  In this paper we provide an overview of the class of inverse semigroups $S$
such that every congruence on $S$ relates at least one idempotent to a
non-idempotent; such inverse semigroups are called $E$-disjunctive. This
overview includes the study of the inverse semigroup theoretic structure of
$E$-disjunctive semigroups; a large number of natural examples; some asymptotic
results establishing the rarity of such inverse semigroups; and a general
structure theorem for all inverse semigroups where the building blocks are
$E$-disjunctive.
