• Communications of the Korean Mathematical Society
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• v.26 no.1
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• pp.1-12
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• 2011

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).

• Communications of the Korean Mathematical Society
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• v.24 no.4
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• pp.481-500
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• 2009

The concepts of *-relation of a ($\Gamma$-)semigroup and $\bar{\Gamma}$-adequate transversal of a ($\Gamma$-)abundant semigroup are defined in this note. Then we develop a matrix type theory for abundant semigroups. We give some equivalent conditions of a Rees matrix semigroup being abundant and some equivalent conditions of an abundant Rees matrix semigroup having an adequate transversal. Then we obtain some Rees matrix representations for abundant semigroups with adequate transversals by the above theories.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.1-8
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• 2014

In this paper, we give congruences on an abundant semigroup with a quasi-ideal S-adequate transversal $S^{\circ}$ by the congruence pair abstractly which consists of congruences on the structure component parts R and ${\Lambda}$. We prove that the set of all congruences on this kind of semigroups is a complete lattice.