On a class of countable Boolean inverse monoids and Matui's spatial realization theorem

TL;DR

This paper introduces a class of non-commutative Boolean inverse monoids related to étale groupoids, reinterpreting Matui's theorem within this algebraic framework and connecting it to dynamical systems and Thompson groups.

Contribution

It defines a new class of inverse monoids generalizing Boolean algebras and symmetric inverse monoids, linking them to étale groupoids and dynamical systems, and reinterpreting Matui's theorem.

Findings

Inverse monoids generalize Boolean algebras and symmetric inverse monoids.

They relate to étale topological groupoids via a non-commutative Stone duality.

Matui's theorem is reinterpreted as a statement about inverse monoids.

Abstract

We introduce a class of inverse monoids that can be regarded as non-commutative generalizations of Boolean algebras. These inverse monoids are related to a class of \'etale topological groupoids, under a non-commutative generalization of classical Stone duality. Furthermore, and significantly for this paper, they arise naturally in the theory of dynamical systems as developed by Matui. We are thereby able to reinterpret a theorem of Matui on a class of \'etale groupoids, in the spirit of Rubin's theorem, as an equivalent theorem about a class of inverse monoids. The inverse monoids in question may be viewed as the countably infinite generalizations of finite symmetric inverse monoids. Their groups of units therefore generalize the finite symmetric groups and include amongst their number the Thompson groups $G_{n,1}$.