Semigroup Forum

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

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

Research Article


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.


Extended Green’s relations Abundant semigroups Restriction semigroups Generalized inverses 


  1. 1.
    Billhardt, B., Giraldes, E., Marques-Smith, P., Martins, P.: Associate inverse subsemigroups of regular semigroups. Semigroup Forum 79, 101–118 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Blyth, T.S., McFadden, R.: Unit orthodox semigroups. Glasg. Math. J. 24, 39–42 (1983)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blyth, T.S., Giraldes, E., Marques-Smith, P.: Associate subgroups of orthodox semi-groups. Glasg. Math J. 36, 163–171 (1994)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Clifford, A.H.: Semigroups admitting relative inverses. Ann. Math. 42(4), 1037–1049 (1941)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    de Barros, C.M.: Sur les catégories ordonnées régulières. Cah. Topol. Géom. Différ. Catég. 11, 23–55 (1969)MATHGoogle Scholar
  7. 7.
    Drazin, M.P.: A partial order in completely regular semigroups. J. Algebra 98(2), 362–374 (1986)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fountain, J.B.: Right PP monoids with central idempotents. Semigroup Forum 13, 229–237 (1977)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fountain, J.B.: Adequate semigroups. Proc. Edinb. Math. Soc. (2) 22(2), 113–125 (1979)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fountain, J.B.: Abundant semigroups. Proc. Lond. Math. Soc. (3) 44(1), 103–129 (1982)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fountain, J.B., Lawson, M.V.: The translational hull of an adequate semigroup. Semigroup Forum 32(1), 79–86 (1985)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fountain, J.B., Petrich, M.: Completely \(0\)-simple semigroups of quotients. J. Algebra 101, 365–402 (1986)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gomes, G.M.S., Gould, V.: Proper weakly left ample semigroups. Int. J. Algebra Comput. 9, 72–139 (1999)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gould, V.: Straight left orders. Stud. Sci. Math. Hung. 30, 355–373 (1995)MathSciNetMATHGoogle Scholar
  19. 19.
    Gould, V.: Semigroups of left quotients: existence, straightness and locality. J. Algebra 267(2), 514–541 (2003)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Green, J.A.: On the structure of semigroups. Ann. Math. 54(2), 163–172 (1951)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Hartwig, R.: How to partially order regular elements. Math. Jpn. 25, 1–13 (1980)MathSciNetMATHGoogle Scholar
  25. 25.
    Hickey, J.B.: A class of regular semigroups with regularity-preserving elements. Semigroup Forum 81(1), 145–161 (2010)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Hollings, C.: From right PP monoids to restriction semigroups: a survey. Eur. J. Pure Appl. Math. 2(1), 21–57 (2009)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jones, P.R.: On lattices of varieties of restriction semigroups. Semigroup Forum 86(2), 337–361 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kilp, M.: Commutative monoids all of whose principal ideals are projective. Semigroup Forum 6, 334–339 (1973)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lallement, G., Petrich, M.: A generalization of the Rees theorem in semigroups. Acta Sci. Math. (Szeged) 30, 113–132 (1969)MathSciNetMATHGoogle Scholar
  32. 32.
    Lawson, M.V.: The structure of type A semigroups. Q. J. Math. Oxf. Ser. (2) 37(147), 279–298 (1986)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Lawson, M.V.: Abundant Rees matrix semigroups. J. Aust. Math. Soc. Ser. A 42(1), 132–142 (1987)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Lawson, M.V.: Rees matrix semigroups. Proc. Edinb. Math. Soc. (2) 33(1), 23–37 (1990)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Lawson, M.V.: Semigroups and ordered categories I: the reduced case. J. Algebra 141, 422–462 (1991)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Lyapin, E.S.: Semigroups. Translations of Mathematical Monographs, vol. 3. American Mathematical Society, Providence, RI (1963)MATHGoogle 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)MATHGoogle Scholar
  40. 40.
    Miller, D.D., Clifford, A.H.: Regular \(\cal{D}\)-classes in semigroups. Trans. Am. Math. Soc. 82(1), 270–280 (1956)MathSciNetMATHGoogle Scholar
  41. 41.
    Mitsch, H.: A natural partial order on semigroups. Proc. Am. Math. Soc. 97(3), 384–388 (1986)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Nambooripad, K.: The natural partial order on a regular semigroup. Proc. Edinb. Math. Soc. 23, 249–260 (1980)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Petrich, M.: Lectures in Semigroups. Wiley, New York (1977)MATHGoogle Scholar
  44. 44.
    Petrich, M.: A structure theorem for completely regular semigroups. Proc. Am. Math. Soc. 99(4), 617–622 (1987)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Petrich, M.: On weakly ample semigroups. J. Aust. Math. Soc. 97, 404–417 (2014)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Université Paris NanterreNanterreFrance

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