Skip to main content

The commuting graph of the symmetric inverse semigroup

Israel Journal of Mathematics volume 207, pages 103–149 (2015)Cite this article

Abstract

The commuting graph of a finite non-commutative semigroup S, denoted G(S), is a simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. Let I(X) be the symmetric inverse semigroup of partial injective transformations on a finite set X. The semigroup I(X) has the symmetric group Sym(X) of permutations on X as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of Sym(X). In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of G(Sym(X)), and in 2011, Dolžan and Oblak claimed that this upper bound is in fact the exact value.

The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup I(X). We calculate the clique number of G(I(X)), the diameters of the commuting graphs of the proper ideals of I(X), and the diameter of G(I(X)) when |X| is even or a power of an odd prime. We show that when |X| is odd and divisible by at least two primes, then the diameter of G(I(X)) is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of I(X) of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of I(X). The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.

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

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, Journal of Algebra 298 (2006), 468–492.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. M. André, J. Araújo and J. Konieczny, Regular centralizers of idempotent transformations, Semigroup Forum 82 (2011), 307–318.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. M. André, V. H. Fernandes and J. D. Mitchell, Largest 2-generated subsemigroups of the symmetric inverse semigroup, Proceedings of the Edinburgh Mathematical Society 50 (2007), 551–561.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Araújo and J. Konieczny, Automorphism groups of centralizers of idempotents, Journal of Algebra 269 (2003), 227–239.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Araújo and J. Konieczny, Semigroups of transformations preserving an equivalence relation and a cross-section, Communications in Algebra 32 (2004), 1917–1935.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Araújo, M. Kinyon and J. Konieczny, Minimal paths in the commuting graphs of semigroups, European Journal of Combinatorics 32 (2011), 178–197.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Araújo and J. Konieczny, A method for finding new sets of axioms for classes of semigroups, Archive for Mathematical Logic 51 (2012), 461–474.

    Article  MATH  MathSciNet  Google Scholar 

  8. C. Bates, D. Bundy, S. Perkins and P. Rowley, Commuting involution graphs for symmetric groups, Journal of Algebra 266 (2003), 133–153.

    Article  MATH  MathSciNet  Google Scholar 

  9. E. A. Bertram, Some applications of graph theory to finite groups, Discrete Mathematics 44 (1983), 31–43.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Bundy, The connectivity of commuting graphs, Journal of Combinatorial Theory. Series A 113 (2006), 995–1007.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. M. Burns and B. Goldsmith, Maximal order abelian subgroups of symmetric groups, Bulletin of the London Mathematical Society 21 (1989), 70–72.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. R. Darafsheh, Groups with the same non-commuting graph, Discrete Applied Mathematics 157 (2009), 833–837.

    Article  MATH  MathSciNet  Google Scholar 

  13. D. Dolžan and P. Oblak, Commuting graphs of matrices over semirings, Linear Algebra and its Applications 435 (2011), 1657–1665.

    Article  MATH  MathSciNet  Google Scholar 

  14. V. H. Fernandes, Presentations for some monoids of partial transformations on a finite chain: a survey, in Semigroups, Algorithms, Automata and Languages (Coimbra, 2001), World Scientific, River Edge, NJ, 2002, pp. 363–378.

    Chapter  Google Scholar 

  15. A. G. Ganyushkin and T. V. Kormysheva, On nilpotent subsemigroups of a finite symmetric inverse semigroup, Matematicheskie Zametki 56 (1994), 29–35, 157 (Russian); translation in Mathematical Notes 56 (1994), 896–899 (1995).

    MathSciNet  Google Scholar 

  16. O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups. An Introduction, Algebra and Applications, Vol. 9, Springer-Verlag, London, 2009.

    Book  MATH  Google Scholar 

  17. The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.4.12, 2008, http://www.gap-system.org.

  18. M. Giudici and C. Parker, There is no upper bound for the diameter of the commuting graph of a finite group, Journal of Combinatorial Theory. Series A 120 (2013), 1600–1603.

    Article  MathSciNet  Google Scholar 

  19. J. M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, New York, 1995.

    MATH  Google Scholar 

  20. A. Iranmanesh and A. Jafarzadeh, On the commuting graph associated with the symmetric and alternating groups, Journal of Algebra and its Applications 7 (2008), 129–146.

    Article  MATH  MathSciNet  Google Scholar 

  21. J. Konieczny, Green’s relations and regularity in centralizers of permutations, Glasgow Mathematical Journal 41 (1999), 45–57.

