A characterization of adequate semigroups by forbidden subsemigroups

TL;DR

This paper characterizes adequate semigroups, which are amiable with commuting idempotents, by identifying forbidden subsemigroups that distinguish them from nonadequate semigroups.

Contribution

It provides a precise characterization of adequate semigroups through forbidden subsemigroup conditions, enhancing understanding of their structural properties.

Findings

Adequate semigroups are exactly amiable semigroups without certain subsemigroups.

Identifies two specific nonadequate semigroups as forbidden subsemigroups.

Offers a structural criterion for classifying adequate semigroups.

Abstract

A semigroup is \emph{amiable} if there is exactly one idempotent in each $\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. A semigroup is \emph{adequate} if it is amiable and if its idempotents commute. We characterize adequate semigroups by showing that they are precisely those amiable semigroups which do not contain isomorphic copies of two particular nonadequate semigroups as subsemigroups.