A refinement of Stone duality to skew Boolean algebras

Abstract

Abstract. We establish two duality theorems which refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category ofétale spaces over locally compact Boolean spaces whose morphisms areétale space cohomomorphisms over continuous proper maps. In the second theorem we prove that the category of lefthanded skew Boolean ∩-algebras whose morphisms are proper skew Boolean ∩-algebra homomorphisms is equivalent to the category ofétale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injectiveétale space cohomomorphisms over continuous proper maps.

Free Skew Boolean Intersection Algebras and Set Partitions

“…Thus every finite left-handed SBIA is isomorphic to some 2 m 2 × 3 m 3 L × · · · × (n + 1) m n+1 L , where m i ≥ 0 for all i under a convention that n 0 L ≃ {0}. The following results can be derived from [2, Proposition 3.8] as well as from the duality theory of [8].…”

Free skew Boolean algebras

“…In the left-handed case, Proposition 3.10 follows from the description of morphisms in the duality theory of SBAs [16,18] applied to the case where S/D is finite and thus its dual topology is discrete.…”

“…As a result we obtain: ]. See also the descriptions of ∩-morphisms between SBAs with intersections given in [1] and [16].) For example, in the case of 2 µ 2 ×3 µ 3 L ×· · ·×m µm L , the kernel ideal is a sub-product of primitive factors so that the map, to within isomorphism, is just a projection onto the product of the complementary factors.…”

“…We apply Theorem 6.6 to give an alternative construction, suggested by [16], of a free left-handed algebra on any non-empty set X. We begin with the set {0, 1} X of all maps X → {0, 1}.…”

“…Various algebraic structures with close ties to skew Boolean algebras have been recently studied in [3,7,15]. Stone duality has been extended from Boolean algebras to skew Boolean algebras in [1,16,17,18]. It is remarkable, however, that thus far free SBAs have received no systematic study.…”