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.
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