[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Structure theory of regular semigroups

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

Abstract

This survey aims to give an overview of several substantial developments of the last 50 years in the structure theory of regular semigroups and to shed light on their impact on other parts of semigroup theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auinger, K.: On Existence Varieties of Regular Semigroups, Semigroups, Algorithms, Automata and Languages (Coimbra, 2001), pp. 65–89. World Sci. Publ, River Edge (2002)

    Book  MATH  Google Scholar 

  2. Auinger, K., Oliveira, L.: On the variety of strict pseudosemilattices. Stud. Sci. Math. Hung. 50, 207–241 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Billhardt, B.: On embeddability into a semidirect product of a band by a group. J. Algebra 206, 40–50 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Billhardt, B., Szendrei, M.B.: Weakly \(E\)-unitary locally inverse semigroups. J. Algebra 267, 559–576 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Billhardt, B., Szittyai, I.: On embedabbility of idempotent separating extensions of inverse semigroups. Semigroup Forum 61, 26–31 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brittenham, M., Margolis, S.W., Meakin, J.: Subgroups of the free idempotent generated semigroups need not be free. J. Algebra 321, 3026–3042 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Broeksteeg, R.: The set of idempotents of a completely regular semigroup as a binary algebra. Bull. Austral. Math. Soc. 50, 91–107 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clifford, A.H.: The fundamental representation of a regular semigroup. Semigroup Forum 10, 84–92 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. Clifford, A.H.: The fundamental representation of a completely regular semigroup. Semigroup Forum 12, 341–346 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dékány, T., Szendrei, M.B., Szittyai, I.: \(E\)-solid locally inverse semigroups, Acta Sci. Math. accepted

  11. Dolinka, I., Gray, R.D., Ruškuc, N.: On regularity and the word problem for free idempotent generated semigroups. Proc. Lond. Math. Soc. (3) 114, 401–432 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dolinka, I., Ruškuc, N.: Every group is a maximal subgroup of the free idempotent generated semigroup over a band. Int. J. Algebra Comput. 23, 573–581 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Easdown, D.: Biordered sets of bands. Semigroup Forum 29, 241–246 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. Easdown, D.: Biordered sets of eventually regular semigroups. Proc. Lond. Math. Soc. (3) 49, 483–503 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Easdown, D.: Biordered sets come from semigroups. J. Algebra 96, 581–591 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Easdown, D.: Biorder-preserving coextensions of fundamental semigroups. Proc. Edinb. Math. Soc. 31, 463–467 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Easdown, D., Hall, T.E.: Reconstructing some idempotent-generated semigroups from their biordered sets. Semigroup Forum 29, 207–216 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. Easdown, D., Jordan, P., Roberts, B.: Biordered sets and fundamental semigroups. Semigroup Forum 81, 85–101 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Edwards, P.M.: Eventually regular semigroups. Bull. Aust. Math. Soc. 28, 23–38 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Edwards, P.M.: Fundamental semigroups. Proc. R. Soc. Edinb. Sect. A 99, 313–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fountain, J., Pin, J.-E., Weil, P.: Covers for monoids. J. Algebra 271, 529–586 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gould, V.: Notes on restriction semigroups and related structures; formerly (Weakly) left \(E\)-ample semigroups. http://www-users.york.ac.uk/~varg1/restriction.pdf

  23. Gould, V., Yang, D.: Every group is a maximal subgroup of a naturally occurring free idempotent generated semigroup. Semigroup Forum 89, 125–134 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gray, R., Ruskuc, N.: On maximal subgroups of free idempotent generated semigroups. Isr. J. Math. 189, 147–176 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Grillet, P.A.: The structure of regular semigroups. I. A representation. Semigroup Forum 8, 177–183 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  26. Grillet, P.A.: The structure of regular semigroups. II. Cross-connections. Semigroup Forum 8, 254–259 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  27. Grillet, P.A.: The structure of regular semigroups. III. The reduced case. Semigroup Forum 8, 260–265 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  28. Grillet, P.A.: The structure of regular semigroups. IV. The general case. Semigroup Forum 8, 368–373 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  29. Grillet, P.A.: Semigroups. An Introduction to the Structure Theory. Monographs and Textbooks in Pure and Applied Mathematics, vol. 193. Marcel Dekker Inc., New York (1995)

    MATH  Google Scholar 

  30. Hall, T.E.: On orthodox semigroups and uniform and antiuniform bands. J. Algebra 16, 204–217 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hall, T.E.: On regular semigroups. J. Algebra 24, 1–24 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hartmann, M.: Almost factorizable orthodox semigroups. Semigroup Forum 74, 106–124 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hartmann, M., Szendrei, M.B.: \(E\)-unitary almost factorizable orthodox semigroups. Semigroup Forum 84, 157–175 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Howie, J.M.: An Introduction to Semigroup Theory. L.M.S. Monographs, vol. 7. Academic Press, London-New York (1976)

    MATH  Google Scholar 

  35. Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs, New Series, vol. 12. The Clarendon Press, New York (1995)

    MATH  Google Scholar 

  36. Jones, P.R.: An introduction to existence varieties of regular semigroups. Southeast Asian Bull. Math. 19, 107–118 (1995)

    MathSciNet  MATH  Google Scholar 

  37. Kad’ourek, J.: On some existence varieties of locally inverse semigroups. Int. J. Algebra Comput. 6, 761–788 (1996)

    Article  MathSciNet  Google Scholar 

  38. Lawson, M.V.: Inverse Semigroups. The Theory of Partial Symmetries. World Sci. Publishing Co., Inc, River Edge (1998)

