strongly symmetric.
Assume that the right-handed skew lattice S/L is not strongly symmetric.
8
Algebra univers.
K. Cvetko-Vah
Then
y ∨ x VVV
LLL VVVV
{
{
LLL VVVVV
{
VVVV
{
LLL
VVVV
{{
{
L
VV
{
{
yC
x ∧ (y ∨ x)
(x ∧ y) ∨ x
CC
r
hhhh
r
h
CC
h
r
h
r
hh
CC
rrr hhhhhh
CC
rhrhrhhhhh
x∧y
is a copy of NCR
5 in S and S is not cancellative by Theorem 3.6.
4. The minimal cardinality of symmetric skew lattices that are not
strongly symmetric
As described in the previous section, results proved in [6] allow us to characterize the variety of symmetric skew lattices as well as the variety of cancellative skew lattices by a finite number of finite (five- or seven-element) forbidden
characterizing sub-algebras. In this paper we shall provide a characterization
of strongly symmetric skew lattices by minimal forbidden sub-algebras. We
shall see that minimal symmetric non-strongly symmetric skew lattices can be
infinite, and characterize them.
A partial algebra (S; ∧, ∨) is a non-empty set S where ∧ and ∨ are partial
operations on S.
Proposition 4.1. A right-handed symmetric skew lattice S is strongly symmetric if and only if S contains no partial sub-algebra isomorphic to the partial
algebra PR
8 given by the Cayley tables:
∧
0
1
2
3
4
5
6
7
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
0
0
2
2
2
0
2
2
3
1
1
3
3
3
1
3
3
4
1
1
4
4
4
1
4
4
5
5
1
1
1
1
5
5
5
6
0
0
2
7
1
1
4
3
2 4
5
6 7
6 7
∨
0
1
2
3
4
5
6
7
0
0
1
2
3
4
5
6
7
1
0
1
2
3
4
5
6
7
2
2
3
2
3
4
7
6
7
3
2
3
2
3
4
7
6
7
4
5
4
2
3
4
7
6
7
5
6
7
7
5
6
7
6 7
6 6
7 7
6 6
7 7
7 7
7 7
6 6
7 7
Setting a = 2 and b = 5, one obtains a∧b = 1, b∧a = 0, a∨b = 6, b∨a = 7,
α := (a ∧ b) ∨ a = 3 and β := a ∧ (b ∨ a) = 4, and the configuration for PR
8 is
Vol. 00, XX
On Strongly Symmetric Skew Lattices
9
given by:
b ∨ aB
a∨b
BB
ww
{{
BB
w
{
w
{
BB
w
w
{{
BB
w
{
{
ww
αG
aC
β
b
GG
CC
||
GG
|
CC
GG
||
CC
GG
||
C
||
a∧b
b∧a
(5)
Proof. Assume that S is a skew lattice satisfying the assumptions of the proposition but is not strongly symmetric. Hence there exist a and b such that
(a ∧ b) ∨ a 6= a ∧ (b ∨ a). Let α = (a ∧ b) ∨ a and β = a ∧ (b ∨ a). Absorption
and α 6= β imply a ∧ b 6= b ∧ a and a ∨ b 6= b ∨ a. If a and b lie in the same
D-class or in comparable D-classes, then they generate a skew chain. But all
skew chains are cancellative and hence strongly symmetric. (See [6] or [4].)
So the eight elements of the above partial skew lattice are indeed mutually
distinct and all horizontal edges in the configuration are clear. Clearly b ∧ a <
a < a ∨ b, a ∧ b < b < b ∨ a, a ∧ b < α and β < b ∨ a. To see α < b ∨ a
first observe that b ∨ a = (a ∧ b) ∨ a ∨ b by symmetry. Therefore α ∧ (b ∨ a) =
((a ∧ b) ∨ a) ∧ (b ∨ a) = ((a ∧ b) ∨ a) ∧ ((a ∧ b) ∨ a ∨ b) = (a ∧ b) ∨ a by absorption.
That a ∧ b < β is proved in a dual fashion.
This completes part of the Cayley tables for ∧ and ∨. To complete the
tables we use right-handedness and argumentations like 5 ∧ 3 = 1 ∧ 5 ∧ 3 =
(a ∧ b) ∧ ((a ∧ b) ∨ a) = a ∧ b = 1.
Corollary 4.2. If S is a symmetric skew lattice that is not strongly symmetric
then |S| ≥ 10.
