Condition (K) for inverse semigroups and the ideal structure of their $C^*$-algebras

TL;DR

This paper explores the relationship between the ideal structure of inverse semigroups and their associated tight $C^*$-algebras, introducing conditions analogous to graph algebra conditions and applying them to self-similar graph actions.

Contribution

It develops analogues of Conditions (L) and (K) for inverse semigroups and relates ideals in semigroups to open invariant sets in their groupoids.

Findings

Established connections between semigroup ideals and groupoid open invariant sets

Introduced Conditions (K) and (L) analogues for inverse semigroups

Applied results to inverse semigroups of self-similar graph actions

Abstract

Inspired by results for graph $C^*$-algebras, we investigate connections between the ideal structure of an inverse semigroup $S$ and that of its tight $C^*$-algebra by relating ideals in $S$ to certain open invariant sets in the associated tight groupoid. We also develop analogues of Conditions (L) and (K) for inverse semigroups, which are related to certain congruences on $S$. We finish with applications to the inverse semigroups of self-similar graph actions and some relevant comments on the authors' earlier uniqueness theorems for inverse semigroups.