Advertisement

Inductive groupoids and cross-connections of regular semigroups

  • P. A. Azeef Muhammed
  • M. V. Volkov
Article
  • 10 Downloads

Abstract

There are two major structure theorems for an arbitrary regular semigroup using categories, both due to Nambooripad. The first construction using inductive groupoids departs from the biordered set structure of a given regular semigroup. This approach belongs to the realm of the celebrated Ehresmann–Schein–Nambooripad Theorem and its subsequent generalisations. The second construction is a generalisation of Grillet’s work on cross-connected partially ordered sets, arising from the principal ideals of the given semigroup. In this article, we establish a direct equivalence between these two seemingly different constructions. We show how the cross-connection representation of a regular semigroup may be constructed directly from the inductive groupoid of the semigroup, and vice versa.

Key words and phrases

regular semigroup biordered set inductive groupoid crossconnection normal category 

Mathematics Subject Classification

20M10 20M17 20M50 18B40 06A75 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We are very grateful to J. Meakin, University of Nebraska-Lincoln, for reading an initial draft of the article and helping us with several enlightening suggestions. We also thank the referee for the careful reading of the article and the detailed comments which helped us to improve the manuscript.

References

  1. 1.
    Armstrong, S.: Structure of concordant semigroups. J. Algebra 118, 205–260 (1988)MathSciNetCrossRefGoogle Scholar
  2. 2.
    K. Auinger, The bifree locally inverse semigroup on a set, J. Algebra, 166 (1994), 630–650MathSciNetCrossRefGoogle Scholar
  3. 3.
    K. Auinger, On the bifree locally inverse semigroup, J. Algebra, 178 (1995), 581–613MathSciNetCrossRefGoogle Scholar
  4. 4.
    P. A. Azeef Muhammed, Cross-connections and variants of the full transformation semigroup, Acta Sci. Math. (Szeged) (2018), to appear; arXiv:1703.04139
  5. 5.
    P. A. Azeef Muhammed, Cross-connections of linear transformation semigroups, Semigroup Forum (2018),  https://doi.org/10.1007/s00233-018-9942-5.
  6. 6.
    P. A. Azeef Muhammed, K. S. S. Nambooripad, and P. G. Romeo, Cross-connection structure of concordant semigroups (2018), submitted; arXiv:1806.11031
  7. 7.
    P. A. Azeef Muhammed and A. R. Rajan, Cross-connections of the singular transformation semigroup, J. Algebra Appl., 17(2018), 1850047Google Scholar
  8. 8.
    P. A. Azeef Muhammed and M. V. Volkov, Inductive groupoids and cross-connections (2018), in preparationGoogle Scholar
  9. 9.
    M. Brittenham, S. W. Margolis, and J. C. Meakin, Subgroups of free idempotent generated semigroups need not be free, J. Algebra, 321 (2009), 3026–3042MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. H. Clifford, The partial groupoid of idempotents of a regular semigroup, Semigroup Forum, 10 (1975), 262–268MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. 1, Mathematical Surveys, 7, American Mathematical Society (Providence, RI, 1961)Google Scholar
  12. 12.
    Y. Dandan and V. Gould, Free idempotent generated semigroups over bands and biordered sets with trivial products, Internat. J. Algebra Comput., 26 (2016), 473–507MathSciNetCrossRefGoogle Scholar
  13. 13.
    I. Dolinka and R. Gray, Maximal subgroups of free idempotent generated semigroups over the full linear monoid, Trans. Amer. Math. Soc., 366 (2014), 419–455MathSciNetCrossRefGoogle Scholar
  14. 14.
    D. Easdown, Biordered sets come from semigroups, J. Algebra, 96 (1985), 581–591MathSciNetCrossRefGoogle Scholar
  15. 15.
    D. Easdown, M. V. Sapir, and M. V. Volkov, Periodic elements of the free idempotent generated semigroup on a biordered set, Internat. J. Algebra Comput., 20 (2010), 189–194MathSciNetCrossRefGoogle Scholar
  16. 16.
    C. Ehresmann, Gattungen von lokalen Strukturen, Jahresber. Deutsch. Math.-Verein., 60 (1957), 49–77MathSciNetzbMATHGoogle Scholar
  17. 17.
    C. Ehresmann, Categories inductives et pseudogroupes, Ann. Inst. Fourier (Grenoble), 10 (1960), 307–336MathSciNetCrossRefGoogle Scholar
