Groupoids viewed as inverse semigroups

TL;DR

This paper shows that every groupoid can be transformed into an inverse semigroup by adding a single element, revealing a deep connection between groupoids and inverse semigroups with implications for C*-algebras.

Contribution

It establishes a simple method to view groupoids as inverse semigroups, highlighting the orthogonality of idempotents and linking their C*-algebras.

Findings

Every groupoid becomes an inverse semigroup with one added element.

All idempotents in such inverse semigroups are mutually orthogonal.

C*-algebras of discrete groupoids are also C*-algebras of inverse semigroups.

Abstract

A simple observation, showing that every groupoid becomes an inverse semigroup after adding one element. In such inverse semigroups all idempotents are mutually orthogonal. This fact implies that every C*-algebra of a discrete groupoid is a C*-algebra of an inverse semigroup.