Page 160 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 160
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Proof. This follows from Theorem 4.4.14 and the two previous results. £
Does being a minimal skew Boolean cover of its maximal lattice image characterize an
ω-algebra? We briefly explore this question, beginning with a theorem. Its fairly straightforward
proof is omitted.
Theorem 4.5.15. Given an atomic skew Boolean algebra S with atomic D-classes Di for
i in index set I, then S has a trivial center and hence is a skew Boolean cover of S/D if and only if
|Di| ≥ 2 for all i. S is a minimal skew Boolean cover of S/D if and only if |Di| = 2 for i ∈ I. S ≅
ω(S/D) must occur whenever S is also left-handed and complete. £
Example 4.5.2. A skew Boolean algebra S that is a minimal skew Boolean cover of S/D
but not isomorphic to any ω(S/D) is given by taking the subset lattice P(X) of an infinite set X
and letting S be the atomic subalgebra of ω(P(X)) consisting of all pairs (A, Aʹ) for Aʹ finite.
Given (A, Aʹ) for A infinite and Aʹ finite, the D-class of (A, Aʹ) has |A| many elements in S, but
|P(A)| many elements in ω(P(X)). Since S and ω(P(X)) share P(X) as a maximal lattice image,
and |A| < |P(A)| for all subsets of X, S cannot be isomorphic to any ω(B).
The functor ω and its left adjoint
In the following, 〈2〉 and 〈3L〉 denote the respective varieties of generalized Boolean
algebras and left-handed skew Boolean algebras viewed as categories. The ω-construction is the
object stage of a functor ω: 〈2〉 → 〈3L〉. Given homomorphism f: B → Bʹ in 〈2〉, its ω-image in
〈3L〉 is the homomorphism fω: ω(B) → ω(Bʹ) in 〈3L〉 given by fω(a, aʹ) = (f(a), f(aʹ)).
Theorem 4.5.16. ω: 〈2〉 → 〈3L〉 preserves limits. Thus it has a left adjoint, Ω: 〈3L〉 → 〈2〉.
Proof. Observe first that ω preserves products: indeed ω(∏i ∈IBi) ≅ ∏i ∈Iω(Bi) under the map
(〈ai〉, 〈aiʹ 〉) → 〈(ai), (aiʹ)〉. Next, note that ω preserves equalizers: given f, g: B → Bʹ,
equ(fω, gω) = (equ(f, g))ω where equ(f, g) is the standard equalizer given by the inclusion of the
maximal subalgebra of B on which f and g agree. Hence ω preserves limits.
Next we show that every skew Boolean algebra S has a solution set of homomorphisms F
= {fi: S → ω(Bi)⎮i ∈ ∩} such that every homomorphism f: S → ω(B) in 〈3L〉 can be written as a
composite f = hω o fi for some fi ∈ F and some homomorphism h:Bi → B in 〈2〉. So given a
homomorphism f: S → ω(B) of skew Boolean algebras, consider the subalgebra Bʹ of B that is
determined by the subset union
∪{{a, aʹ}⎮(a, aʹ) = f(x) ∈ ω(B) for some x ∈ S}.
Clearly f[S] ⊆ ω(Bʹ). Moreover |Bʹ| ≤ max{ℵ0, |S|}. Hence a solution set for S is given by the
set of all homomorphisms f: S → ω(Bj) where {Bj⎪j ∈ J} is a maximal class of mutually
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Proof. This follows from Theorem 4.4.14 and the two previous results. £
Does being a minimal skew Boolean cover of its maximal lattice image characterize an
ω-algebra? We briefly explore this question, beginning with a theorem. Its fairly straightforward
proof is omitted.
Theorem 4.5.15. Given an atomic skew Boolean algebra S with atomic D-classes Di for
i in index set I, then S has a trivial center and hence is a skew Boolean cover of S/D if and only if
|Di| ≥ 2 for all i. S is a minimal skew Boolean cover of S/D if and only if |Di| = 2 for i ∈ I. S ≅
ω(S/D) must occur whenever S is also left-handed and complete. £
Example 4.5.2. A skew Boolean algebra S that is a minimal skew Boolean cover of S/D
but not isomorphic to any ω(S/D) is given by taking the subset lattice P(X) of an infinite set X
and letting S be the atomic subalgebra of ω(P(X)) consisting of all pairs (A, Aʹ) for Aʹ finite.
Given (A, Aʹ) for A infinite and Aʹ finite, the D-class of (A, Aʹ) has |A| many elements in S, but
|P(A)| many elements in ω(P(X)). Since S and ω(P(X)) share P(X) as a maximal lattice image,
and |A| < |P(A)| for all subsets of X, S cannot be isomorphic to any ω(B).
The functor ω and its left adjoint
In the following, 〈2〉 and 〈3L〉 denote the respective varieties of generalized Boolean
algebras and left-handed skew Boolean algebras viewed as categories. The ω-construction is the
object stage of a functor ω: 〈2〉 → 〈3L〉. Given homomorphism f: B → Bʹ in 〈2〉, its ω-image in
〈3L〉 is the homomorphism fω: ω(B) → ω(Bʹ) in 〈3L〉 given by fω(a, aʹ) = (f(a), f(aʹ)).
Theorem 4.5.16. ω: 〈2〉 → 〈3L〉 preserves limits. Thus it has a left adjoint, Ω: 〈3L〉 → 〈2〉.
Proof. Observe first that ω preserves products: indeed ω(∏i ∈IBi) ≅ ∏i ∈Iω(Bi) under the map
(〈ai〉, 〈aiʹ 〉) → 〈(ai), (aiʹ)〉. Next, note that ω preserves equalizers: given f, g: B → Bʹ,
equ(fω, gω) = (equ(f, g))ω where equ(f, g) is the standard equalizer given by the inclusion of the
maximal subalgebra of B on which f and g agree. Hence ω preserves limits.
Next we show that every skew Boolean algebra S has a solution set of homomorphisms F
= {fi: S → ω(Bi)⎮i ∈ ∩} such that every homomorphism f: S → ω(B) in 〈3L〉 can be written as a
composite f = hω o fi for some fi ∈ F and some homomorphism h:Bi → B in 〈2〉. So given a
homomorphism f: S → ω(B) of skew Boolean algebras, consider the subalgebra Bʹ of B that is
determined by the subset union
∪{{a, aʹ}⎮(a, aʹ) = f(x) ∈ ω(B) for some x ∈ S}.
Clearly f[S] ⊆ ω(Bʹ). Moreover |Bʹ| ≤ max{ℵ0, |S|}. Hence a solution set for S is given by the
set of all homomorphisms f: S → ω(Bj) where {Bj⎪j ∈ J} is a maximal class of mutually
158