Page 128 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 128
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Given complete skew Boolean algebras S and T, any homomorphism f: S → T sends
commutative subsets to commutative subsets. If f also preserves suprema and infima, then f is a
complete homomorphism of complete skew Boolean algebras. We state the following fuller
description of the P(A, B) process.
Theorem 4.1.14. Given maps α: Aʹ → A and β: B → Bʹ, P(α, β): P(A, B) → P(Aʹ, Bʹ)
defined by P(α, β)f = βfα, for all f ∈P(A, B), is a homomorphism of complete Boolean skew
algebras. Moreover, if Set and CSBA denote the respective categories of sets and complete skew
Boolean algebras, then P: Set × Set → CSBA is a bifunctor from the category of sets to the
category of complete skew Boolean algebras.
4.2 Finiteness, orthosums and free algebras
We recast the assertions of Theorem 4.1.4 and Corollary 4.1.5 in more detail for the case
where S/D is finite and especially when the algebra S itself is finite. It will be helpful to explore
things in a slightly more general context, proceeding as follows. Two element a and b in a skew
Boolean algebra are orthogonal if a ∧ b = 0 (and thus b∧a = 0 and also a∨b = b∨a). A finite set
of elements {a1, … , an} is an orthogonal set if the ai are pairwise orthogonal, so that
a1 ∨ … ∨ an = aσ1 ∨ … ∨ aσn
for all permutations σ on {1, 2, … , n}. In this situation, a1 ∨ … ∨ an is denoted by a1 + … + an
∑or n ai . (Indeed, such notation will assume orthogonality.) Such a sum is referred to as an
1
orthogonal sum, or an orthosum for short.
A family of D-classes {D1, … , Dr} is orthogonal when elements from distinct classes
are orthogonal. For this it is sufficient that some transversal set {d1, … , dr} is orthogonal. When
this occurs, the Di are the primitive D-classes in the subalgebra they generate. In general:
Proposition 4.2.1. Given an orthogonal family of nonzero D-classes {D1, … , Dr} and
two orthosums a1 + … + ar and b1 + … + br where ai, bi ∈ Di:
i) (a1 + … + ar) ∨ (b1 + … + br) = (a1∨b1) + (a2∨b2) + … + (ar∨br);
ii) (a1 + … + ar) ∧ (b1 + … + br) = (a1∧b1) + (a2∧b2) + … + (ar∧br);
iii) (a1 + … + ar) \ (b1 + … + br) = (a1\b1) + (a2\b2) + … + (ar\br);
iv) a1 + … + ar = b1 + … + br iff a1 = b1, a2 = b2, … , and ar = br.
Indeed (i) – (iv) extend to the subalgebra ∑1r Di0 = {x1 + … + xr⎪xi ∈ Di0 } generated from the
union D1 ∪ … ∪ Dr where elements from distinct Di0 are also orthogonal. £
126
Given complete skew Boolean algebras S and T, any homomorphism f: S → T sends
commutative subsets to commutative subsets. If f also preserves suprema and infima, then f is a
complete homomorphism of complete skew Boolean algebras. We state the following fuller
description of the P(A, B) process.
Theorem 4.1.14. Given maps α: Aʹ → A and β: B → Bʹ, P(α, β): P(A, B) → P(Aʹ, Bʹ)
defined by P(α, β)f = βfα, for all f ∈P(A, B), is a homomorphism of complete Boolean skew
algebras. Moreover, if Set and CSBA denote the respective categories of sets and complete skew
Boolean algebras, then P: Set × Set → CSBA is a bifunctor from the category of sets to the
category of complete skew Boolean algebras.
4.2 Finiteness, orthosums and free algebras
We recast the assertions of Theorem 4.1.4 and Corollary 4.1.5 in more detail for the case
where S/D is finite and especially when the algebra S itself is finite. It will be helpful to explore
things in a slightly more general context, proceeding as follows. Two element a and b in a skew
Boolean algebra are orthogonal if a ∧ b = 0 (and thus b∧a = 0 and also a∨b = b∨a). A finite set
of elements {a1, … , an} is an orthogonal set if the ai are pairwise orthogonal, so that
a1 ∨ … ∨ an = aσ1 ∨ … ∨ aσn
for all permutations σ on {1, 2, … , n}. In this situation, a1 ∨ … ∨ an is denoted by a1 + … + an
∑or n ai . (Indeed, such notation will assume orthogonality.) Such a sum is referred to as an
1
orthogonal sum, or an orthosum for short.
A family of D-classes {D1, … , Dr} is orthogonal when elements from distinct classes
are orthogonal. For this it is sufficient that some transversal set {d1, … , dr} is orthogonal. When
this occurs, the Di are the primitive D-classes in the subalgebra they generate. In general:
Proposition 4.2.1. Given an orthogonal family of nonzero D-classes {D1, … , Dr} and
two orthosums a1 + … + ar and b1 + … + br where ai, bi ∈ Di:
i) (a1 + … + ar) ∨ (b1 + … + br) = (a1∨b1) + (a2∨b2) + … + (ar∨br);
ii) (a1 + … + ar) ∧ (b1 + … + br) = (a1∧b1) + (a2∧b2) + … + (ar∧br);
iii) (a1 + … + ar) \ (b1 + … + br) = (a1\b1) + (a2\b2) + … + (ar\br);
iv) a1 + … + ar = b1 + … + br iff a1 = b1, a2 = b2, … , and ar = br.
Indeed (i) – (iv) extend to the subalgebra ∑1r Di0 = {x1 + … + xr⎪xi ∈ Di0 } generated from the
union D1 ∪ … ∪ Dr where elements from distinct Di0 are also orthogonal. £
126
