Skip to main content
SpringerLink
Log in

Generation of infinite factorizable inverse monoids

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

Abstract

We investigate the generation of factorizable inverse monoids, paying special attention to the factorizable parts of the symmetric and dual symmetric inverse monoids. Key ideas covered include rank, relative rank, Sierpiński rank, and the semigroup Bergman property. The results for finite monoids are well-known or follow quickly from well-known facts, so most of the paper concerns the infinite case.

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. Chen, S.Y., Hsieh, S.C.: Factorizable inverse semigroups. Semigroup Forum 8(4), 283–297 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Easdown, D., East, J., FitzGerald, D.G.: Braids and factorizable inverse monoids. In: Araújo, I.M., Branco, M.J.J., Fernandes, V.H., Gomes, G.M.S. (eds.) Semigroups and Languages, pp. 86–105. World Scientific, Singapore (2002)

    Google Scholar 

  3. Easdown, D., East, J., FitzGerald, D.G.: Presentations of factorizable inverse monoids. Acta Sci. Math. (Szeged) 71, 509–520 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Easdown, D., East, J., FitzGerald, D.G.: A presentation of the dual symmetric inverse monoid. Int. J. Algebra Comput. 18(2), 357–374 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. East, J.: Dual reflection monoids. In preparation

  6. East, J.: The factorizable braid monoid. Proc. Edinb. Math. Soc. (2) 49(3), 609–636 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. East, J.: Braids and partial permutations. Adv. Math. 213(1), 440–461 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. East, J.: On a class of factorizable inverse monoids associated with braid groups. Commun. Algebra 36(8), 3155–3190 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. East, J.: Infinite dual symmetric inverse monoids. Preprint (2011)

  10. Everitt, B., Fountain, J.: Partial symmetry, reflection monoids and coxeter groups. Adv. Math. 223(5), 1782–1814 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. FitzGerald, D.G.: A presentation for the monoid of uniform block permutations. Bull. Aust. Math. Soc. 68, 317–324 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. FitzGerald, D.G.: Factorizable inverse monoids. Semigroup Forum 80(3), 484–509 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. FitzGerald, D.G., Leech, J.: Dual symmetric inverse semigroups and representation theory. J. Aust. Math. Soc. 64, 146–182 (1998)

    Article  MathSciNet  Google Scholar 

  14. Galvin, F.: Generating countable sets of permutations. J. Lond. Math. Soc. (2) 51(2), 230–242 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Higgins, P., Howie, J., Mitchell, J., Ruškuc, N.: Countable versus uncountable ranks in infinite semigroups of transformations and relations. Proc. Edinb. Math. Soc. (2) 46(3), 531–544 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Howie, J., Ruškuc, N., Higgins, P.: On relative ranks of full transformation semigroups. Commun. Algebra 26(3), 733–748 (1998)

    Article  MATH  Google Scholar 

  17. Jech, T.: Set Theory. The Third Millenium Edition. Springer, Berlin (1999)

    MATH  Google Scholar 

  18. Lawson, M.V.: Inverse Semigroups. The Theory of Partial Symmetries. World Scientific, River Edge (1998)

    Book  MATH  Google Scholar 

  19. Lipscombe, S.: Symmetric Inverse Semigroups. Am. Math. Soc., Providence (1996)

    Google Scholar 

  20. Maltcev, V.: On a new approach to the dual symmetric inverse monoid \(\mathcal{I}_{X}^{*}\). Int. J. Algebra Comput. 17(3), 567–591 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Maltcev, V., Mitchell, J., Ruškuc, N.: The Bergman property for semigroups. J. Lond. Math. Soc. (2) 80(1), 212–232 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. McAlister, D.B.: Embedding inverse semigroups in coset semigroups. Semigroup Forum 20, 255–267 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  23. Moore, E.H.: Concerning the abstract groups of order k! and \(\frac {1}{2}k!\) holohedrically isomorphic with the symmetric and alternating substitution groups on k letters. Proc. Lond. Math. Soc. 28, 357–366 (1897)

    Article  MATH  Google Scholar 

  24. Péress, Y.: Generating uncountable transformation semigroups. Ph.D. thesis, University of St Andrews (2009)

  25. Petrich, M.: Inverse Semigroups. Pure and Applied Mathematics. Wiley, New York (1984)

    Google Scholar 

  26. Schein, B.: Cosets in groups and semigroups. In: Howie, J.M., Munn, W.D., Weinert, H.J. (eds.) Semigroups with Applications, pp. 205–221. World Scientific, Singapore (1992); pp. 86–105 (2002)

    Google Scholar 

  27. Tirasupa, Y.: Factorizable transformation semigroups. Semigroup Forum 18(1), 15–19 (1979)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computing and Mathematics, University of Western Sydney, Locked Bag 1797, Penrith, NSW, 2751, Australia

    James East

Authors
  1. James East
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James East.

Additional information

Communicated by Mark V. Lawson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

East, J. Generation of infinite factorizable inverse monoids. Semigroup Forum 84, 267–283 (2012). https://doi.org/10.1007/s00233-011-9339-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-011-9339-1

Keywords

Search

Navigation