Proof. If S is not strongly symmetric then at least one of S/R, S/L is not
strongly symmetric. If S/L is not strongly symmetric then S/L contains a
R
partial sub-algebra T isomorphic to PR
8 . Observe that P8 is not a skew lattice.
Indeed, all cosets in the bottom class are of cardinality 2, which implies that
the cardinalities of D-classes Da and Db must be even. Hence we need at least
one more element in each of the two classes. Therefore |S| ≥ |S/L| ≥ |T | ≥ 10.
The left-handed case follows by the dual of Proposition 4.1.
Theorem 4.3. There exists a symmetric, non-strongly symmetric skew lattice
of cardinality 10.
10
Algebra univers.
K. Cvetko-Vah
Proof. This is proved by Mace4 (see McCune [13]), where the following example of a symmetric, non-strongly symmetric skew lattice S of minimal cardinality was found:
∧
0
1
2
3
4
5
6
7
8
9
0
0
4
4
0
4
0
0
4
0
0
1
2
1
2
1
2
2
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
3
0
7
4
3
4
8
3
7
8
0
4
4
4
4
4
4
4
4
4
4
4
5
5
2
2
5
2
5
5
2
5
5
6
9
1
2
6
2
5
6
1
5
9
7
4
7
4
7
4
4
7
7
4
4
8
8
4
4
8
4
8
8
4
8
8
9
9
2
2
9
2
9
9
2
9
9
∨
0
1
2
3
4
5
6
7
8
9
0
0
6
5
3
0
5
6
3
8
9
1
3
1
1
3
7
6
6
7
3
6
2
0
1
2
3
4
5
6
7
8
9
3
3
6
6
3
3
6
6
3
3
6
4
0
1
2
3
4
5
6
7
8
9
5
0
6
5
3
0
5
6
3
8
9
6
3
6
6
3
3
6
6
3
3
6
7
3
1
1
3
7
6
6
7
3
6
8
0
6
9
3
8
5
6
3
8
9
9
0
6
9
3
8
5
6
3
8
9
Taking a = 0 and b = 1 one obtains (a ∧ b) ∨ a = 6 and a ∧ (b ∨ a) = 8. The
corresponding configuration is given below.
p 6 TNNTTT p 3 TWTWTWTWW
ppp pppNpNpNTNTNTTTT TTWTWTWTWTWWWW
p
p
TT WW
NNN TTTT
pp
pp
NNN TTTT TTTTTWTWWWWWWW
ppp ppppp
p
TTT
TTT
WWWW
N
p
p
p
1 NNN 7 NNN
5 jjj 9 jjj 0 ggggg 8
p
p
NNN
N
p
jj
jj gg
NNN NNNNN
ppp jjjjj jjjjgjggggg
NNN
NNNpppjpjpjjjj jjjgjgjgjgggg
j gg
NN ppjpjjNjN
j
jjgjgjgg
4g
2
Propostition 4.1 yields several interesting implications.
Corollary 4.4. A distributive, symmetric skew lattice is strongly symmetric.
Proof. Let S be a right-handed, symmetric skew lattice. If S is not strongly
R
symmetric, then S contains a copy of PR
8 . Using the notation of P8 we have
[(a∧b)∨a]∧(b∨a) = α∧(b∨a) = α 6= β = (a∧b)∨β = [(a∧b)∧(b∨a)]∨[a∧(b∨a)],
since b ∨ a > both α, β > a ∧ b, showing that S is not distributive. Thus
right-handed distributive, symmetric skew lattices are strongly symmetric by
contraposition. The left-handed counterpart holds by a dual argument and
the corollary follows from Lemma 3.2.
Corollary 4.5. Skew lattices in rings and skew Boolean algebras are strongly
symmetric.
In general, distributive skew lattices need not be symmetric, much less
strongly symmetric.
Vol. 00, XX
On Strongly Symmetric Skew Lattices
11
Recall that a skew lattice S is categorical if given a > b > c and aDa0 , cDc0
in S such that c0 = a0 ∧ c ∧ a0 and a0 = c0 ∨ a ∨ c0 , then a0 ∧ b ∧ a0 = c0 ∨ b ∨ c0 .
a0
c0 ∨ a ∨ c0
p
p
ppp
p
p
p
ppp ?
0
0
a ∧b∧a
c0 ∨ b ∨ c0
NNN
NNN
NNN
NN
c0
a0 ∨ c ∨ a0
a
b
c
Categorical skew lattices need not be symmetric. Given symmetry, however,
we have:
Corollary 4.6. If S is a symmetric, categorical skew lattice, then S is strongly
symmetric.