    Article  MATH  MathSciNet  Google Scholar 

  22. J. Konieczny, Semigroups of transformations commuting with idempotents, Algebra Colloquium 9 (2002), 121–134.

    MATH  MathSciNet  Google Scholar 

  23. J. Konieczny, Semigroups of transformations commuting with injective nilpotents, Mathematica Bohemica 128 (2003), 179–186.

    MATH  MathSciNet  Google Scholar 

  24. J. Konieczny, Regular, inverse, and completely regular centralizers of permutations, Communications in Algebra 32 (2004), 1551–1569.

    Article  MATH  MathSciNet  Google Scholar 

  25. J. Konieczny, Centralizers in the semigroup of injective transformations on an infinite set, Bulletin of the Australian Mathematical Society 82 (2010), 305–321.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Konieczny and S. Lipscomb, Centralizers in the semigroup of partial transformations, Mathematica Japonica 48 (1998), 367–376.

    MATH  MathSciNet  Google Scholar 

  27. L. G. Kovács and C. E. Praeger, Finite permutation groups with large abelian quotients, Pacific Journal of Mathematics 136 (1989), 283–292.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. V. Lawson, Inverse Semigroups. The Theory of Partial Symmetries, World Scientific Publishing, River Edge, NJ, 1998.

    Book  MATH  Google Scholar 

  29. S. Lipscomb, Symmetric Inverse Semigroups, Mathematical Surveys and Monographs, Vol. 46, American Mathematical Society, Providence, RI, 1996.

    MATH  Google Scholar 

  30. B. H. Neumann, A problem of Paul Erdős on groups, Australian Mathematical Society. Journal. Series A. Pure Mathematicas and Statistics 21 (1976), 467–472.

    Article  MATH  Google Scholar 

  31. A. L. T. Paterson, Groupoids, Inverse Semigroups, and their Operator Algebras, Progress in Mathematics, Vol. 170, Birkhäuser Boston, Boston, MA, 1999.

    Book  MATH  Google Scholar 

  32. M. Petrich, Inverse Semigroups, John Wiley & Sons, New York, 1984.

    MATH  Google Scholar 

  33. A. S. Rapinchuk and Y. Segev, Valuation-like maps and the congruence subgroup property, Inventiones Mathematicae 144 (2001), 571–607.

    Article  MATH  MathSciNet  Google Scholar 

  34. A. S. Rapinchuk, Y. Segev and G. M. Seitz, Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable, Journal of the American Mathematical Society 15 (2002), 929–978.

    Article  MATH  MathSciNet  Google Scholar 

  35. Y. Segev, The commuting graph of minimal nonsolvable groups, Geometriae Dedicata 88 (2001), 55–66.

    Article  MATH  MathSciNet  Google Scholar 

  36. L. H. Soicher, The GRAPE package for GAP, Version 4.3, 2006, http://www.maths.qmul.ac.uk/~leonard/grape/.

  37. H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.

    MATH  Google Scholar 

  38. X. Yang, A classification of maximal inverse subsemigroups of the finite symmetric inverse semigroups, Communications in Algebra 27 (1999), 4089–4096.

    Article  MATH  MathSciNet  Google Scholar 

  39. X. Yang, Extensions of Clifford subsemigroups of the finite symmetric inverse semigroup, Communications in Algebra 33 (2005), 381–391.

    Article  MATH  MathSciNet  Google Scholar 

  40. L. Zhang and W. Shi, Recognition of the projective general linear group PGL(2, q) by its noncommuting graph, Journal of Algebra and its Applications 10 (2011), 201–218.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidade Aberta, R. Escola Politécnica, 147, 1269-001, Lisboa, Portugal

    João Araújo

  2. Centro de Álgebra, Universidade de Lisboa, 1649-003, Lisboa, Portugal

    João Araújo & Wolfram Bentz

  3. Department of Mathematics, University of Mary Washington, Fredericksburg, VA, 22401, USA

    Konieczny Janusz

Authors
  1. João Araújo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfram Bentz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Konieczny Janusz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to João Araújo.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Araújo, J., Bentz, W. & Janusz, K. The commuting graph of the symmetric inverse semigroup. Isr. J. Math. 207, 103–149 (2015). https://doi.org/10.1007/s11856-015-1173-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-015-1173-9

Keywords

  • Inverse Semigroup
  • Regular Semigroup
  • Commutative Semigroup
  • Clique Number
  • Nilpotent Semigroup