# ABUNDANT SEMIGROUPS WITH QUASI-IDEAL S-ADEQUATE TRANSVERSALS

• Kong, Xiangjun (SCHOOL OF MATHEMATICS AND STATISTICS LANZHOU UNIVERSITY) ;
• Wang, Pei (SCHOOL OF COMPUTER SCIENCE QUFU NORMAL UNIVERSITY)
• Published : 2011.01.31

#### 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).

#### Acknowledgement

Supported by : National Natural Science Foundation of China, Natural Science Foundation of Shandong Province

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