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

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 tight completion of $S$ and show that it arises from Exel's tight groupoid under non-commutative Stone duality.