Page 197 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 197
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 = b0, b1, b2, … , 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
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 = b0, b1, b2, … , 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
