Completions of Generalized Restriction P-Restriction Semigroups

Abstract

Generalized restriction P-restriction semigroups are common generalizations of restriction semigroups and generalized inverse \(*\)-semigroups. Gomes and Szendrei (resp. Ohta and Imaoka) have shown that every restriction semigroup (every generalized inverse \(*\)-semigroup) can be embedded in a complete, infinitely distributive restriction semigroup (resp. a \(*\)-complete, infinitely distributive generalized inverse \(*\)-semigroup). The main aim of this paper is to obtain an entirely corresponding result for generalized restriction P-restriction semigroups. Specifically, among other things, we show that every generalized restriction P-restriction semigroup can be (2,1,1)-embedded in a complete, infinitely distributive generalized restriction P-restriction semigroup. Our results generalize and enrich the corresponding results of Gomes, Szendrei, Ohta and Imaoka.

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Acknowledgements

The authors express their profound gratitude to the referee for the valuable comments and suggestions which not only improve the present paper but also give some new methods and thoughts for the future study. In particular, the referee has pointed out that some results of Sect. 3 can be deduced by using the fact that every generalized restriction P-restriction semigroup is a subdirect product of a restriction semigroup and a full (2,1,1)-subsemigroup of a generalized inverse \(*\)-semigroup. (This fact is essentially given by Jones in Proposition 5.5 of [11].) Thanks also go to the editor for the timely communications. This research is supported partially by Nature Science Foundation of China (11661082) and the Postgraduate Students Research Innovation Fund of Yunnan Normal University (2019).

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Correspondence to Shoufeng Wang.

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Yan, P., Wang, S. Completions of Generalized Restriction P-Restriction Semigroups. Bull. Malays. Math. Sci. Soc. 43, 3651–3673 (2020). https://doi.org/10.1007/s40840-020-00888-w

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Keywords

  • Generalized restriction P-restriction semigroup
  • Permissible set
  • Completion

Mathematics Subject Classification

  • 20M10