Joins and covers in inverse semigroups and tight C*-algebras

We show Exel's tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the \(C^*\)-algebra of a finitely-aligned category of paths, developed by Spielberg, is the tight \(C^*\)-algebra of a natural inverse semigroup. This includes as a special case finitely-aligned higher-rank graphs: i.e., for such a higher-rank graph \(\Lambda\), the tight \(C^*\)-algebra of the inverse semigroup associated to \(\Lambda\) is the same as the \(C^*\)-algebra of \(\Lambda\). DOI: 10.1017/S000497271300111