On a problem of M. Kambites regarding abundant semigroups

TL;DR

This paper investigates a question in semigroup theory about whether certain finite abundant semigroups with unique idempotents necessarily have commuting idempotents, and provides a counterexample to this conjecture.

Contribution

The authors use ideal extensions to demonstrate that the converse of a known property does not hold in finite abundant semigroups.

Findings

Counterexample to Kambites' question

Idempotents need not commute in certain finite abundant semigroups

Use of ideal extensions to analyze semigroup properties

Abstract

A semigroup is \emph{regular} if it contains at least one idempotent in each $\mathcal{R}$-class and in each $\mathcal{L}$-class. A regular semigroup is \emph{inverse} if satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each $\mathcal{R}$-class and in each $\mathcal{L}$-class, or (ii) the idempotents commute. Analogously, a semigroup is \emph{abundant} if it contains at least one idempotent in each $\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. An abundant semigroup is \emph{adequate} if its idempotents commute. In adequate semigroups, there is a unique idempotent in each $\mathcal{R}^*$ and $\mathcal{L}^*$-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each $\mathcal{R}^*$ and $\mathcal{L}^*$-class, must the idempotents commute? In this note we use ideal extensions to provide a negative answer to this question.