E-disjunctive inverse semigroups

Abstract

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.