Proof. Assume that S is right-handed. (The left-handed case is proved dually
while the general case follows by Leech’s Second Decomposition Theorem.)
By the assumption, S is symmetric. If S is not strongly symmetric then it
contains a partial sub-algebra T isomorphic to PR
8 . Then a ∧ b < α < b ∨ a
and a ∧ b < β < b ∨ a. Also b ∧ a < a < a ∨ b. If S is categorical then
a ∧ (b ∨ a) = (a ∧ b) ∨ a, which is in contradiction with α 6= β.
A natural question arises: Does strongly symmetric imply categorical? This
is answered in the negative by the following example, again found by Mace4.
Example 4.7. The skew lattice S given by the Cayley tables below is strongly
symmetric but it is not categorical.
∧
0
1
2
3
4
5
6
7
0
0
1
2
3
4
5
6
7
1
3
1
2
3
1
5
6
7
2
2
5
2
2
5
5
2
5
3
3
1
2
3
1
5
6
7
4
0
1
2
3
4
5
6
7
5
2
5
2
2
5
5
2
5
6
6
1
2
3
7
5
6
7
7
6
1
2
3
7
5
6
7
∨
0
1
2
3
4
5
6
7
0
0
0
0
0
0
0
0
0
1
4
1
1
1
4
1
1
1
The corresponding configuration is:
0
3>
6
>>
>>
>>
2
4>
>>
>>
>>
1
7
5
2
0
6
2
3
0
2
6
3
3
0
3
3
3
0
3
3
3
4
4
4
4
4
4
4
4
4
5
4
1
5
7
4
5
1
7
6
0
6
6
6
0
6
6
6
7
4
7
7
7
4
7
7
7
12
K. Cvetko-Vah
Algebra univers.
Since S has only 8 elements, it follows from Corollary 4.2 that S is strongly
symmetric. In fact, S is evan cancellative since it is a skew chain. To see that
S is not categorical, consider a = 0, b = 6 and c = 2. Taking a0 = 4 and c0 = 5
one obtains a0 ∧ c ∧ a0 = c0 , c0 ∨ a ∨ c0 = a0 but a0 ∧ b ∧ a0 = 7 6= 1 = c0 ∨ b ∨ c0 .
5. Minimal symmetric skew lattices that are not strongly symmetric
Description 5.1. In Leech [11] the following symmetric, right-handed skew
lattices were constructed. Let m and n, 1 ≤ m, n ≤ ∞ be given. Define a
skew lattice Sm,n with the D-class structure given by the Hasse diagram
J A
}} AAA
}
AA
}
AA
}}
}}
B
AA
AA
}}
AA
}
AA
}}
A }}}
M
as follows. Let J = {0+ , 1+ }, M = {0− , 1− }, A = Z2m if m is finite and A = Z
otherwise; and B = Z2n if n is finite and B = Z otherwise. We write elements
of B in prime form (e.g., 50 ) in order to distinguish them from elements of A.
The natural ordering of Sm,n is given by
0+ ≥ 2i, (2j)0 ≥ 0− and 1+ ≥ 2i + 1, (2j + 1)0 ≥ 1− .
Note that the skew lattice given in the proof of Theorem 4.3 is exactly S1,2 .
Both J and M form a single coset with respect to any of the three other
classes. J-cosets in A and B have the form {2i − 1, 2i}, while M -cosets in A
and B have the form {2j, 2j + 1}. The skew lattice Sm,n is generated by any
two elements i ∈ A, j 0 ∈ B of different parity. See [11] for details.
Lemma 5.2. The following hold in Sm,n , where e, e0 [o, o0 ] are even [odd]
integers:
i)
ii)
iii)
iv)
v)
e ∧ (o0 ∨ e) = e ∧ 1+ = e − 1 and (e ∧ o0 ) ∨ e = 1− ∨ e = e + 1.
o0 ∧ (e ∨ o0 ) = o0 ∧ 0+ = o0 + 1 and (o0 ∧ e) ∨ o0 = 0− ∨ o0 = o0 − 1.
e0 ∧ (o ∨ e0 ) = e0 ∧ 1+ = e0 − 1 and (e0 ∧ o) ∨ e0 = 1− ∨ e0 = e0 + 1.
o ∧ (e0 ∨ o) = o ∧ 0+ = o + 1 and (o ∧ e0 ) ∨ o = 0− ∨ o = o − 1.
x ∨ y = y ∨ x iff x ∧ y = y ∧ x, with both occuring when x = y or when x
and y lie in distinct D-classes but have the same parity.
