Skip to main content

Semigroups of transformations whose restrictions belong to a given semigroup

Abstract

For a set X, denote by T(X) the semigroup of full transformations on X. For any subset Y of X and any subsemigroup \({\mathbb {S}}(Y)\) of T(Y), denote by \(T_{{\mathbb {S}}(Y)}(X)\) the semigroup of all transformations \(\alpha \in T(X)\) such that \(\alpha |_{Y}\in {\mathbb {S}}(Y)\), where \(\alpha |_Y\) is the restriction of \(\alpha \) to Y. In this paper, we describe the regular elements of \(T_{{\mathbb {S}}(Y)}(X)\) and determine when \(T_{{\mathbb {S}}(Y)}(X)\) is a regular semigroup [inverse semigroup, completely regular semigroup]. With the assumption that \({\mathbb {S}}(Y)\) contains the identity \({{\,\mathrm{id}\,}}_{{\tiny Y}}\), we describe Green’s relations in \(T_{{\mathbb {S}}(Y)}(X)\) in terms of the corresponding Green’s relations in \({\mathbb {S}}(Y)\). We apply these general results to obtain more concrete results for the semigroup \(T_{\Gamma (Y)}(X)\), where \(\Gamma (Y)\) is the semigroup of full injective transformations on Y. We also discuss generalizations and extensions of the semigroup \(T_{{\mathbb {S}}(Y)}(X)\).

This is a preview of subscription content, access via your institution.

References

  1. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, Mathematical Surveys, No. 7, American Mathematical Society, Providence, Rhode Island, (1964) (Vol. I) and 1967 (Vol. II)

  2. Honyam, P., Sanwong, J.: Semigroups of transformations with invariant set. J. Korean Math. Soc. 48, 289–300 (2011)

    MathSciNet  Article  Google Scholar 

  3. Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  4. Konieczny, J.: Centralizers in the semigroup of injective transformations on an infinite set. Bull. Aust. Math. Soc. 82, 305–321 (2010)

    MathSciNet  Article  Google Scholar 

  5. Magill, K.D., Jr.: Subsemigroups of \(S(X)\). Math. Jpn. 11, 109–115 (1966)

    MathSciNet  MATH  Google Scholar 

  6. Mendes-Gonçalves, S., Sullivan, R.P.: The ideal structure of semigroups of transformations with restricted range. Bull. Aust. Math. Soc. 83, 289–300 (2011)

    MathSciNet  Article  Google Scholar 

  7. Nenthein, S., Youngkhong, P., Kemprasit, Y.: Regular elements of some transformation semigroups. Pure Math. Appl. 16, 307–314 (2005)

    MathSciNet  MATH  Google Scholar 

  8. Petrich, M.: Inverse Semigroups. John Wiley & Sons, New York (1984)

    MATH  Google Scholar 

  9. Petrich, M., Reilly, N.R.: Completely Regular Semigroups. John Wiley & Sons, New York (1999)

    MATH  Google Scholar 

  10. Sun, L., Sun, J.: A note on naturally ordered semigroups of transformations with invariant set. Bull. Aust. Math. Soc. 91, 264–267 (2015)

    MathSciNet  Article  Google Scholar 

  11. Sun, L., Wang, L.: Natural partial order in semigroups of transformations with invariant set. Bull. Aust. Math. Soc. 87, 94–107 (2013)

    MathSciNet  Article  Google Scholar 

  12. Symons, J.S.V.: Some results concerning a transformation semigroup. J. Aust. Math. Soc. Ser. A 19, 413–425 (1975)

    MathSciNet  Article  Google Scholar 

  13. Tinpun, K., Koppitz, J.: Generating sets of infinite full transformation semigroups with restricted range. Acta Sci. Math. (Szeged) 82, 55–63 (2016)

    MathSciNet  Article  Google Scholar 

  14. Tinpun, K., Koppitz, J.: Relative rank of the finite full transformation semigroup with restricted range. Acta Math. Univ. Comenian. 85, 347–356 (2016)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to the referee for pointing out that the semigroup \(T_{T(Y)}(X)\) has been studied by various authors.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Janusz Konieczny.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Victoria Gould.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Konieczny, J. Semigroups of transformations whose restrictions belong to a given semigroup. Semigroup Forum 104, 109–124 (2022). https://doi.org/10.1007/s00233-021-10227-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-021-10227-5

Keywords

  • Semigroups of transformations
  • Restrictions of transformations
  • Injective transformations
  • Regularity
  • Green’s relations