  18. 18.
    G. M. Gomes and V. Gould, Fundamental Ehresmann semigroups, Semigroup Forum, 63 (2001), 11–33MathSciNetCrossRefGoogle Scholar
  19. 19.
    V. Gould, Restriction and Ehresmann semigroups, in: Proceedings of the International Conference on Algebra 2010: Advances in Algebraic Structures, World Scientific (2011), p. 265Google Scholar
  20. 20.
    V. Gould and C. Hollings, Restriction semigroups and inductive constellations, Comm. Algebra, 38 (2009), 261–287MathSciNetCrossRefGoogle Scholar
  21. 21.
    V. Gould and Y. Wang, Beyond orthodox semigroups, J. Algebra, 368 (2012), 209–230MathSciNetCrossRefGoogle Scholar
  22. 22.
    R. Gray and N. Ruskuc, Maximal subgroups of free idempotent generated semigroups over the full transformation monoid, Proc. London Math. Soc., 104 (2012), 997–1018MathSciNetCrossRefGoogle Scholar
  23. 23.
    R. Gray and N. Ruskuc, On maximal subgroups of free idempotent generated semigroups, Israel J. Math., 189 (2012), 147–176MathSciNetCrossRefGoogle Scholar
  24. 24.
    P. A. Grillet, Structure of regular semigroups: A representation, Semigroup Forum, 8 (1974), 177–183MathSciNetCrossRefGoogle Scholar
  25. 25.
    P. A. Grillet, Structure of regular semigroups: Cross-connections, Semigroup Forum, 8 (1974), 254–259MathSciNetCrossRefGoogle Scholar
  26. 26.
    P. A. Grillet, Structure of regular semigroups: The reduced case, Semigroup Forum, 8 (1974), 260–265MathSciNetCrossRefGoogle Scholar
  27. 27.
    P. A. Grillet, Semigroups: An Introduction to the Structure Theory, CRC Pure and Applied Mathematics, Taylor & Francis (1995)zbMATHGoogle Scholar
  28. 28.
    T. E. Hall, On regular semigroups, J. Algebra, 24 (1973), 1–24MathSciNetCrossRefGoogle Scholar
  29. 29.
    T. E. Hall, Identities for existence varieties of regular semigroups, Bull. Austral. Math. Soc., 40 (1989), 59–77MathSciNetCrossRefGoogle Scholar
  30. 30.
    R. E. Hartwig, How to partially order regular elements, Math. Japon., 25 (1980), 1–13MathSciNetzbMATHGoogle Scholar
  31. 31.
    P. J. Higgins, Notes on Categories and Groupoids, Van Nostrand (Reinhold, 1971)Google Scholar
  32. 32.
    C. Hollings, From right PP-monoids to restriction semigroups: a survey, European J. Pure Appl. Math., 2 (2009), 21–57MathSciNetzbMATHGoogle Scholar
  33. 33.
    C. Hollings, The Ehresmann-Schein-Nambooripad theorem and its successors, European J. Pure Appl. Math., 5 (2012), 414–450MathSciNetzbMATHGoogle Scholar
  34. 34.
    J. Kadourek and M. Szendrei, A new approach in the theory of orthodox semigroups, Semigroup Forum, 40 (1990), 257–296MathSciNetCrossRefGoogle Scholar
  35. 35.
    M. V. Lawson, Semigroups and ordered categories. I. the reduced case, J. Algebra, 141 (1991), 422–462MathSciNetCrossRefGoogle Scholar
  36. 36.
    M. V. Lawson, Enlargements of regular semigroups, Proc. Edinb. Math. Soc. (2), 39 (1996), 425–460MathSciNetCrossRefGoogle Scholar
  37. 37.
    M. V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific Pub. Co., Inc. (1998)Google Scholar
  38. 38.
    M. V. Lawson, Ordered groupoids and left cancellative categories, Semigroup Forum, 68 (2004), 458–476MathSciNetCrossRefGoogle Scholar
  39. 39.
    S. Lukose and A. R. Rajan, Ring of normal cones, Indian J. Pure Appl. Math., 41 (2010), 663–681MathSciNetCrossRefGoogle Scholar
  40. 40.
    S. MacLane, Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, Springer-Verlag (New York, 1971)Google Scholar
  41. 41.
    J. C. Meakin, On the structure of inverse semigroups, Semigroup Forum, 12 (1976), 6–14MathSciNetCrossRefGoogle Scholar
  42. 42.
    J. C. Meakin, The structure mappings on a regular semigroup, Proc. Edinb. Math. Soc. (2), 21 (1978), 135–142MathSciNetCrossRefGoogle Scholar
  43. 43.
    J. C. Meakin, Structure mappings, coextensions and regular four-spiral semigroups, Trans. Amer. Math. Soc., 255 (1979), 111–134MathSciNetCrossRefGoogle Scholar
  44. 44.
    J. C. Meakin and A. R. Rajan, Tribute to K. S. S. Nambooripad, Semigroup Forum, 91 (2015), 299–304Google Scholar
  45. 45.
    H. Mitsch, A natural partial order for semigroups, Proc. Amer. Math. Soc., 97 (1986), 384–388MathSciNetCrossRefGoogle Scholar
  46. 46.