    Book  MATH  Google Scholar 

  39. McAlister, D.B.: Rees matrix covers for locally inverse semigroups. Trans. Am. Math. Soc. 277, 727–738 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  40. McAlister, D.B.: Rees matrix covers for regular semigroups. J. Algebra 89, 264–279 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  41. McElwee, B.: Subgroups of the free semigroup on a biordered set in which principal ideals are singletons. Commun. Algebra 30, 5513–5519 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Meakin, J.: The structure mappings on a regular semigroup. Proc. Edinb. Math. Soc. 21, 135–142 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  43. Meakin, J.: The free local semilattice on a set. J. Pure Appl. Algebra 27, 263–275 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  44. Muhammed, P.A.A., Volkov, M.V.: Inductive groupoids and cross-connections of regular semigroups. Acta Math. Hung. 157, 80–120 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  45. Muhammed, P.A.A., Volkov, M.V.: The tale of two categories: inductive groupoids and cross-connections. arXiv:1901.05731v2, (2019)

  46. Nambooripad, K.S.S.: Structure of regular semigroups. PhD thesis, University of Kerala (India) (1973)

  47. Nambooripad, K.S.S.: Structure of regular semigroups. I. Fundamental regular semigroups. Semigroup Forum 9, 354–363 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  48. Nambooripad, K.S.S.: Structure of regular semigroups. II. The general case. Semigroup Forum 9, 364–371 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  49. Nambooripad, K.S.S.: Structure of Regular Semigroups. I, vol. 22(224). American Mathematical Society, Providence (1979)

    MATH  Google Scholar 

  50. Nambooripad, K.S.S.: Pseudosemilattices and biordered sets. I. Simon Stevin 55, 103–110 (1981)

    MathSciNet  MATH  Google Scholar 

  51. Nambooripad, K.S.S.: Structure of Regular Semigroups II. Cross-Connections, vol. 15. Centre for Mathematical Sciences, Trivandrum (1989)

    MATH  Google Scholar 

  52. Nambooripad, K.S.S.: Theory of Cross-Connections, vol. 28. Centre for Mathematical Sciences, Trivandrum (1994)

    MATH  Google Scholar 

  53. Nambooripad, K.S.S.: Theory of Regular Semigroups. Sayahna Foundation, Thiruvananthapuram (2018)

    Google Scholar 

  54. Nambooripad, K.S.S., Pastijn, F.: Subgroups of free idempotent generated regular semigroups. Semigroup Forum 21, 1–7 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  55. Pastijn, F.: The structure of pseudo-inverse semigroups. Trans. Am. Math. Soc. 273, 631–655 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  56. Pastijn, F., Petrich, M.: Straight locally inverse semigroups. Proc. Lond. Math. Soc. (3) 49, 307–328 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  57. Pastijn, F., Petrich, M.: Regular Semigroups as Extensions. Research Notes in Mathematics, vol. 136. Pitman, Boston (1985)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  59. Petrich, M., Reilly, N.R.: Completely Regular Semigroups. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 23. Wiley, New York (1999)

    MATH  Google Scholar 

  60. Šaĭn, B.M.: On the theory of generalized groups and generalized heaps (Russian) theory of Semigroups and Appl. I (Russian), 286–324, Izdat. Saratov. Univ., Saratov, (1965); translation in: Schein, B.M.: On the theory of inverse semigroups and generalized grouds, American Mathematical Society Translations (2) 113, 89–122 (1979)

  61. Szendrei, M.B.: Extensions of regular orthogroups by groups. J. Aust. Math. Soc. Ser. A 59, 28–60 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  62. Szendrei, M.B.: Almost factorizable locally inverse semigroups. Int. J. Algebra Comput. 21, 1037–1052 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  63. Trotter, P.G.: Congruence extensions in regular semigroups. J. Algebra 137, 166–179 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  64. Trotter, P.G.: Covers for regular semigroups and an application to complexity. J. Pure Appl. Algebra 105, 319–328 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  65. Trotter, P.G.: e-Varieties of Regular Semigroups, Semigroups, Automata and Languages (Porto, 1994), pp. 247–262. World Sci. Publ., River Edge (1996)

    MATH  Google Scholar 

  66. Veeramony, R.: Proper pseudo-inverse semigroups. Simon Stevin 58, 65–86 (1984)

    MathSciNet  MATH  Google Scholar 

  67. Wang, Y.: Beyond regular semigroups. Semigroup Forum 92, 414–448 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  68. Yamada, M.: On a certain class of regular semigroups. In: Proceedings of the Symposium of Regular Semigroups (DeKalb), pp. 146–179 (1979)

Download references

Author information

Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, 6720, Hungary

    Mária B. Szendrei

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13–15, Budapest, 1053, Hungary

    Mária B. Szendrei

Authors
  1. Mária B. Szendrei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mária B. Szendrei.

Additional information

Communicated by László Márki.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

M. B. Szendrei was partially supported by the National Research, Development and Innovation Office, Grants K115518 and K128042.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Szendrei, M.B. Structure theory of regular semigroups. Semigroup Forum 100, 119–140 (2020). https://doi.org/10.1007/s00233-019-10055-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-019-10055-8

Keywords

Search

Navigation