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Fundamental Regular Semigroups with Quasi-ideal Regular \(*\)-Transversals

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Abstract

Let \(S\) be a semigroup and “\(*\)” a unary operation on \(S\) which satisfies the following identities

$$\begin{aligned} xx^*x=x, x^*x x^*=x^*, x^{***}=x^*, (xy^*)^*=y^{**}x^*, (x^*y)^*=y^*x^{**}. \end{aligned}$$

Then, \(S^*=\{x^*|x\in S\}\) is called a regular \(*\)-transversal of \(S\) in the literatures. Following Munn and Hall’s idea, in this paper we construct fundamental regular semigroups with quasi-ideal regular \(*\)-transversals by which fundamental representations of regular semigroups with quasi-ideal regular \(*\)-transversals are obtained.

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Acknowledgments

This study is initiated in the author’s Ph.D. dissertation, and the author wishes to thank his supervisor Professor Yuqi Guo for his encouragement and help. The author also expresses his profound gratitude to the referees for the valuable comments, which improve the presentation of this article. Thanks also go to Professor K. P. Shum for the timely communications. This paper is supported jointly by a Nature Science Foundation of Yunnan Province (2012FB139) and Nature Science Foundations of China (11226049; 11301470).

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Authors and Affiliations

  1. Department of Mathematics, Yunnan Normal University, Kunming, 650500, Yunnan, People’s Republic of China

    Shou-feng Wang

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  1. Shou-feng Wang
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Correspondence to Shou-feng Wang.

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Communicated by Kar Ping Shum.

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Wang, Sf. Fundamental Regular Semigroups with Quasi-ideal Regular \(*\)-Transversals. Bull. Malays. Math. Sci. Soc. 38, 1067–1083 (2015). https://doi.org/10.1007/s40840-014-0070-4

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  • DOI: https://doi.org/10.1007/s40840-014-0070-4

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