E-disjunctive inverse semigroups

TL;DR

This work analyzes $E$-disjunctive inverse semigroups, defined by the absence of nontrivial idempotent-pure congruences, and develops a comprehensive structure theory around them. It establishes that every inverse semigroup has an $E$-disjunctive quotient and that any inverse semigroup embeds into and is a homomorphic image of an $E$-disjunctive one, via wreath-product constructions and the syntactic congruence on $E(S)$. The authors introduce a general Q-construction, $Q(T, ext{Y}, ext{X})$, and prove a Q-theorem that any inverse semigroup is isomorphic to such a semigroup built from its maximum $E$-disjunctive image $T=S/ ho$ and its idempotents, extending McAlister-type representations. They provide a wide range of examples (graph, symmetric and dual symmetric inverse semigroups, finite monogenic families, Thompson-group-like monoids) and classify when these are $E$-disjunctive, along with asymptotic rarity results for monogenic cases. The framework connects idempotent-pure congruences, maximal $E$-disjunctive images, and preactions, offering structural decompositions with potential applications to related areas such as Leavitt path algebras and graph algebras.

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.

Figures (3)

• Figure 1: Egg-box diagram of an $E$-disjunctive inverse semigroup with a non-trivial idempotent-pure right congruence.
• Figure 2: Egg-box diagram of a semigroup that attains the bound in \ref{['idem_bound_thm']} when $\kappa = 3$
• Figure 3: Diagram demonstrating the arguments in the second claim within the proof of \ref{['thm:extend_preactions']}. The dashed lines indicate an application of \ref{['dfn:preaction_connnected_groupoid*']} starting with the arrow labelled $\underline{s_3}_q$ and proceeding anti-clockwise.

Theorems & Definitions (114)

• Definition 2.1: $E$-disjunctive.
• Example 2.2
• Definition 2.3: Syntactic congruence.
• Lemma 2.4
• proof
• Lemma 2.5: cf. Remark III.4.15($\delta$) in petrich_book
• Lemma 2.6
• Lemma 2.7
• proof
• Lemma 2.8