### Abstract

This paper continues the study of 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 étale 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 (à la Rubin) on a class of étale 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 Thompson groups (Formula presented.).

Original language | English |
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Pages (from-to) | 1-26 |

Number of pages | 26 |

Journal | Semigroup Forum |

Early online date | 21 Jun 2017 |

DOIs | |

Publication status | E-pub ahead of print - 21 Jun 2017 |

### Keywords

- Inverse semigroups
- Stone duality
- Étale topological groupoids

### ASJC Scopus subject areas

- Algebra and Number Theory