A non-commutative generalization of Stone duality

TL;DR

This paper extends Stone duality to boolean inverse monoids and boolean groupoids, establishing a dual equivalence that generalizes classical results and connects to structures like the Cuntz groupoid and Thompson groups.

Contribution

It introduces a non-commutative generalization of Stone duality, linking boolean inverse monoids with boolean groupoids, and explores their relationships with well-known algebraic structures.

Findings

Boolean inverse monoids are dually equivalent to boolean groupoids.

The boolean inverse monoid from the Cuntz groupoid is the strong orthogonal completion of the polycyclic monoid.

The group of units in this monoid is a Thompson group.

Abstract

We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid associated with the Cuntz groupoid is the strong orthogonal completion of the polycyclic (or Cuntz) monoid and so its group of units is a Thompson group.