Advertisement

Semigroup Forum

, Volume 94, Issue 3, pp 738–776 | Cite as

On \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups and their subclasses

  • Xavier Mary
Research Article

Abstract

We study semigroups that behave nicely with respect to a distinguished subset of idempotents E, both in terms of the extended Green’s relations \(\widetilde{\mathcal {K}}_E\) and as unary semigroups. New structure theorems are given, notably in the case of central idempotents. Finally, the decomposition theorems are applied to the study of regular semigroups with particular generalized inverses.

Keywords

Extended Green’s relations Abundant semigroups Restriction semigroups Generalized inverses 

References

  1. 1.
    Billhardt, B., Giraldes, E., Marques-Smith, P., Martins, P.: Associate inverse subsemigroups of regular semigroups. Semigroup Forum 79, 101–118 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blyth, T.S., McFadden, R.: Unit orthodox semigroups. Glasg. Math. J. 24, 39–42 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blyth, T.S., Giraldes, E., Marques-Smith, P.: Associate subgroups of orthodox semi-groups. Glasg. Math J. 36, 163–171 (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, Y., He, Y., Shum, K.P.: Projectively condensed semigroups, generalized completely regular semigroups and projective orthomonoids. Acta Math. Hungar. 119(3), 281–305 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Clifford, A.H.: Semigroups admitting relative inverses. Ann. Math. 42(4), 1037–1049 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    de Barros, C.M.: Sur les catégories ordonnées régulières. Cah. Topol. Géom. Différ. Catég. 11, 23–55 (1969)zbMATHGoogle Scholar
  7. 7.
    Drazin, M.P.: A partial order in completely regular semigroups. J. Algebra 98(2), 362–374 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Du, L., Guo, Y.Q., Shum, K.P.: Some remarks on \((l)\)-Green’s relations and strongly RPP semigroups. Acta Math. Sci. Ser. B Engl. Ed. 31(4), 1591–1599 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    El-Qallali, A.: Structure theory for abundant and related semigroups. Ph.D. thesis, University of York (1980)Google Scholar
  10. 10.
    Fountain, J.B.: A class of right PP monoids. Q. J. Math. Oxf. (2) 28, 285–305 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fountain, J.B.: Right PP monoids with central idempotents. Semigroup Forum 13, 229–237 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fountain, J.B.: Adequate semigroups. Proc. Edinb. Math. Soc. (2) 22(2), 113–125 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fountain, J.B.: Abundant semigroups. Proc. Lond. Math. Soc. (3) 44(1), 103–129 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fountain, J.B., Lawson, M.V.: The translational hull of an adequate semigroup. Semigroup Forum 32(1), 79–86 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fountain, J.B., Petrich, M.: Completely \(0\)-simple semigroups of quotients. J. Algebra 101, 365–402 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fountain, J.B., Gomes, G.M.S., Gould, V.: A Munn type representation for a class of \(E\)-semiadequate semigroups. J. Algebra 218(2), 693–714 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gomes, G.M.S., Gould, V.: Proper weakly left ample semigroups. Int. J. Algebra Comput. 9, 72–139 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gould, V.: Straight left orders. Stud. Sci. Math. Hung. 30, 355–373 (1995)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gould, V.: Semigroups of left quotients: existence, straightness and locality. J. Algebra 267(2), 514–541 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gould, V., Szendrei, M.B.: Proper restriction semigroups, semidirect products and W-products. Acta Math. Hung. 141(1–2), 36–57 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Green, J.A.: On the structure of semigroups. Ann. Math. 54(2), 163–172 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grillet, P.A.: Commutative Semigroups. Advances in Mathematics, vol. 2. Kluwer Academic Publishers, Dordrecht (2001)CrossRefGoogle Scholar
  23. 23.
