  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.
