Abundant semigroups with quasi-ideal S-adequate transversals
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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).
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).
참고문헌 (12)
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