Page 158 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 158
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
In general, for any B the atoms of ω(B) have either the form (x, 0) or the form (x, x)
where x is an atom of B. When B is both complete and atomic, then for all (a, aʹ) ∈ω(B),
(a, aʹ) = sup[{(x, x) ⎢x ∈A(B), x ≤ aʹ} ∪ {(x, 0) ⎢x ∈A(B), x ≤ a \ aʹ}].
Thus ω(B) is also atomic in which case ω(B) is isomorphic to a product of primitive algebras, all
of which are copies of 3L since each atomic D-class of ω(B) has exactly two elements. The
converse holds in general: if a skew Boolean algebra S is both complete and atomic, then so is its
maximal (generalized) Boolean algebra image, S/D. £
Corollary 4.5.10. Every left-handed skew Boolean algebra S can be embedded as a
skew Boolean algebra into some omega algebra.
Proof. By Corollary 4.1.7, every left-handed skew Boolean algebra can be embedded in a power
of 3L which in turn is isomorphic to the omega algebra of the same power of 2. o
The ∩-version of Corollary 4.5.10 fails since, as seen in Section 4.4, as an ∩-algebra 3L
does not generate the variety 〈ℵ0L 〉∩ of all left-handed skew Boolean ∩-algebras. But we have:
Theorem 4.5.11. As algebras with ∩, all omega algebras lie in 〈3L〉∩. A skew Boolean
∩-algebra S can be embedded in some omega algebra if and only if it lies in 〈3L〉∩.
Proof. In the variety 〈ℵ0L 〉∩, primitive images of omega algebras are isomorphic to omega
algebras (by Lemma 4.5.13 below) and thus are copies of 3L. Hence nontrivial omega algebras
are subdirect products of copies of 3L and must lie in 〈3L〉∩. In general a skew Boolean ∩-algebra
lies in 〈3L〉∩ if and only if it can be embedded as an ∩-algebra in a power of 3L. But such powers
are copies of omega algebras by Theorem 4.5.9. £
Skew Boolean covers of generalized Boolean algebras
Recall that the center of a skew lattice S is the set ZS = {a ∈S⎮a∧x = x∧a for all x ∈S} or
equivalently, the set {a ∈S⎮ a∨x = x∨a for all b ∈ S}. ZS is also the union of all singleton D-
classes of S. (See Theorem 2.2.2.) For skew lattices ZS can be empty, but for a skew Boolean
algebra S, at least {0} ⊆ ZS. If ZS = {0}, then S has a trivial center. The center of a skew
Boolean algebra S is always an ideal of S. A skew Boolean cover of a generalized Boolean
algebra B is any skew Boolean algebra S with trivial center such that S/D ≅ B. In this case, S is a
minimal cover if the center ZS/θ of S/θ is nontrivial for all nontrivial congruences θ ⊆ D.
Theorem 4.5.12. For any nontrivial generalized Boolean algebra B, the derived omega
algebra ω(B) forms a minimal, skew Boolean cover of B.
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In general, for any B the atoms of ω(B) have either the form (x, 0) or the form (x, x)
where x is an atom of B. When B is both complete and atomic, then for all (a, aʹ) ∈ω(B),
(a, aʹ) = sup[{(x, x) ⎢x ∈A(B), x ≤ aʹ} ∪ {(x, 0) ⎢x ∈A(B), x ≤ a \ aʹ}].
Thus ω(B) is also atomic in which case ω(B) is isomorphic to a product of primitive algebras, all
of which are copies of 3L since each atomic D-class of ω(B) has exactly two elements. The
converse holds in general: if a skew Boolean algebra S is both complete and atomic, then so is its
maximal (generalized) Boolean algebra image, S/D. £
Corollary 4.5.10. Every left-handed skew Boolean algebra S can be embedded as a
skew Boolean algebra into some omega algebra.
Proof. By Corollary 4.1.7, every left-handed skew Boolean algebra can be embedded in a power
of 3L which in turn is isomorphic to the omega algebra of the same power of 2. o
The ∩-version of Corollary 4.5.10 fails since, as seen in Section 4.4, as an ∩-algebra 3L
does not generate the variety 〈ℵ0L 〉∩ of all left-handed skew Boolean ∩-algebras. But we have:
Theorem 4.5.11. As algebras with ∩, all omega algebras lie in 〈3L〉∩. A skew Boolean
∩-algebra S can be embedded in some omega algebra if and only if it lies in 〈3L〉∩.
Proof. In the variety 〈ℵ0L 〉∩, primitive images of omega algebras are isomorphic to omega
algebras (by Lemma 4.5.13 below) and thus are copies of 3L. Hence nontrivial omega algebras
are subdirect products of copies of 3L and must lie in 〈3L〉∩. In general a skew Boolean ∩-algebra
lies in 〈3L〉∩ if and only if it can be embedded as an ∩-algebra in a power of 3L. But such powers
are copies of omega algebras by Theorem 4.5.9. £
Skew Boolean covers of generalized Boolean algebras
Recall that the center of a skew lattice S is the set ZS = {a ∈S⎮a∧x = x∧a for all x ∈S} or
equivalently, the set {a ∈S⎮ a∨x = x∨a for all b ∈ S}. ZS is also the union of all singleton D-
classes of S. (See Theorem 2.2.2.) For skew lattices ZS can be empty, but for a skew Boolean
algebra S, at least {0} ⊆ ZS. If ZS = {0}, then S has a trivial center. The center of a skew
Boolean algebra S is always an ideal of S. A skew Boolean cover of a generalized Boolean
algebra B is any skew Boolean algebra S with trivial center such that S/D ≅ B. In this case, S is a
minimal cover if the center ZS/θ of S/θ is nontrivial for all nontrivial congruences θ ⊆ D.
Theorem 4.5.12. For any nontrivial generalized Boolean algebra B, the derived omega
algebra ω(B) forms a minimal, skew Boolean cover of B.
156
