E-disjunctive inverse semigroups

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.