General Non-Commutative Locally Compact Locally Hausdorff Stone Duality

TL;DR

This paper generalizes Stone duality to a broader class of topological groupoids and inverse semigroups, removing previous restrictions and establishing a new duality framework.

Contribution

It extends classical Stone duality to non-zero-dimensional locally compact Hausdorff spaces and étale groupoids, linking them with inverse semigroups.

Findings

Established a duality between étale groupoids and inverse semigroups.

Removed the zero-dimensionality restriction in classical Stone duality.

Extended the duality to locally compact Hausdorff spaces.

Abstract

We extend the classical Stone duality between zero dimensional compact Hausdorff spaces and Boolean algebras. Specifically, we simultaneously remove the zero dimensionality restriction and extend to \'etale groupoids, obtaining a duality with an elementary class of inverse semigroups.