Advertisement

Completions of Generalized Restriction P-Restriction Semigroups

  • Pan Yan
  • Shoufeng WangEmail author
Article
  • 12 Downloads

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.

Keywords

Generalized restriction P-restriction semigroup Permissible set Completion 

Mathematics Subject Classification

20M10 

Notes

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).

References

  1. 1.
    Auinger, K.: Free locally inverse \(\ast \)-semigroup. Czechoslov. Math. J. 43, 523–545 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Burgess, W.D.: Completions of semilattices of cancellative semigroups. Glasg. Math. J. 21, 29–37 (1980)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gomes, G.M.S., Szendrei, M.B.: Almost factorizable weakly ample semigroups. Commun. Algebra 35, 3503–3523 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gould, V.: Restriction and Ehresmann semigroups. In: Proceedings of the International Conference on Algebra (2010), pp. 265–288. World Sci. Publ., Hackensack, NJ (2012)Google Scholar
  5. 5.
    Howie, J.M.: An Introduction to Semigroup Theory. Academic Press, London (1976)zbMATHGoogle Scholar
  6. 6.
    Hall, T.E., Imaoka, T.: Representations and amalgamation of generalized inverse \(\ast \)-semigroups. Semigroup Forum 58, 126–141 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hollings, C.: From right PP monoids to restriction semigroups: a survey. Eur. J. Pure Appl. Math. 2, 21–57 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Imaoka, T.: On fundamental regular \(\ast \)-semigroups. Mem. Fac. Sci. Shimane Univ. 14, 19–23 (1980)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Imaoka, T.: Representations of generalized inverse \(\ast \)-semigroups. Acta Sci. Math. (Szeged) 61, 171–180 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Jones, P.R.: A common framework for restriction semigroups and regular \(\ast \)-semigroups. J. Pure Appl. Algebra 216, 618–632 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jones, P.R.: Varieties of \(P\)-restriction semigroups. Commun. Algebra 42, 1811–1834 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jones, P.R.: Almost perfect restriction semigroups. J. Algebra 445, 193–220 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kudryavtseva, G.: Partial monoid actions and a class of restriction semigroups. J. Algebra 429, 342–370 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lawson, M.V.: Covering and embeddings of inverse semigroups. Proc. Edinb. Math. Soc. 36, 399–419 (1993)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lawson, M.V.: Almost factorizable inverse semigroups. Glasg. Math. J. 36, 97–111 (1994)CrossRefGoogle Scholar
  16. 16.
    Lawson, M.V.: Inverse Semigroups. World Scientific, Singapre (1998)CrossRefGoogle Scholar
  17. 17.
    Leech, J.: Inverse monoids with a natural semilattice ordering. Proc. Lond. Math. Soc. 70, 146–182 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    McAlister, D.B., Reilly, N.R.: E-unitary covers for inverse semigroups. Pac. J. Math. 68, 161–174 (1977)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nordahl, T.E., Scheiblich, H.E.: Regular \(\ast \)-semigroups. Semigroup Forum 16, 369–377 (1978)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ohta, H., Imaoka, T.: Completions of generalized inverse \(\ast \)-semigroups. RIMS Kokyuroku 1604, 114–119 (2008)Google Scholar
  21. 21.
    Petrich, M.: Inverse Semigroups. Wiley, New York (1984)zbMATHGoogle Scholar
  22. 22.
    Pastijn, F.J., Oliveira, L.: Maximal dense ideal extensions of locally inverse semigroups. Semigroup Forum 72, 441–458 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Qallali, A.E., Fountain, J.: Proper covers for left ample semigroups. Semigroup Forum 71, 411–427 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Schein, B.M.: Completions, translational hulls and ideal extensions of inverse semigroups. Czechoslov. Math. J. 23, 575–610 (1973)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Scheiblich, H.E.: Generalized inverse semigroups with involution. Rocky Mt. J. Math. 12, 205–211 (1982)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Szendrei, M.B.: Free \(\ast \)-orthodox semigroups. Simon Stevin 59, 175–201 (1985)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Shoji, K.: Completions and injective hulls of \(E\)-reflexive inverse semigroups. Semigroup Forum 36, 55–68 (1987)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Szendrei, M.B.: Embedding into almost left factorizable restriction semigroups. Commun. Algebra 41, 1458–1483 (2013)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wang, S.F.: On algebras of \(P\)-Ehresmann semigroups and their associate partial semigroups. Semigroup Forum 95, 569–588 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wang, S.F.: An Ehresmann–Schein–Nambooripad-type theorem for a class of \(P\)-restriction semigroups. Bull. Malays. Math. Sci. Soc. 42, 535–568 (2019)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wang, S.F.: An Ehresmann–Schein–Nambooripad theorem for locally Ehresmann \(P\)-Ehresmann semigroups. Periodica Mathematica Hungarica, to appearGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020

Authors and Affiliations

  1. 1.Department of MathematicsYunnan Normal UniversityKunmingPeople’s Republic of China

Personalised recommendations