Abundant Semigroups with Weakly Simplistic RGQA Transversals

Abstract

As the {\it real} common generalisations of both orthodox transversals and adequate transversals in the abundant case, the concept of {\it refined generalised quasi-adequate transversals}, for short, {\it RGQA transversals} was introduced by Kong and Wang. In this paper, for the RGQA transversal, the necessary and sufficient condition for the sets $I$ and $\Lambda$ to be bands is investigated. It is shown that the sets $I$ and $\Lambda$ are both bands if and only if the RGQA transversal is weakly simplistic,
and the RGQA transversal $S^o$ being weakly simplistic is different from $S^o$ being a quasi-ideal nor the abundant semigroup $S$ satisfying the regularity condition. Finally, by means of a quasi-adequate semigroup and a band, a structure theorem for an abundant semigroup with a weakly simplistic RGQA transversal is established.