On Refined Generalised Quasi-adequate Transversals

Abstract

Some properties and characterisations for abundant semigroups with generalised quasi-adequate transversals are explored.
In such semigroups, an interesting property $[\forall a,b \in RegS,\ V_{S^o}(a)\cap V_{S^o}(b)\neq \emptyset\Rightarrow V_{S^o}(a) = V_{S^o}(b)]$ is investigated and thus the concept of {\it refined generalised quasi-adequate transversals}, for short, {\it RGQA transversals} is introduced.
It is shown that RGQA transversals are the {\it real} common generalisations of both orthodox transversals and adequate transversals in the abundant case.
Finally, by means of two abundant semigroups $R$ and $L$, a spined product structure theorem for an abundant semigroup with a quasi-ideal RGQA transversal is established.