Advertisement

Semigroup Forum

, Volume 93, Issue 1, pp 111–130 | Cite as

The structure of a graph inverse semigroup

  • Zachary Mesyan
  • J. D. Mitchell
Research Article

Abstract

Given any directed graph E one can construct a graph inverse semigroup G(E), where, roughly speaking, elements correspond to paths in the graph. In this paper we study the semigroup-theoretic structure of G(E). Specifically, we describe the non-Rees congruences on G(E), show that the quotient of G(E) by any Rees congruence is another graph inverse semigroup, and classify the G(E) that have only Rees congruences. We also find the minimum possible degree of a faithful representation by partial transformations of any countable G(E), and we show that a homomorphism of directed graphs can be extended to a homomorphism (that preserves zero) of the corresponding graph inverse semigroups if and only if it is injective.

Keywords

Inverse semigroup Directed graph Congruence 

Notes

Acknowledgments

We would like to thank Benjamin Steinberg for pointing us to relevant literature. We are also grateful to the referee for helpful suggestions about improving the paper and for noticing a gap in the proof of an earlier version of Lemma 9.

References

  1. 1.
    Abrams, G., Pino, G.A.: The Leavitt path algebra of a graph. J. Algebra 293, 319–334 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ara, P., Moreno, M.A., Pardo, E.: Nonstable \(K\)-theory for graph algebras. Algebra Represent. Theory 10, 157–178 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ash, C.J., Hall, T.E.: Inverse semigroups on graphs. Semigroup Forum 11, 140–145 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Costa, A., Steinberg, B.: A categorical invariant of flow equivalence of shifts. Ergod. Theory Dyn. Syst. 36, 470–513 (2016)Google Scholar
  5. 5.
    Easdown, D.: The minimal faithful degree of a fundamental inverse semigroup. Bull. Aust. Math. Soc. 35, 373–378 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Frucht, R.: Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compos. Math. 6, 239–250 (1939)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  8. 8.
    Jones, D.G.: Polycyclic monoids and their generalisations. Ph. D. Thesis, Heriot-Watt University (2011)Google Scholar
  9. 9.
    Jones, D.G., Lawson, M.V.: Graph inverse semigroups: their characterization and completion. J. Algebra 409, 444–473 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Krieger, W.: On subshifts and semigroups. Bull. London Math. Soc. 38, 617–624 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kumjian, A., Pask, D., Raeburn, I., Renault, J.: Graphs, groupoids, and Cuntz-Krieger algebras. J. Funct. Anal. 144, 505–541 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kumjian, A., Pask, D., Raeburn, I.: Cuntz-Krieger algebras of directed graphs. Pac. J. Math. 184, 161–174 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lawson, M.V.: Inverse semigroups: the theory of partial symmetries. World Scientific, Singapore (1998)CrossRefzbMATHGoogle Scholar
  14. 14.
    Mesyan, Z., Mitchell, J.D., Morayne, M., Péresse, Y.: Topological graph inverse semigroups. arXiv:1306.5388
  15. 15.
    Mitchell, J.D., et al.: Semigroups - GAP package, Version 2.4. (April 2015). http://www-groups.mcs.st-andrews.ac.uk/~jamesm/semigroups.php
  16. 16.
    Nivat, M., Perrot, J.-F.: Une généralisation du monoïde bicyclique. C. R. Acad. Sci. Paris 271, 824–827 (1970)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Paterson, A.L.T.: Graph inverse semigroups, groupoids and their C*-algebras. J. Oper. Theory 48, 645–662 (2002)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ColoradoColorado SpringsUSA
  2. 2.Mathematical InstituteUniversity of St AndrewsSt AndrewsScotland, UK

Personalised recommendations