Skip to main content
SpringerLink
Log in

Factorizable inverse monoids

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

An Erratum to this article was published on 17 March 2010

Abstract

Factorizable inverse monoids constitute the algebraic theory of those partial symmetries which are restrictions of automorphisms; the formal definition is that each element is the product of an idempotent and an invertible. This class of monoids has theoretical significance, and includes concrete instances which are important in various contexts. This survey is organised around the idea of group acts on semilattices and contains a large range of examples. Topics also include methods for construction of factorizable inverse monoids, and aspects of their inner structure, morphisms, and presentations.

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, 283–297 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  2. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. Mathematical Surveys, No. 7, vols. I, II. Am. Math. Soc., Providence (1964, 1967)

    Google Scholar 

  3. D’Alarcao, H.: Factorizable as a finiteness condition. Semigroup Forum 20, 281–282 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dombi, E.R., Ruškuc, N.: On generators and presentations of semidirect products in inverse semigroups. Bull. Aust. Math. Soc. 79, 353–365 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Easdown, D., Lavers, T.: The inverse braid monoid. Adv. Math. 186, 438–455 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Easdown, D., East, J., FitzGerald, D.G.: Braids and factorizable inverse monoids. In: Semigroups and Languages, pp. 86–105. World Scientific, Singapore (2004)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  8. East, J.: Factorizable inverse monoids of cosets of subgroups of a group. Commun. Algebra 34, 2659–2665 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Everitt, B., Fountain, J.: Partial symmetry, reflection monoids and Coxeter groups. Preprint (2008). arXiv:0812.2789

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

    Article  MATH  MathSciNet  Google Scholar 

  12. FitzGerald, D.G., Leech, J.: Dual symmetric inverse monoids and representation theory. J. Aust. Math. Soc. Ser. A 64, 345–367 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fountain, J.: Unique factorisation monoids. Personal communication (2008)

  14. Gilbert, N.D., Wazzan, S.: Zappa-Szép products of bands and groups. Semigroup Forum 77, 438–455 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Howie, J.M.: Fundamentals of Semigroup Theory. L.M.S. Monographs, New Series, No. 12. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  16. Joubert, G.: Contribution á L’étude des Catégories Ordonnées. Application aux Structures Feuillètes. Topologie et Géométrie Différentielle, vol. 8. Centre Nat. Recherche Sci., Paris (1966)

    Google Scholar 

  17. Kaarli, K.: Arithmetical affine complete varieties and inverse monoids. Stud. Sci. Math. Hung. Akad. Kiadó 45, 13–28 (2008)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Kudryavtseva, G., Mazorchuk, V.: Partialization of categories and inverse braid-permutation monoids. Int. J. Algebra Comput. 18, 989–1017 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lawson, M.V.: Almost factorisable inverse semigroups. Glasg. Mat. J. 36, 97–111 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  22. Lawson, M.V.: Ordered groupoids and left cancellative categories. Semigroup Forum 68, 458–476 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Leech, J.: Inverse monoids with a natural semilattice ordering. Proc. Lond. Math. Soc. (3) 70, 146–182 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lipscomb, S.L.: The alternating semigroups: generators and congruences. Semigroup Forum 44, 96–106 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lipscomb, S.L.: The infinite alternating semigroup. Semigroup Forum 55, 118–130 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lipscomb, S.L., Konieczny, J.: The class equation and counting in factorizable monoids. Proc. Am. Math. Soc. 131, 3345–3351 (2003) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  27. McAlister, D.B.: Groups, semilattices and inverse semigroups I, II. Trans. Am. Math. Soc. 192, 227–244 (1974); ibid. 196, 351–370 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  28. McAlister, D.B.: Some covering and embedding theorems for inverse semigroups. J. Aust. Math. Soc. Ser. A 22, 188–211 (1976)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  30. McAlister, D.B.: Semigroups generated by a group and an idempotent. Commun. Algebra 26, 515–547 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  31. McAlister, D.B., Reilly, N.R.: E-unitary covers for inverse semigroups. Pac. J. Math. 68, 161–174 (1977)

    MATH  MathSciNet  Google Scholar 

  32. Mills, J.E.: Combinatorially factorizable inverse monoids. Semigroup Forum 59, 220–232 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  34. Putcha, M.S.: Linear Algebraic Monoids. London Mathematical Society Lecture Note Series, vol. 133. Cambridge University Press, Cambridge (1988)

    MATH  Google Scholar 

  35. Rédei, L.: Zur Theorie der faktorisierbaren Gruppen I. Acta Math. Acad. Sci. Hung. 1, 74–98 (1950)

    Article  MATH  Google Scholar 

  36. Renner, L.E.: Linear Algebraic Monoids. Encyclopedia of Mathematical Sciences, vol. 134. Springer, Berlin (2005). Invariant Theory and Algebraic Transformation Groups, V

    MATH  Google Scholar 

  37. Schein, B.M.: Semigroups of strong subsets. Volž. Mat. Sb. 4, 180–186 (1966) (in Russian)

    Google Scholar 

  38. Solomon, L.: An introduction to reductive monoids. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups (Proc. of NATO Adv. Sci. Inst., York 1993), pp. 295–352. Kluwer Academic, Dordrecht (1995)

    Google Scholar 

  39. Szendrei, M.B.: A note on Birget-Rhodes expansion of groups. J. Pure Appl. Algebra 58, 93–99 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  40. Szép, J.: Zur Theorie der faktorisierbaren Gruppe. Publ. Math. Debr. 2, 43–45 (1951)

    MATH  Google Scholar 

  41. Tirasupa, Y.: Weakly factorizable inverse semigroups. Semigroup Forum 18, 283–291 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  42. Tolo, K.: Factorizable semigroups. Pac. J. Math. 31, 523–535 (1969)

    MATH  MathSciNet  Google Scholar 

  43. Zappa, G.: Sulla costruzione dei gruppi prodotto di duedati sottogruppi permutabili tra loro. In: Atti Secondo Congresso Un. Mat. Ital., Bologna 1940, pp. 119–125. Edizioni Cremonense, Rome (1942).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, 7001, Australia

    D. G. FitzGerald

Authors
  1. D. G. FitzGerald
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. G. FitzGerald.

Additional information

Communicated by Norman R. Reilly.

In memory of Douglas Munn.

An erratum to this article can be found at http://dx.doi.org/10.1007/s00233-010-9216-3

Rights and permissions

Reprints and permissions

About this article

Cite this article

FitzGerald, D.G. Factorizable inverse monoids. Semigroup Forum 80, 484–509 (2010). https://doi.org/10.1007/s00233-009-9177-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-009-9177-6

Keywords

Search

Navigation