Skip to main content
SpringerLink
Log in

On the graph condition regarding the \(F\)-inverse cover problem

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

Auinger and Szendrei, (J Pure Appl Algebra 204:493–506, 2006), have shown that every finite inverse monoid has an \(F\)-inverse cover if and only if each finite graph admits a locally finite group variety with a certain property. We study this property and prove that the class of graphs for which a given group variety has the required property is closed downwards in the minor ordering, and can therefore be described by forbidden minors. We find these forbidden minors for all varieties of Abelian groups, thus describing the graphs for which such a group variety satisfies the above mentioned condition.

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

Fig. 1

Similar content being viewed by others

References

  1. Auinger, K., Szendrei, M.B.: On \(F\)-inverse covers of inverse monoids. J. Pure Appl. Algebra 204, 493–506 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dehornoy, P.: Braids, Self-distributivity, Progress in Mathematics, vol. 192. Birkhäuser, Basel (2000)

    Book  MATH  Google Scholar 

  3. Dehornoy, P.: The geometry monoid of left self-distributivity. J. Pure Appl. Algebra 160, 123–156 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Henckell, K., Rhodes, J.: The theorem of Knast, the PG=BG and type II conjectures. In: Monoids and Semigoups with Applications, Berkeley, CA, 1989, pp. 453–463. World Scientific, River Edge (1991)

  5. Kaarli, K., Márki, L.: A characterization of the inverse monoid of bi-congruences of certain algebras. Int. J. Algebra Comput. 19(6), 791–808 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kellendonk, J., Lawson, M.V.: Partial actions of groups. Int. J. Algebra Comput. 14, 87–114 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Koshkam, M., Skandalis, G.: Regular representation of groupoid \(C^\ast \)-algebras and applications to inverse semigroups. J. Reine Angew. Math. 546, 47–72 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Lawson, M.V.: Inverse Semigroups: the Theory of Partial Symmetries. World Scientific, Singapore (1998)

    Book  MATH  Google Scholar 

  9. Nica, A.: On a groupoid construction for actions of certain inverse semigroups. Int. J. Math. 5, 249–372 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Petrich, M.: Inverse Semigroups. Wiley, New York (1984)

    MATH  Google Scholar 

  11. Robertson, N., Seymour, P.D.: Graph Minors. XX. Wagner’s conjecture. J. Combin. Theory Ser. B 98, 325–357 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Steinberg, B.: Strong Morita equivalence of inverse semigroups. Houston J. Math. 37(3), 895–927 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

I would like to thank my supervisor, Mária B. Szendrei for introducing me to the problem and giving me helpful advice during my work. This research was supported by the Hungarian National Foundation for Scientific Research Grant no. K104251.

Author information

Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720, Szeged, Hungary

    Nóra Szakács

Authors
  1. Nóra Szakács
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nóra Szakács.

Additional information

Communicated by László Márki.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Szakács, N. On the graph condition regarding the \(F\)-inverse cover problem. Semigroup Forum 92, 551–558 (2016). https://doi.org/10.1007/s00233-015-9713-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-015-9713-5

Keywords

Search

Navigation