# Tarski monoids: Matui's spatial realization theorem

## TL;DR

This paper introduces Tarski monoids, a class of inverse monoids that generalize Boolean algebras and symmetric groups, linking them to etale groupoids and dynamical systems, and reinterpreting Matui's theorem in this context.

## Contribution

It defines Tarski monoids as non-commutative analogs of Boolean algebras and relates them to etale groupoids, extending Matui's theorem to this new algebraic setting.

## Key findings

- Two simple Tarski monoids are isomorphic iff their groups of units are isomorphic.
- Tarski monoids generalize finite symmetric inverse monoids and include classical Thompson groups.
- Reinterpretation of Matui's theorem in terms of Tarski monoids.

## Abstract

We introduce a class of inverse monoids, called Tarski monoids, that can be regarded as non-commutative generalizations of the unique countable, atomless Boolean algebra. These inverse monoids are related to a class of etale topological groupoids under a non-commutative generalization of classical Stone duality and, significantly, 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 as an equivalent theorem about a class of Tarski monoids: two simple Tarski monoids are isomorphic if and only if their groups of units are isomorphic. The inverse monoids in question may also be viewed as countably infinite generalizations of finite symmetric inverse monoids. Their groups of units therefore generalize the finite symmetric groups and include amongst their number the classical Thompson groups.