E-disjunctive inverse semigroups

TL;DR

This paper studies a special class of inverse semigroups called E-disjunctive, exploring their structure, examples, rarity, and providing a general structure theorem for inverse semigroups built from them.

Contribution

It offers a comprehensive overview of E-disjunctive inverse semigroups, including their structure, examples, rarity, and a unifying structure theorem.

Findings

E-disjunctive inverse semigroups are rare.

A large number of natural examples are identified.

A general structure theorem for inverse semigroups built from E-disjunctive components.

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.