Abundant semigroups with quasi-ideal $S$-adequate transversals
Commun. Korean Math. Soc. 2011 Vol. 26, No. 1, 1-12
https://doi.org/10.4134/CKMS.2011.26.1.1
Published online March 1, 2011
Xiangjun Kong and Pei Wang
Lanzhou University, Qufu Normal University
Abstract : 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
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