Quasi-ideal transversals of abundant semigroups and spined products

TL;DR

This paper introduces a simplified structural framework for quasi-ideal transversals of abundant semigroups using spined products, extending existing theories and providing new structure theorems for inverse transversals.

Contribution

It offers a new, simpler approach to understanding quasi-ideal transversals in abundant semigroups through spined products, generalizing previous results.

Findings

New structure theorem for quasi-ideal transversals

Extension of results to multiplicative transversals

Additional structure theorems for inverse transversals

Abstract

We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant semigroups in terms of spined products, which is similar in nature to that given by Saito for weakly multiplicative inverse transversals of regular semigroups. As a consequence we deduce a similar result for multiplicative transversals of abundant semigroups and also consider the case when the semigroups are in fact regular and provide some new structure theorems for inverse transversals.