    W. D. Munn, Fundamental inverse semigroups, Q. J. Math., 21 (1970), 157–170MathSciNetCrossRefGoogle Scholar
  47. 47.
    K. S. S. Nambooripad, Structure of Regular Semigroups, PhD thesis, University of Kerala (India, 1973)Google Scholar
  48. 48.
    K. S. S. Nambooripad, Structure of regular semigroups. I. Fundamental regular semigroups, Semigroup Forum, 9 (1975), 354–363MathSciNetCrossRefGoogle Scholar
  49. 49.
    K. S. S. Nambooripad, Structure of regular semigroups. II. The general case, Semigroup Forum, 9 (1975), 364–371MathSciNetCrossRefGoogle Scholar
  50. 50.
    K. S. S. Nambooripad, Relations between cross-connections and biordered sets, Semigroup Forum, 16 (1978), 67–82MathSciNetCrossRefGoogle Scholar
  51. 51.
    K. S. S. Nambooripad, Structure of Regular Semigroups. I, Mem. Amer. Math. Soc., 224, American Mathematical Society (Providence, RI, 1979)Google Scholar
  52. 52.
    K. S. S. Nambooripad, The natural partial order on a regular semigroup, Proc. Edinb. Math. Soc. (2), 23 (1980), 249–260MathSciNetCrossRefGoogle Scholar
  53. 53.
    K. S. S. Nambooripad, Structure of Regular Semigroups. II. Cross-connections, Publication No. 15, Centre for Mathematical Sciences (Thiruvananthapuram, 1989)Google Scholar
  54. 54.
    K. S. S. Nambooripad, Theory of Cross-connections, Publication No. 28, Centre for Mathematical Sciences (Thiruvananthapuram, 1994)Google Scholar
  55. 55.
    K. S. S. Nambooripad, Cross-connections, in: Proceedings of the International Symposium on Semigroups and Applications, University of Kerala (Thiruvananthapuram, 2007), pp. 1–25Google Scholar
  56. 56.
    K. S. S. Nambooripad, Cross-connections (2014), www.sayahna.org/crs/
  57. 57.
    K. S. S. Nambooripad, Theory of Regular Semigroups, Sayahna Foundation (Thiruvananthapuram, 2018)Google Scholar
  58. 58.
    K. S. S. Nambooripad and F. Pastijn, Subgroups of free idempotent generated regular semigroups, Semigroup Forum, 21 (1980), 1–7MathSciNetCrossRefGoogle Scholar
  59. 59.
    M. S. Putcha, Linear Algebraic Monoids, London Mathematical Society Lecture Note Series, 133, Cambridge University Press (1988)Google Scholar
  60. 60.
    A. R. Rajan, Certain categories derived from normal categories, in: Semigroups, Algebras and Operator Theory (Kochi, India, February 2014), (P. G. Romeo, J. C. Meakin, and A. R. Rajan, editors), Springer (India, 2015), pp. 57–66.Google Scholar
  61. 61.
    A. R. Rajan, Structure theory of regular semigroups using categories, in: Algebra and its Applications: ICAA (Aligarh, India, December 2014), (S. T. Rizvi, A. Ali, and V. D. Filippis, editors), Springer (Singapore, 2016), pp. 259–264Google Scholar
  62. 62.
    A. R. Rajan, Inductive groupoids and normal categories of regular semigroups, Algebra and its Applications (International Conference at AMU, Aligarh, India, 2016), (M. Ashraf, V. D. Filippis, and S. T. Rizvi, editors), De Gruyter (Boston, Berlin, 2018), pp. 193–200Google Scholar
  63. 63.
    P. G. Romeo, Cross connections of Concordant Semigroups, PhD thesis, University of Kerala (India, 1993).Google Scholar
  64. 64.
    P. G. Romeo, Concordant semigroups and balanced categories, Southeast Asian Bull. Math., 31 (2007), 949–961MathSciNetzbMATHGoogle Scholar
  65. 65.
    B. M. Schein, On the theory of generalised groups and generalised heaps, in: The Theory of Semigroups and its Applications, Saratov State University, Russia (1965), pp. 286–324 (in Russian)Google Scholar
  66. 66.
    B. M. Schein, On the theory of inverse semigroups and generalised grouds, Amer. Math. Soc. Transl. Ser. 2, 113 (1979), 89–122 (English translation)Google Scholar
  67. 67.
    M. Szendrei, The bifree regular E-solid semigroups, Semigroup Forum, 52 (1996), 61–82MathSciNetCrossRefGoogle Scholar
  68. 68.
    S. Wang, An Ehresmann–Schein–Nambooripad-type theorem for a class of P-restriction semigroups, Bull. Malays. Math. Sci. Soc. (2017), 1–34Google Scholar
  69. 69.
    Y. Wang, Beyond regular semigroups, Semigroup Forum, 92 (2016), 414–448MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Institute of Natural Sciences and MathematicsUral Federal UniversityEkaterinburgRussia

Personalised recommendations