Page 190 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 190
athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
We proceed with several lemmas, the first of which is evident.
Lemma 5.5.2. Left-handed skew lattices that satisfy (5.2.1) are characterized by:
x∧y∧x = x∧y, x∨y∨x = y∨x and x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). (5.5.1L)
Dually, right-handed skew lattices that satisfy (5.2.1) are characterized by:
x∧y∧x = y∧x, x∨y∨x = x∨y and (y ∨ z) ∧ x = (y ∧ x) ∨ (z ∧ x). (5.5.1R)
Lemma 5.5.3. In a left-handed totally preordered skew lattice, if a∧(b∨c) ≠ a∧b ∨ a∧c,
then a ≻ b ≻ c. Thus, for a left-handed skew lattice S, the following are equivalent:
a) S is linearly distributive.
b) a∧(b ∨ c) = (a ∧ b) ∨ (a ∧ c) for all a ≻ b ≻ c in S.
c) a∧(b ∨ c) = (a ∧ b) ∨ (a ∧ c) for all a ≻ b ≻ c in S.
Left-handed linearly distributive skew lattices are thus characterized by:
x∧[(y∧x) ∨ (z∧y∧x)] = (x∧y) ∨ (x∧z∧y). (5.5.2L)
Dually, right-handed linearly distributive skew lattices are characterized by:
[(x∧y∧z) ∨ (x∧y)]∧x = (y∧z∧x) ∨ (y∧x). (5.5.2R)
Proof. If say b ≻ a, then a∧(b∨c) = a and (a∧b) ∨ (a∧c) = a ∨ (a∧c) = a. If c ≻ a, then
a∧(b∨c) = a again, and (a∧b) ∨ (a∧c) = (a∧b) ∨ a = (a∧b∧a) ∨ a = a. Thus inequality only
occurs when a ≻ b, c. But even here, a ≻ c ≻ b gives us a∧(b∨c) = a∧c and (a∧c) ≻ (a∧b) so that
(a∧b) ∨ (a∧c) = a∧c also. Thus, to completely avoid a∧(b∨c) = a∧b ∨ a∧c we are only left with
a ≻ b ≻ c. £
In particular primitive skew lattices are distributive. It turns out that linear distributivity
is characterized succinctly by either of the dual pair of identities in the following lemma.
188
We proceed with several lemmas, the first of which is evident.
Lemma 5.5.2. Left-handed skew lattices that satisfy (5.2.1) are characterized by:
x∧y∧x = x∧y, x∨y∨x = y∨x and x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). (5.5.1L)
Dually, right-handed skew lattices that satisfy (5.2.1) are characterized by:
x∧y∧x = y∧x, x∨y∨x = x∨y and (y ∨ z) ∧ x = (y ∧ x) ∨ (z ∧ x). (5.5.1R)
Lemma 5.5.3. In a left-handed totally preordered skew lattice, if a∧(b∨c) ≠ a∧b ∨ a∧c,
then a ≻ b ≻ c. Thus, for a left-handed skew lattice S, the following are equivalent:
a) S is linearly distributive.
b) a∧(b ∨ c) = (a ∧ b) ∨ (a ∧ c) for all a ≻ b ≻ c in S.
c) a∧(b ∨ c) = (a ∧ b) ∨ (a ∧ c) for all a ≻ b ≻ c in S.
Left-handed linearly distributive skew lattices are thus characterized by:
x∧[(y∧x) ∨ (z∧y∧x)] = (x∧y) ∨ (x∧z∧y). (5.5.2L)
Dually, right-handed linearly distributive skew lattices are characterized by:
[(x∧y∧z) ∨ (x∧y)]∧x = (y∧z∧x) ∨ (y∧x). (5.5.2R)
Proof. If say b ≻ a, then a∧(b∨c) = a and (a∧b) ∨ (a∧c) = a ∨ (a∧c) = a. If c ≻ a, then
a∧(b∨c) = a again, and (a∧b) ∨ (a∧c) = (a∧b) ∨ a = (a∧b∧a) ∨ a = a. Thus inequality only
occurs when a ≻ b, c. But even here, a ≻ c ≻ b gives us a∧(b∨c) = a∧c and (a∧c) ≻ (a∧b) so that
(a∧b) ∨ (a∧c) = a∧c also. Thus, to completely avoid a∧(b∨c) = a∧b ∨ a∧c we are only left with
a ≻ b ≻ c. £
In particular primitive skew lattices are distributive. It turns out that linear distributivity
is characterized succinctly by either of the dual pair of identities in the following lemma.
188
