Non-commutative Stone duality

TL;DR

This paper establishes a duality between Boolean inverse semigroups and Boolean groupoids, extending classical Stone duality from Boolean algebras and topological spaces to a non-commutative setting.

Contribution

It introduces Boolean groupoids and demonstrates their duality with Boolean inverse semigroups, generalizing classical Stone duality to a non-commutative framework.

Findings

Boolean inverse semigroups are dual to Boolean groupoids

Generalizes classical Stone duality to non-commutative structures

Provides explicit duality construction

Abstract

We show explicitly that Boolean inverse semigroups are in duality with what we term Boolean groupoids. This generalizes the classical Stone duality, which we refer to as commutative Stone duality, between generalized Boolean algebras and locally compact Hausdorff $0$-dimensional spaces.