# The universal Boolean inverse semigroup presented by the abstract Cuntz-Krieger relations

## TL;DR

This paper introduces the Exel completion, a universal Boolean inverse semigroup derived from any inverse semigroup using abstract Cuntz-Krieger relations, linking inverse semigroup theory with non-commutative topology.

## Contribution

It establishes a universal construction of Boolean inverse semigroups from inverse semigroups via abstract Cuntz-Krieger relations, connecting to Exel's tight groupoid.

## Key findings

- The Exel completion is a universal Boolean inverse semigroup for any inverse semigroup.
- It is constructed using abstract Cuntz-Krieger relations.
- The construction relates to non-commutative Stone duality and Exel's tight groupoid.

## Abstract

This paper is a contribution to the theory of what might be termed $0$-dimensional non-commutative spaces. We prove that associated with each inverse semigroup $S$ is a Boolean inverse semigroup presented by the abstract versions of the Cuntz-Krieger relations. We call this Boolean inverse semigroup the Exel completion of $S$ and show that it arises from Exel's tight groupoid under non-commutative Stone duality.