Page 192 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 192
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Since any totally pre-ordered context is symmetric, (5.2.1) and (5.2.2) are equivalent in
such contexts. Thus linear distributivity is also characterized by (5.5.4) by the dual argument. £
There is more to linear distributivity than just what occurs in totally preordered contexts.
Indeed (5.5.3) implies that for each a in a skew lattice S, the map x |→ a∧x∧a defines a homo-
morphic retraction of the principal ideal S∧a∧S onto the set a∧S∧a of all x ≤ a. The identity
states directly that this map preserves joins, with meets preserved due to regularity. Likewise,
(5.5.4) implies that x |→ a∨x∨a defines a homomorphic retraction of the principal filter S∨a∨S
onto the set a∨S∨a of all x ≥ a. Thus all three aspects of distributivity are equivalent for any
skew lattice. There is more. While skew diamonds need not be distributive, linearly distributive
skew diamonds are.
Corollary 5.5.6. Linearly distributive skew diamonds are distributive.
Proof. Given a linearly distributive skew diamond T with D-classes J > A, B > M, we check out
(5.2.1) on T. Given x ∈J, then y∧x∧y = y and z∧x∧z = z so that (5.5.3) reduces to (5.2.1). For x in
M, (5.2.1) immediately reduces to x = x ∨ x. So let x be in an intermediate D-class, say say A.
We consider several possible cases.
1) y D z. Here y ∨ z = z ∧ y so that regularity gives
x∧(y ∨ z)∧x = x∧(z ∧ y)∧x = (x∧z∧x) ∧ (x∧y∧x) = (x∧y∧x) ∨ (x∧z∧x).
2) y or z is in A or J, say y. Here y ∨ z ≻ x and our equation reduces to the absorption
identity x = x ∨ (x∧z∧x). Thus we may assume that y and z are in distinct D-classes, but other
than A and J.
3) So let, say y ∈ M and z ∈ B. Then (y ∨ z ∨ y) D z and Case (1) gives
x∧(y ∨ z)∧x = x∧[(y ∨ z ∨ y) ∨ z]∧x = (x∧(y ∨ z ∨ y)∧x) ∨ (x ∨ z ∧x).
The corollary will follow if we can show that x∧(y ∨ z ∨ y)∧x = x∧y∧x. Since x∧(y ∨ z ∨ y), y and
(y ∨ z ∨ y)∧x are in M,
x∧(y ∨ z ∨ y)∧x = x ∧ (y ∨ z ∨ y) ∧ (y ∨ z ∨ y) ∧ x
= [x∧(y ∨ z ∨ y)] ∧ y ∧ [(y ∨ z ∨ y)∧x] = x∧y∧x,
with the equalities due respectively to ∧ being idempotenct, uvw = uw holding in any rectangular
band and absorption. (5.2.2) is seen in dual manner. £
Since distributive skew lattice are both linearly distributive and quasi-distributive, a
natural question is: Does linear distributivity plus quasi-distributivity imply distributivity? While
true for skew diamonds, it is not true in general. Spinks’ examples (in Theorem 1.3.10) each
190
Since any totally pre-ordered context is symmetric, (5.2.1) and (5.2.2) are equivalent in
such contexts. Thus linear distributivity is also characterized by (5.5.4) by the dual argument. £
There is more to linear distributivity than just what occurs in totally preordered contexts.
Indeed (5.5.3) implies that for each a in a skew lattice S, the map x |→ a∧x∧a defines a homo-
morphic retraction of the principal ideal S∧a∧S onto the set a∧S∧a of all x ≤ a. The identity
states directly that this map preserves joins, with meets preserved due to regularity. Likewise,
(5.5.4) implies that x |→ a∨x∨a defines a homomorphic retraction of the principal filter S∨a∨S
onto the set a∨S∨a of all x ≥ a. Thus all three aspects of distributivity are equivalent for any
skew lattice. There is more. While skew diamonds need not be distributive, linearly distributive
skew diamonds are.
Corollary 5.5.6. Linearly distributive skew diamonds are distributive.
Proof. Given a linearly distributive skew diamond T with D-classes J > A, B > M, we check out
(5.2.1) on T. Given x ∈J, then y∧x∧y = y and z∧x∧z = z so that (5.5.3) reduces to (5.2.1). For x in
M, (5.2.1) immediately reduces to x = x ∨ x. So let x be in an intermediate D-class, say say A.
We consider several possible cases.
1) y D z. Here y ∨ z = z ∧ y so that regularity gives
x∧(y ∨ z)∧x = x∧(z ∧ y)∧x = (x∧z∧x) ∧ (x∧y∧x) = (x∧y∧x) ∨ (x∧z∧x).
2) y or z is in A or J, say y. Here y ∨ z ≻ x and our equation reduces to the absorption
identity x = x ∨ (x∧z∧x). Thus we may assume that y and z are in distinct D-classes, but other
than A and J.
3) So let, say y ∈ M and z ∈ B. Then (y ∨ z ∨ y) D z and Case (1) gives
x∧(y ∨ z)∧x = x∧[(y ∨ z ∨ y) ∨ z]∧x = (x∧(y ∨ z ∨ y)∧x) ∨ (x ∨ z ∧x).
The corollary will follow if we can show that x∧(y ∨ z ∨ y)∧x = x∧y∧x. Since x∧(y ∨ z ∨ y), y and
(y ∨ z ∨ y)∧x are in M,
x∧(y ∨ z ∨ y)∧x = x ∧ (y ∨ z ∨ y) ∧ (y ∨ z ∨ y) ∧ x
= [x∧(y ∨ z ∨ y)] ∧ y ∧ [(y ∨ z ∨ y)∧x] = x∧y∧x,
with the equalities due respectively to ∧ being idempotenct, uvw = uw holding in any rectangular
band and absorption. (5.2.2) is seen in dual manner. £
Since distributive skew lattice are both linearly distributive and quasi-distributive, a
natural question is: Does linear distributivity plus quasi-distributivity imply distributivity? While
true for skew diamonds, it is not true in general. Spinks’ examples (in Theorem 1.3.10) each
190
