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Enumeration of finite inverse semigroups

  • Martin E. MalandroEmail author
Research Article
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Abstract

We give an efficient algorithm for the enumeration up to isomorphism of the inverse semigroups of order n, and we count the number S(n) of inverse semigroups of order \(n\le 15\). This improves considerably on the previous highest-known value S(9). We also give a related algorithm for the enumeration up to isomorphism of the finite inverse semigroups S with a given underlying semilattice of idempotents E, a given restriction of Green’s \(\mathrel {\mathcal {D}}\)-relation on S to E, and a given list of maximal subgroups of S associated to the elements of E.

Keywords

Inverse semigroup Enumeration Semilattice Maximal subgroup Green’s relations 

Notes

Acknowledgements

We are grateful to Sam Houston State University (SHSU) and the IT@Sam department at SHSU for their assistance in building and maintaining the server on which we obtained our computational results. We are also thankful to the anonymous referee whose suggestions have helped improve the presentation and readability of the paper.

References

  1. 1.
    Besche, H.U., Eick, B., O’Brien, E.: The Small Groups Library (2002). http://www.gap-system.org/Packages/sgl.html. Accessed 15 June 2015
  2. 2.
    Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. Mathematical Surveys No. 7, vol. 1. AMS, Providence (1961)Google Scholar
  3. 3.
    Distler, A., Jefferson, C., Kelsey, T., Hotthoff, L.: The semigroups of order 10. Principles and Practice of Constraint Programming. Lecture Notes in Computer Science, pp. 883–899. Springer, Berlin (2012)Google Scholar
  4. 4.
    Distler, A., Kelsey, T.: The semigroups of order 9 and their automorphism groups. Semigroup Forum 88, 93 (2014).  https://doi.org/10.1007/s00233-013-9504-9 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Distler, A., Mitchellm, J.D.: Smallsemi—a GAP package, version 0.6.5 (2012). https://www.gap-system.org/Packages/smallsemi.html. Accessed 6 Dec 2012
  6. 6.
    Distler, A.: Classification and Enumeration of Finite Semigroups. Ph.D. thesis, University of St Andrews (2010)Google Scholar
  7. 7.
    Ehresmann, C.: Oeuvres complétes et commentées. In Cah. Topol. Géom. Différ. Catég. (Amiens) (1980)Google Scholar
  8. 8.
    Forsythe, G.E.: SWAC computes 126 distinct semigroups of order 4. Proc. Am. Math. Soc. 6(3), 443–447 (1955)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gent, I.P., Jefferson, C., Miguel, I.: Minion: a fast scalable constraint solver. In: Proceedings of 17th European Conference on Artificial Intelligence 2006 (ECAI 06), pp. 98–102 (2006)Google Scholar
  10. 10.
    Green, J.A.: On the structure of semigroups. Ann. Math. 54, 163–172 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Heitzig, J., Reinhold, J.: Counting finite lattices. Algebra Universalis 48(1), 43–53 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jipsen, P., Lawless, N.: Generating all finite modular lattices of a given size. Algebra Univers. 74, 253 (2015).  https://doi.org/10.1007/s00012-015-0348-x MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, Singapore (1998)CrossRefzbMATHGoogle Scholar
  14. 14.
    McKay, B.D., Piperno, A.: Practical graph isomorphism. II. J. Symb. Comput. 60, 94–112 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nambooripad, K.S.S.: Structure of Regular Semigroups I, No. 224, vol. 22. Memoirs of the American Mathematical Society, Providence (1979)zbMATHGoogle Scholar
  16. 16.
    Satoh, S., Yama, K., Tokizawa, M.: Semigroups of order 8. Semigroup Forum 49(1), 7–29 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Schein, B.M.: On the theory of inverse semigroups and generalized grouds. Am. Math. Soc. Transl. Ser. 2 2(113), 89–123 (1979)zbMATHGoogle Scholar
  18. 18.
    Stein, W.A., et al.: Sage Mathematics Software (Version 5.6.0). The Sage Development Team (2013). http://www.sagemath.org. Accessed 1 Feb 2013
  19. 19.
    Steinberg, B.: Möbius functions and semigroup representation theory II: character formulas and multiplicities. Adv. Math. 217, 1521–1557 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    The GAP Group: GAP–Groups, Algorithms, and Programming, Version 4.6.5 (2013)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSam Houston State UniversityHuntsvilleUSA

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