Skip to main content
SpringerLink
Log in

The normal subsemigroups of the monoid of injective maps

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

We consider the monoid \(\operatorname{Inj}(M)\) of injective self-maps of a set M and want to determine its normal subsemigroups by numerical invariants. This was established by Mesyan (in Isr. J. Math. 189:287–305, 2012) if M is countable. Here we obtain an explicit description of all normal subsemigroups of \(\operatorname{Inj}(M)\) for any uncountable set M.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baer, R.: Die Kompositionsreihe der Gruppe aller eineindeutigen Abbildungen einer unendlichen Menge auf sich. Stud. Math. 5, 15–17 (1934)

    Google Scholar 

  2. Clifford, A.H., Preston, G.P.: Algebraic Theory of Semigroups, Vol. II. Math. Surveys, vol. 7. Amer. Math. Soc., Providence (1967)

    Book  Google Scholar 

  3. Droste, M.: Cubes of conjugacy classes covering the infinite symmeric groups. Trans. Am. Math. Soc. 288, 381–393 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Droste, M.: Squares of conjugacy classes in the infinite symmeric groups. Trans. Am. Math. Soc. 303, 503–515 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Droste, M., Göbel, R.: On a theorem of Baer, Schreier, and Ulam for permutations. J. Algebra 58, 282–290 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Droste, M., Göbel, R.: Products of conjugate permutations. Pac. J. Math. 94, 47–60 (1981)

    Article  MATH  Google Scholar 

  7. Hotzel, E.: Maximality properties of some subsemigroups of Baer-Levi semigroups. Semigroup Forum 51, 153–190 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jech, T.: Set Theory. Springer, Berlin (2000)

    Google Scholar 

  9. Lindsey, D., Madison, B.: Conjugation of injections by permutations. Semigroup Forum 12, 63–70 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  10. Maltcev, V., Mitchell, J.D., Ruškuc, N.: The Bergman property for semigroups. J. Lond. Math. Soc. 80, 212–232 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mesyan, Z.: Conjugation of injections by permutations. Semigroup Forum 81, 297–324 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mesyan, Z.: Monoids of injective maps closed under conjugation by permutations. Isr. J. Math. 189, 287–305 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Moran, G.: On planar Eulerian graphs and permutations. Trans. Am. Math. Soc. 287, 323–341 (1985)

    MATH  Google Scholar 

  14. Moran, G.: The products of conjugacy classes in some infinite groups. Isr. J. Math. 50, 54–74 (1985)

    Article  MATH  Google Scholar 

  15. Moran, G.: Conjugacy classes whose squares are infinite symmetric groups. Trans. Am. Math. Soc. 316, 493–521 (1989)

    Article  MATH  Google Scholar 

  16. Sit, W.Y., Sui, M.-K.: On the subsemigroups of ℕ. Math. Mag. 48, 225–227 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  17. Schreier, J., Ulam, S.: Über die Permutationsgruppe der natürlichen Zahlenfolge. Stud. Math. 4, 134–141 (1933)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Universität Leipzig, 04009, Leipzig, Germany

    Manfred Droste

  2. Fakultät für Mathematik, Universität Duisburg-Essen, 45117, Essen, Germany

    Rüdiger Göbel

Authors
  1. Manfred Droste
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rüdiger Göbel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manfred Droste.

Additional information

Communicated by Jean-Eric Pin.

The work was supported by the project No. 693-98.6/2007 of the German-Israeli Foundation for Scientific Research and Development.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Droste, M., Göbel, R. The normal subsemigroups of the monoid of injective maps. Semigroup Forum 87, 298–312 (2013). https://doi.org/10.1007/s00233-013-9478-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-013-9478-7

Keywords

Search

Navigation