Theorem 5.3. Sm,n is a symmetric skew lattice generated from any pair
a ∈ A and b ∈ B of opposite parity. It is strongly symmetric if and only if
m = n = 1.
Vol. 00, XX
On Strongly Symmetric Skew Lattices
13
Proof. That S is symmetric is clear from (v) above. That it is generated
as stated follows from (i) through (iv) above, which also show that strong
symmetry requires x+1 = x−1 to hold in A and B, which in turn is equivalent
to both being copies of Z2 .
Consider next the following list of skew lattices:
R2 = ({a, b}; ∨, ∧) with operations defined by x ∧ y = y and x ∨ y = x.
L2 = ({c, d}; ∨, ∧) with operations defined by x ∧ y = x and x ∨ y = y.
D4 = R2 × L2 .
2 is the lattice on {1, 0} with operations determined by 1 > 0.
Tm,n is the left-handed dual of Sm,n with operations x ∨T y = y ∨S x and
x ∧T y = y ∧S x.
Theorem 5.4 (Leech [11]). The nontrivial symmetric skew lattices on two
generators, classified to within isomorphism, are as follows.
i) The strongly symmetric cases: R2 , L2 , D4 , 2, 2 × R2 , 2 × L2 , 2 × D4 , 2 ×
2, 2×2×R2 (∼
= S1,1 ), 2×2×L2 (∼
= T1,1 ) and 2×2×D4 (∼
= S1,1 ×2×2 T1,1 ).
ii) The non-strongly symmetric cases: all Sm,n and all Tm,n with m ≥ n
and m + n ≥ 3, plus all fibre products Sm,n ×2×2 Tp,q with m ≥ n and
m + n + p + q ≥ 5.
Proof. That these provide all the symmetric skew lattices on two distinct generators is Theorems 4.10 and 4.12 of [11]. Since R2 , L2 and 2 are easily seen
to be strongly symmetric, the rest of (i) follows. (ii) is due to the previous
theorem.
Corollary 5.5. The skew lattice 2 × 2 × D4 is a free strongly symmetric skew
lattice on two generators. In particular it is free on (1, 0, a, c) and (0, 1, b, d).
The (i)–(ii) division of Theorem 5.4 was in [11] a division of symmetric skew
lattices on two generators that were or were not categorical. One would be
also correct if “categorical” were replaced by “distributive” in this remark. In
addition, (i) given all nontrivial cancellative skew lattices on two generators. In
all cases, 2 × 2 × D4 is the free algebra. In any case, we may now present both
positive and negative characterizations of strongly symmetric skew lattices.
On the positive side, to within isomorphism we have the precise situation for
skew lattices on two generators in rings (see [14]) and by inference in skew
Boolean algebras (see [3]).
Corollary 5.6. A skew lattice is strongly symmetric if and only if all subalgebras on two distinct generators are isomorphic to one of the following:
R2 , L2 , D4 , 2, 2 × R2 , 2 × L2 , 2 × D4 , 2 × 2, 2 × 2 × R2 , 2 × 2 × L2 , 2 × 2 × D4 .
Corollary 5.7. A symmetric skew lattice is strongly symmetric if and only if
it contains no copies of Sm,n or Tm,n where m + n ≥ 3.
14
K. Cvetko-Vah
Algebra univers.
Proof. Since Sm,n ×2×2 Tp,q contains copies of both Sm,n and Tp,q , this result
follows from Theorem 5.4.
Corollary 5.8. The smallest symmetric skew lattices that are not strongly
symmetric are copies of the 10-element cases, S2,1 or T2,1 .
Corollary 5.9. A skew lattice is strongly symmetric if and only if it contains
L,0
no copies of Sm,n or Tm,n where m + n ≥ 3 nor copies of NSR,0
7 , NS7 ,
L,1
R,1
NS7 or NS7 .
We should mention also that to within isomorphis the two minimal noncategorical skew lattices arise as the skew chain J > A = Z4 > M in either S2,1
or T2,1 .
Acknowledgment
The author is grateful to Michael Kinyon and Jonathan Leech as well as
the anonymous referee for careful reading of the manuscript and valuable suggestions that led to significant improvement of the text.
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Karin Cvetko-Vah
Vol. 00, XX
On Strongly Symmetric Skew Lattices
Department of Mathematics// Faculty for Mathematics and Physics// University of
Ljubljana// Jadranska 19, 1000 Ljubljana, Slovenia
15