    Guo, X., Guo, Y., Shum, K.P.: Left abundant semigroups. Commun. Algebra 32(6), 2061–2085 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hartwig, R.: How to partially order regular elements. Math. Jpn. 25, 1–13 (1980)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hickey, J.B.: A class of regular semigroups with regularity-preserving elements. Semigroup Forum 81(1), 145–161 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hollings, C.: From right PP monoids to restriction semigroups: a survey. Eur. J. Pure Appl. Math. 2(1), 21–57 (2009)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs. New Series, 12. The Clarendon Press, Oxford University Press, New York (1995)Google Scholar
  28. 28.
    Jackson, M., Stokes, T.: An invitation to C-semigroups. Semigroup Forum 62(2), 279–310 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jones, P.R.: On lattices of varieties of restriction semigroups. Semigroup Forum 86(2), 337–361 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kilp, M.: Commutative monoids all of whose principal ideals are projective. Semigroup Forum 6, 334–339 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lallement, G., Petrich, M.: A generalization of the Rees theorem in semigroups. Acta Sci. Math. (Szeged) 30, 113–132 (1969)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Lawson, M.V.: The structure of type A semigroups. Q. J. Math. Oxf. Ser. (2) 37(147), 279–298 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lawson, M.V.: Abundant Rees matrix semigroups. J. Aust. Math. Soc. Ser. A 42(1), 132–142 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lawson, M.V.: Rees matrix semigroups. Proc. Edinb. Math. Soc. (2) 33(1), 23–37 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lawson, M.V.: Semigroups and ordered categories I: the reduced case. J. Algebra 141, 422–462 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Liber, A.E.: On the theory of generalized groups. Dokl. Akad. Nauk. SSSR 97, 25–28 (1954). (Russian)MathSciNetGoogle Scholar
  37. 37.
    Lopez Jr., A.M.: The maximal right quotient semigroup of a strong semilattice of semigroups. Pac. J. Math. 71(2), 477–485 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Lyapin, E.S.: Semigroups. Translations of Mathematical Monographs, vol. 3. American Mathematical Society, Providence, RI (1963)zbMATHGoogle Scholar
  39. 39.
    Ma, S.Y., Ren, X.M., Yuan, Y.: On completely \(\widetilde{\cal{J}}\)-simple semigroups. Acta Math. Sin. (Chin. Ser.) 54(4), 643–650 (2011)zbMATHGoogle Scholar
  40. 40.
    Miller, D.D., Clifford, A.H.: Regular \(\cal{D}\)-classes in semigroups. Trans. Am. Math. Soc. 82(1), 270–280 (1956)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Mitsch, H.: A natural partial order on semigroups. Proc. Am. Math. Soc. 97(3), 384–388 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Nambooripad, K.: The natural partial order on a regular semigroup. Proc. Edinb. Math. Soc. 23, 249–260 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Petrich, M.: Lectures in Semigroups. Wiley, New York (1977)zbMATHGoogle Scholar
  44. 44.
    Petrich, M.: A structure theorem for completely regular semigroups. Proc. Am. Math. Soc. 99(4), 617–622 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Petrich, M.: On weakly ample semigroups. J. Aust. Math. Soc. 97, 404–417 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Ren, X.M., Shum, K.P.: On superabundant semigroups whose set of idempotents forms a subsemigroup. Algebra Colloq. 14(2), 215–228 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Ren, X.M., Shum, K.P., Guo, Y.Q.: A generalized Clifford theorem of semigroups. Sci. China A 53, 1097–1101 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Sutov, E.G.: Potential divisibility of elements in semigroups. Leningr. Gosud. Ped. Inst. Uc. Zap. 166, 105–119 (1958). (Russian)MathSciNetGoogle Scholar
  49. 49.
    Wang, Y., Ren, X.M., Ma, S.Y.: The translational hull of superabundant semigroups with semilattice of idempotents. Sci. Magna 2(4), 75–80 (2006)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Wang, Y.: Beyond regular semigroups. Ph.D. Thesis, University of York (2012)Google Scholar
  51. 51.
    Wang, Y.: Beyond regular semigroups. Semigroup Forum 92(2), 414–448 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Yuan, Y., Gong, C., Ma, S.Y.: The structure of \(U\)-superabundant semigroups and the translational hull of completely \(\widetilde{\cal{J}}\)-simple semigroups. Adv. Math. (China) 1, 35–47 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Université Paris NanterreNanterreFrance

Personalised recommendations