Boolean sets, skew Boolean algebras and a non-commutative Stone duality

TL;DR

This paper introduces a duality between Boolean sets, which are presheaves over Boolean algebras, and etale spaces over Boolean spaces, expanding the understanding of non-commutative Stone duality.

Contribution

It characterizes right-hand skew Boolean algebras via Boolean sets and establishes a duality theorem linking these sets to etale spaces over Boolean spaces.

Findings

Established a duality between Boolean sets and etale spaces.

Provided a new perspective on skew Boolean algebras.

Extended classical Stone duality to a non-commutative setting.

Abstract

We describe right-hand skew Boolean algebras in terms of a class of presheaves of sets over Boolean algebras called Boolean sets, and prove a duality theorem between Boolean sets and etale spaces over Boolean spaces.