INVARIANT SETS AND NORMAL SUBGROUPOIDS OF UNIVERSAL ÉTALE GROUPOIDS INDUCED BY CONGRUENCES OF INVERSE SEMIGROUPS

Fuyuta Komura

doi:10.1017/s1446788721000021

Abstract

AbstractFor a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept for considering quotients of inverse semigroups. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization.

Highlights

The relation among inverse semigroups, étale groupoids and C*-algebras has been revealed by many researchers
He proved that the C*-algebras associated to universal groupoids are isomorphic to inverse semigroup C*-algebras
Paterson showed that the universal groupoid has a universal property about ample actions on totally disconnected spaces

Summary

Introduction

The relation among inverse semigroups, étale groupoids and C*-algebras has been revealed by many researchers. Paterson associated the universal groupoid to an inverse semigroup in [4]. He proved that the C*-algebras associated to universal groupoids are isomorphic to inverse semigroup C*-algebras. In this paper we investigate a relation between inverse semigroups and universal groupoids. Congruences of inverse semigroups induce closed invariant subsets and normal subgroupoids of universal groupoids. The universal groupoid of a quotient inverse semigroup is isomorphic to the restriction and quotient of the original universal groupoid (Theorem 4.3). We prove that the number of fixed points of a Boolean action is less than or equal to the number of certain semigroup homomorphisms (Corollary 5.17)

Preliminaries

Certain least congruences

Applications and examples