Page 191 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 191
V: Further Topics in Skew Lattices
Lemma 5.5.4. Identities (5.2.1) and (5.2.2) respectively imply
x ∧ [(y∧x∧y) ∨ (z∧x∧z)] ∧ x = (x∧y∧x) ∨ (x∧z∧x) (5.5.3)
and
x ∨ [(y∨x∨y) ∧ (z∨x∨z)] ∨ x = (x∨y∨x) ∧ (x∨z∨x). (5.5.4)
For left-handed skew lattices, these identities simplify to:
x ∧ [(y∧x) ∨ (z∧x)] = (x∧y) ∨ (x∧z) (5.5.3L).
and
[(x∨y) ∧ (x∨z)] ∨ x = (y∨x) ∧ (z∨x). (5.5.4L).
Dually, in the right-handed case they simplify to:
[(x∧y) ∨ (x∧z)] ∧ x = (y∧x) ∨ (z∧x) (5.5.3R).
and
x ∨ [(y∨x) ∧ (z∨x)] = (x∨y) ∧ (x∨z) . (5.5.4R).
Proof. Since x∧y∧x∧y∧x = x∧y∧x by regularity, (5.2.1) ⇒ (5.5.3) and also (5.2.2) ⇒ (5.5.4). £
Theorem 5.5.5. For all skew lattices, (5.5.3) and (5.5.4) are equivalent. A skew lattice
satisfies either and hence both if and only if it is linearly distributive.
Proof. For left-handed skew lattices, (5.5.3L) clearly implies (5.5.2L), while (5.5.3R) implies
(5.5.2R). Thus if a skew lattice S satisfies (5.5.3), so do both S/R and S/L, making them linearly
distributive, and hence S also by Theorem 5.5.1
Conversely, assume that S is linearly distributive. First, let S be left-handed also. Then
left-handedness gives the first, third and sixth equalities below.
x ∧ [(y∧x) ∨ (z∧x)] = x ∧ [(z∧x) ∨ (y∧x) ∨ (z∧x)]
= {x ∧ [(z∧x) ∨ (y∧x)]} ∨ [x ∧ (z∧x)]
= {x ∧ [(y∧x) ∨ (z∧x) ∨ (y∧x)]} ∨ [x ∧ (z∧x)]
= {x ∧ [(y∧x) ∨ (z∧x)]} ∨ [x ∧ (y∧x)] ∨ [x ∧ (z∧x)]
= [x ∧ (y∧x)] ∨ [x ∧ (z∧x)]
= (x ∧ y) ∨ (x∧z).
Linear distributivity implies the second and fourth equalities, since e.g., x ≻ (z∧x)∨(y∧x) ≻ (z∧x).
The fifth equality follows upon observing that x ∧ [(y∧x)∨(z∧x)] and (x∧y∧x) ∨ (x∧z∧x) are
L-related, since they are equal in any lattice, and in particular in S/D = S/L. Thus (5.5.3L) holds.
Similarly (5.5.3R) holds in the right-handed linearly distributive case. Again the embedding of S
into S/R × S/L guarantees that all linearly distributive skew lattices satisfy (5.5.3). Thus linear
distributivity is characterized by (5.5.3).
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Lemma 5.5.4. Identities (5.2.1) and (5.2.2) respectively imply
x ∧ [(y∧x∧y) ∨ (z∧x∧z)] ∧ x = (x∧y∧x) ∨ (x∧z∧x) (5.5.3)
and
x ∨ [(y∨x∨y) ∧ (z∨x∨z)] ∨ x = (x∨y∨x) ∧ (x∨z∨x). (5.5.4)
For left-handed skew lattices, these identities simplify to:
x ∧ [(y∧x) ∨ (z∧x)] = (x∧y) ∨ (x∧z) (5.5.3L).
and
[(x∨y) ∧ (x∨z)] ∨ x = (y∨x) ∧ (z∨x). (5.5.4L).
Dually, in the right-handed case they simplify to:
[(x∧y) ∨ (x∧z)] ∧ x = (y∧x) ∨ (z∧x) (5.5.3R).
and
x ∨ [(y∨x) ∧ (z∨x)] = (x∨y) ∧ (x∨z) . (5.5.4R).
Proof. Since x∧y∧x∧y∧x = x∧y∧x by regularity, (5.2.1) ⇒ (5.5.3) and also (5.2.2) ⇒ (5.5.4). £
Theorem 5.5.5. For all skew lattices, (5.5.3) and (5.5.4) are equivalent. A skew lattice
satisfies either and hence both if and only if it is linearly distributive.
Proof. For left-handed skew lattices, (5.5.3L) clearly implies (5.5.2L), while (5.5.3R) implies
(5.5.2R). Thus if a skew lattice S satisfies (5.5.3), so do both S/R and S/L, making them linearly
distributive, and hence S also by Theorem 5.5.1
Conversely, assume that S is linearly distributive. First, let S be left-handed also. Then
left-handedness gives the first, third and sixth equalities below.
x ∧ [(y∧x) ∨ (z∧x)] = x ∧ [(z∧x) ∨ (y∧x) ∨ (z∧x)]
= {x ∧ [(z∧x) ∨ (y∧x)]} ∨ [x ∧ (z∧x)]
= {x ∧ [(y∧x) ∨ (z∧x) ∨ (y∧x)]} ∨ [x ∧ (z∧x)]
= {x ∧ [(y∧x) ∨ (z∧x)]} ∨ [x ∧ (y∧x)] ∨ [x ∧ (z∧x)]
= [x ∧ (y∧x)] ∨ [x ∧ (z∧x)]
= (x ∧ y) ∨ (x∧z).
Linear distributivity implies the second and fourth equalities, since e.g., x ≻ (z∧x)∨(y∧x) ≻ (z∧x).
The fifth equality follows upon observing that x ∧ [(y∧x)∨(z∧x)] and (x∧y∧x) ∨ (x∧z∧x) are
L-related, since they are equal in any lattice, and in particular in S/D = S/L. Thus (5.5.3L) holds.
Similarly (5.5.3R) holds in the right-handed linearly distributive case. Again the embedding of S
into S/R × S/L guarantees that all linearly distributive skew lattices satisfy (5.5.3). Thus linear
distributivity is characterized by (5.5.3).
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