Page 19 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 19
I: Preliminaries
Proof. That χ is a homomorphism follows easily from the associative, commutative and
distributive laws. By cancellation, χ is one-to-one. Upon composing with either coordinate
projections, it clearly it mapped onto each factor. £
Corollary 1.1.10. Every nontrivial distributive lattice is a subdirect product of C1. £
We return to the variety of all lattices. On any lattice, consider the polynomial
M(x, y, z) = (x∨y) ∧ (x∨z) ∧ (y∨z) that was implicit in the proof of Theorem 1.5. M satisfies the
identities
M(x, x, y) = M(x, y, x) = M(y, x, x) = x.
Given an algebra A = (A; f1, …, fr) on which a ternary operation M(x, y, z) satisfying these
identities is polynomial-defined using the operations of A, then Con(A) is distributive. In general,
if a ternary function M can be defined from the functions symbols of a variety V such that M
satisfied these identities on all algebras in V, then the congruence lattices of all algebras in that
variety are distributive and V is said to be congruence distributive.
Boolean lattices and Boolean algebras
Given a lattice (L; ∧, ∨) with maximal and minimal elements 1 and 0, elements x and xʹ
are complements in L if x∨xʹ = 1 and x∧xʹ = 0. If L is distributive, then the complement xʹ of any
element x is unique. Indeed, let xʺ be a second complement of x. Then
xʺ = xʺ ∧ 1 = xʺ ∧ (x ∨ xʹ) = (xʺ ∧ x) ∨ (xʺ ∧ xʹ) = 0 ∨ (xʺ ∧ xʹ) = xʺ ∧ xʹ.
Similarly, xʹ = xʹ ∧ xʺ and xʹ = xʺ follows. Clearly 0 and 1 are mutual complements.
Recall that Boolean lattice is a distributive lattice with maximal and minimal elements 1
and 0, (L; ∧, ∨, 1, 0), such that every x in L has a (necessarily unique) complement xʹ in L. If the
operation ʹ is built into the signature, then (L; ∧, ∨, ʹ, 1, 0) is a Boolean algebra. Boolean
algebras are characterized by the identities for a distributive lattice augmented by the identities
for maximal and minimal elements and the identities for complementation. They also satisfy the
DeMorgan identities: (x ∨ y)ʹ = xʹ ∧ yʹ and (x ∧ y)ʹ = xʹ ∨ yʹ.
Given a Boolean algebra, the difference (or relative complement) of elements x and y is
defined by x \ y = x ∧ yʹ. This operation satisfies the relative DeMorgan identities:
x \ (y ∨ z) = (x\y) ∧ (x\z) and x \ (y ∧ z) = (x\y) ∨ (x\z).
More generally, given any distributive lattice with a maximum 1 and minimum 0, if x and y have
complements, then so do x ∨ y and x ∧ y with (x ∨ y)ʹ = xʹ ∧ yʹ and (x ∧ y)ʹ = xʹ ∨ yʹ.
17
Proof. That χ is a homomorphism follows easily from the associative, commutative and
distributive laws. By cancellation, χ is one-to-one. Upon composing with either coordinate
projections, it clearly it mapped onto each factor. £
Corollary 1.1.10. Every nontrivial distributive lattice is a subdirect product of C1. £
We return to the variety of all lattices. On any lattice, consider the polynomial
M(x, y, z) = (x∨y) ∧ (x∨z) ∧ (y∨z) that was implicit in the proof of Theorem 1.5. M satisfies the
identities
M(x, x, y) = M(x, y, x) = M(y, x, x) = x.
Given an algebra A = (A; f1, …, fr) on which a ternary operation M(x, y, z) satisfying these
identities is polynomial-defined using the operations of A, then Con(A) is distributive. In general,
if a ternary function M can be defined from the functions symbols of a variety V such that M
satisfied these identities on all algebras in V, then the congruence lattices of all algebras in that
variety are distributive and V is said to be congruence distributive.
Boolean lattices and Boolean algebras
Given a lattice (L; ∧, ∨) with maximal and minimal elements 1 and 0, elements x and xʹ
are complements in L if x∨xʹ = 1 and x∧xʹ = 0. If L is distributive, then the complement xʹ of any
element x is unique. Indeed, let xʺ be a second complement of x. Then
xʺ = xʺ ∧ 1 = xʺ ∧ (x ∨ xʹ) = (xʺ ∧ x) ∨ (xʺ ∧ xʹ) = 0 ∨ (xʺ ∧ xʹ) = xʺ ∧ xʹ.
Similarly, xʹ = xʹ ∧ xʺ and xʹ = xʺ follows. Clearly 0 and 1 are mutual complements.
Recall that Boolean lattice is a distributive lattice with maximal and minimal elements 1
and 0, (L; ∧, ∨, 1, 0), such that every x in L has a (necessarily unique) complement xʹ in L. If the
operation ʹ is built into the signature, then (L; ∧, ∨, ʹ, 1, 0) is a Boolean algebra. Boolean
algebras are characterized by the identities for a distributive lattice augmented by the identities
for maximal and minimal elements and the identities for complementation. They also satisfy the
DeMorgan identities: (x ∨ y)ʹ = xʹ ∧ yʹ and (x ∧ y)ʹ = xʹ ∨ yʹ.
Given a Boolean algebra, the difference (or relative complement) of elements x and y is
defined by x \ y = x ∧ yʹ. This operation satisfies the relative DeMorgan identities:
x \ (y ∨ z) = (x\y) ∧ (x\z) and x \ (y ∧ z) = (x\y) ∨ (x\z).
More generally, given any distributive lattice with a maximum 1 and minimum 0, if x and y have
complements, then so do x ∨ y and x ∧ y with (x ∨ y)ʹ = xʹ ∧ yʹ and (x ∧ y)ʹ = xʹ ∨ yʹ.
17
