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Semigroup Forum

, Volume 88, Issue 3, pp 647–669 | Cite as

Classes of semigroups modulo Green’s relation \(\mathcal{H}\)

  • Xavier Mary
RESEARCH ARTICLE

Abstract

Inverse semigroups and orthodox semigroups are either defined in terms of inverses, or in terms of the set of idempotents E(S). In this article, we study analogs of these semigroups defined in terms of inverses modulo Green’s relation \(\mathcal{H}\), or in terms of the set of completely regular elements H(S). Results are obtained both for the regular and the non-regular cases. We then study the interplays between these new classes of semigroups, as well as with various known classes notably of inverse, orthodox, E-solid and cryptic semigroups.

Keywords

Generalized inverses Green’s relations Semigroups 

References

  1. 1.
    Almeida, J., Pin, J.-E., Weil, P.: Semigroups whose idempotents form a subsemigroup. Math. Proc. Camb. Philos. Soc. 111(2), 241–253 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Auinger, K., Hall, T.E., Reilly, N.R., Zhang, S.: Congruences on the lattice of pseudovarieties of finite semigroups. Int. J. Algebra Comput. 7(4), 433–455 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ben Israel, A., Greville, T.N.E.: Generalized Inverses, Theory and Applications, 2nd edn. Springer, New York (2003) zbMATHGoogle Scholar
  4. 4.
    Birget, J.-C., Margolis, S., Rhodes, J.: Semigroups whose idempotents form a subsemigroup. Bull. Aust. Math. Soc. 41(2), 161–184 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Blyth, T.S., Almeida Santos, M.H.: On quasi-orthodox semigroups with inverse transversals. Proc. Edinb. Math. Soc. (2) 40(3), 505–514 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Fountain, J.: E-unitary dense covers of E-dense monoids. Bull. Lond. Math. Soc. 22(4), 353–358 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gould, V.: Left orders in regular H-semigroups. I. J. Algebra 141(1), 11–35 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Green, J.A.: On the structure of semigroups. Ann. Math. 54(1), 163–172 (1951) CrossRefzbMATHGoogle Scholar
  9. 9.
    Hall, T.E.: On regular semigroups. J. Algebra 24, 1–24 (1973) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hall, T.E.: Some properties of local subsemigroups inherited by larger subsemigroups. Semigroup Forum 25(1–2), 35–49 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hartwig, R.E.: Block generalized inverses. Arch. Ration. Mech. Anal. 61(3), 197–251 (1976) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Howie, J.M., Lallement, G.: Certain fundamental congruences on a regular semigroup. Proc. Glasg. Math. Assoc. 7, 145–159 (1964) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs. New Series, vol. 12. Oxford Science Publications, Oxford (1995) zbMATHGoogle Scholar
  14. 14.
    Kadourek, J., Szendrei, M.B.: On existence varieties of E-solid semigroups. Semigroup Forum 59(3), 470–521 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Liber, A.E.: On the theory of generalized groups. Dokl. Akad. Nauk SSSR 97, 25–28 (1954) (In Russian) zbMATHMathSciNetGoogle Scholar
  16. 16.
    McAlister, D.B.: Rees matrix covers for regular semigroups. J. Algebra 89, 264–279 (1984) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Margolis, S.W., Pin, J.-E.: Inverse semigroups and extensions of groups by semilattices. J. Algebra 110(2), 277–297 (1987) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Mary, X.: On generalized inverses and Green’s relations. Linear Algebra Appl. 434(8), 1836–1844 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Mary, X.: Natural generalized inverse and core of an element in semi-groups, rings and Banach and operator algebras. Eur. J. Pure Appl. Math. 5(2), 160–173 (2012) MathSciNetGoogle Scholar
  20. 20.
    Mary, X., Patricio, P.: Generalized invertibilty modulo \(\mathcal{H}\) in semigroups and rings. Linear and Multilinear Algebra 61(8), 1130–1135 (2012) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Miller, D.D., Clifford, A.H.: Regular \(\mathcal{D}\)-classes in semigroups. Trans. Am. Math. Soc. 82(1), 270–280 (1956) zbMATHMathSciNetGoogle Scholar
  22. 22.
    Munn, W.D., Penrose, R.: A note on inverse semigroups. Proc. Camb. Philos. Soc. 51, 396–399 (1955) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Nagy, A.: Special Classes of Semigroups. Advances in Mathematics, vol. 1. Kluwer Academic, Dordrecht (2001) CrossRefzbMATHGoogle Scholar
  24. 24.
    Petrich, M., Reilly, N.R.: A network of congruences on an inverse semigroup. Trans. Am. Math. Soc. 270, 309–325 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Petrich, M.: Inverse Semigroups. Wiley, New York (1984) zbMATHGoogle Scholar
  26. 26.
    Petrich, M.: Onesided inverses for semigroups. Acta Math. Univ. Comen. (NS) 75(1), 1–19 (2006) zbMATHMathSciNetGoogle Scholar
  27. 27.
    Reilly, N.R., Scheiblich, H.E.: Congruences on regular semigroups. Pac. J. Math. 23, 349–360 (1967) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Reilly, N.R.: Free generators in free inverse semigroups. Bull. Aust. Math. Soc. 7, 407–424 (1972) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Reilly, N.R.: Minimal non-cryptic varieties of inverse semigroups. Q. J. Math. 36(4), 467–487 (1985) CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Scheiblich, H.E.: Kernels of inverse semigroups homomorphisms. J. Aust. Math. Soc. A 18, 289–292 (1974) CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Tully, E.J. Jr: \(\mathcal{H}\)-commutative semigroups in which each homomorphism is uniquely determined by its kernel. Pac. J. Math. 45(2), 669–679 (1973) CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Wagner, V.V.: Generalized groups. Dokl. Akad. Nauk SSSR 84, 1119–1122 (1952) (In Russian) Google Scholar
  33. 33.
    Warne, R.J.: Standard regular semigroups. Pac. J. Math. 65(2), 539–562 (1976) CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Yamada, M., Shoji, K.: On \(\mathcal{H}\)-compatible quasi-orthodox semigroups. Mem. Fac. Sci. Shimane Univ. 13, 1–9 (1979) zbMATHMathSciNetGoogle Scholar
  35. 35.
    Yamada, M.: On the structure of regular semigroups in which the maximal subgroups form a band of groups. In: Semigroups, Proc. Conf., Monash Univ., Clayton, 1979, pp. 47–55. Academic Press, New York (1980) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Université Paris-Ouest Nanterre-La Défense, Laboratoire Modal’XParisFrance

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