  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.
