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A General Representation Theorem of a Kind of Super B-Quasi-Ehresmann Semigroups

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Abstract

We first study the structure of a special generalized regular semigroup, namely the B-semiabundant semigroup which can be expressed as the join of the pseudo-varieties of finite groups and finite aperiodic groups. In the literature, the weakly B-semiabundant semigroups have recently been thoughtfully investigated and considered by Wang. One easily observes that the class of good B-semiabundant semigroups is a special class of semigroups embraces all abundant (and hence regular) semigroups. In particular, a super B-quasi-Ehresmann semigroup is an analogy of an orthodox semigroup within the class of B-semiabundant semigroups. Thus, the class of super B-quasi-Ehresmann semigroups is obviously a subclass of the class of good B-quasi-Ehresmann semigroups which contains all orthodox semigroups. Thus, the super B-quasi-Ehresmann semigroup behaves similarly as the Clifford subsemigroups within the class of regular semigroups. Consequently, a super B-quasi-Ehresmann semigroup is now recognized as an important generalized regular semigroup. Our aim in this paper is to describe the properties and intrinsic structure of a super B-quasi-Ehresmann semigroup whose band of projections is right regular, right normal, left semiregular, left seminormal, regular, left quasinormal or normal, respectively. Hence, our representation theorem of the super B-quasi-Ehresmann semigroups improves, strengthens and generalizes the well-known “standard representation theorem of an orthodox semigroup” established by He et al. (Commun. Algebra 33:745–761, 2005). Finally, a general representation theorem in the category of Ehresmann semigroups is given.

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Acknowledgements

The authors would like to thank their colleagues for their helpful discussion and giving them many valuable suggestions toward to the draft of this paper.

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Authors and Affiliations

  1. School of Mathematical Science, Shandong Normal University, Jinan, 250014, People’s Republic of China

    Gang Li

  2. Institute of Mathematics, Yunnan University, Kunming, 650091, People’s Republic of China

    K. P. Shum

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  1. Gang Li
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  2. K. P. Shum
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Correspondence to K. P. Shum.

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Communicated by See Keong Lee.

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Li, G., Shum, K.P. A General Representation Theorem of a Kind of Super B-Quasi-Ehresmann Semigroups. Bull. Malays. Math. Sci. Soc. 41, 427–441 (2018). https://doi.org/10.1007/s40840-017-0521-9

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  • DOI: https://doi.org/10.1007/s40840-017-0521-9

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