Page 193 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 193
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
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
