Page 55 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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II: Skew Lattices
A skew lattice S is quasi-distributive if its lattice image S/D is distributive. Quasi-
distributive skew lattices are a subvariety of skew lattices. Indeed, since (x∧y) ∨ (x∧z) ≺ x ∧ (y∨z)
holds for all lattices, these skew lattices are characterized by the identity:
[x∧(y∨z)] ∧ [(x∧y) ∨ (x∧z)] ∧ [x∧(y∨z)] = x∧(y∨z).
We have the following corollary to Theorem 2.2.9.
Theorem 2.2.10. A skew lattice S is quasi-distributive if and only if neither M3 nor N5 is
a subalgebra of S.
2.3 Normal skew lattices
Recall that a band S is normal if it satisfies any and hence all of:
a) ∀a, b, c ∈ S: abca = acba.
aʹ) ∀a, b, c, d ∈ S: abcd = acbd.
b) ∀a ∈S, aSa = {axa⎮x ∈ S} is a semilattice in S.
bʹ) ∀a, b ∈S, aSb = {axb⎮x ∈ S} is a semilattice in S.
c) Given D-classes A ≥ B in S, ∀a ∈ A, ∃!b ∈ B, a ≥ b.
Likewise, a skew lattice S is normal if either (and thus both) of the following hold:
a) (S: ∧) is a normal band. In particular, abcd = acbd holds on (S: ∧).
b) ∀a ∈ S, a∧S∧a = {a∧x∧a⎮x ∈ S} = {b ∈ S⎮a ≥ b} is a sublattice of S.
Clearly: Normal skew lattices form a subvariety of skew lattices.
Applying the Kimura decomposition we get the elementary but important:
Theorem 2.3.1. A skew lattice S is normal if and only if its left factor S/R is left normal
(a∧b∧c = a∧c∧b) and its right factor S/L is right normal (a∧b∧c = b∧a∧c). £
Being normal has interesting connections with distributivity.
Theorem 2.3.2. [Leech, 1992] Given a skew lattice S, the following are equivalent:
1) a ∧ (b ∨ c) ∧ d = (a ∧ b∧ d) ∨ (a ∧ c ∧ d) holds on S.
2) S is distributive and normal.
3) S/D is distributive and S is normal.
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A skew lattice S is quasi-distributive if its lattice image S/D is distributive. Quasi-
distributive skew lattices are a subvariety of skew lattices. Indeed, since (x∧y) ∨ (x∧z) ≺ x ∧ (y∨z)
holds for all lattices, these skew lattices are characterized by the identity:
[x∧(y∨z)] ∧ [(x∧y) ∨ (x∧z)] ∧ [x∧(y∨z)] = x∧(y∨z).
We have the following corollary to Theorem 2.2.9.
Theorem 2.2.10. A skew lattice S is quasi-distributive if and only if neither M3 nor N5 is
a subalgebra of S.
2.3 Normal skew lattices
Recall that a band S is normal if it satisfies any and hence all of:
a) ∀a, b, c ∈ S: abca = acba.
aʹ) ∀a, b, c, d ∈ S: abcd = acbd.
b) ∀a ∈S, aSa = {axa⎮x ∈ S} is a semilattice in S.
bʹ) ∀a, b ∈S, aSb = {axb⎮x ∈ S} is a semilattice in S.
c) Given D-classes A ≥ B in S, ∀a ∈ A, ∃!b ∈ B, a ≥ b.
Likewise, a skew lattice S is normal if either (and thus both) of the following hold:
a) (S: ∧) is a normal band. In particular, abcd = acbd holds on (S: ∧).
b) ∀a ∈ S, a∧S∧a = {a∧x∧a⎮x ∈ S} = {b ∈ S⎮a ≥ b} is a sublattice of S.
Clearly: Normal skew lattices form a subvariety of skew lattices.
Applying the Kimura decomposition we get the elementary but important:
Theorem 2.3.1. A skew lattice S is normal if and only if its left factor S/R is left normal
(a∧b∧c = a∧c∧b) and its right factor S/L is right normal (a∧b∧c = b∧a∧c). £
Being normal has interesting connections with distributivity.
Theorem 2.3.2. [Leech, 1992] Given a skew lattice S, the following are equivalent:
1) a ∧ (b ∨ c) ∧ d = (a ∧ b∧ d) ∨ (a ∧ c ∧ d) holds on S.
2) S is distributive and normal.
3) S/D is distributive and S is normal.
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