Communications of the
Korean Mathematical Society CKMS

In this paper, the connection of the inverse transversal with the adequate transversal is explored. It is proved that if $S$ is an abundant semigroup with an adequate transversal $S^o$, then $S$ is regular if and only if $S^o$ is an inverse semigroup. It is also shown that adequate transversals of a regular semigroup are just its inverse transversals. By means of a quasi-adequate semigroup and a right normal band, we construct an abundant semigroup containing a quasi-ideal $S$-adequate transversal and conversely, every such a semigroup can be constructed in this manner. It is simpler than the construction of Guo and Shum [9] through an $SQ$-system and the construction of El-Qallali [5] by $W(E, S)$.

Keywords: abundant semigroup, quasi-adequate semigroup, $S$-adequate transversal, quasi-ideal

MSC numbers: 20M10