A refinement of Stone duality to skew Boolean algebras

TL;DR

This paper extends classical Stone duality to skew Boolean algebras, establishing two duality theorems that connect algebraic structures with étale spaces over locally compact Boolean spaces.

Contribution

It introduces two duality theorems that refine Stone duality by relating skew Boolean algebras to étale spaces with specific properties.

Findings

First duality theorem links left-handed skew Boolean algebras to étale spaces over Boolean spaces.

Second duality theorem relates skew Boolean ap-algebras to étale spaces with compact clopen equalizers.

Provides categorical equivalences between algebraic and topological structures.

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 of \'{e}tale spaces over locally compact Boolean spaces whose morphisms are \'{e}tale space cohomomorphisms over continuous proper maps. In the second theorem we prove that the category of left-handed skew Boolean $\cap$-algebras whose morphisms are proper skew Boolean $\cap$-algebra homomorphisms is equivalent to the category of \'{e}tale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective \'{e}tale space cohomomorphisms over continuous proper maps.