Adequate transversals of quasi-adequate semigroups

TL;DR

This paper establishes a structure theorem for adequate transversals in certain quasi-adequate semigroups, generalizing known results and applying to specific classes like left ample adequate transversals.

Contribution

It provides a new structure theorem for adequate transversals of quasi-adequate semigroups, extending classical inverse transversal results and applying semidirect product constructions.

Findings

Structured description of adequate transversals in quasi-adequate semigroups

Generalization of Saito's inverse transversal result

Application to left ample adequate transversals using semidirect products

Abstract

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands.