A local structure theorem for stable, \(\mathcal {J}\)-simple semigroup biacts

  • Xavier MaryEmail author


We describe a class of semigroup biacts that is analogous to the class of completely simple semigroups, and prove a structure theorem for those biacts that is analogous to the Rees–Sushkevitch Theorem. Precisely, we describe stable, \(\mathcal {J}\)-simple biacts in terms of wreath products, translations of completely simple semigroups, biacts over endomorphism monoids of free G-acts, tensor products and matrix biacts. Applications to coproducts and left acts are given.


Semigroup acts Green’s relations Stability Wreath products Endomorphism monoid of free G-acts Completely simple semigroups Rees matrix semigroups 



This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01). It also benefited from very fruitful conversations with members of the Department of Mathematics, University of York, while the present author was visiting the University.


  1. 1.
    Adámek, J., Rosickỳ, J., Vitale, E.: Algebraic Theories, vol. 184, p. 1. Cambridge University Press, Cambridge (2011)zbMATHGoogle Scholar
  2. 2.
    Andersen, O.: Ein Bericht über die Struktur abstrakter Halbgruppen. Staatsexamensarbeit thesis, Hamburg (1952)Google Scholar
  3. 3.
    Avdeyev, A., Kožukhov, I.: Acts over completely 0-simple semigroups. Acta Cybern. 14(4), 523–531 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chen, G. Interpolationseigenschaften von Halbgruppen und \(\Omega \)-Halbgruppen. PhD thesis. Dissertation, TU Wien (1997)Google Scholar
  5. 5.
    Chen, G.: The endomorphism structure of simple faithful S-acts. Semigroup Forum 59(2), 179–182 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Clifford, A.: Matrix representations of completely simple semigroups. Am. J. Math. 64(1), 327–342 (1942)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Clifford, A., Petrich, M.: Some classes of completely regular semigroups. J. Algebra 46(2), 462–480 (1977)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Green, J.A.: On the structure of semigroups. Ann. Math. 54, 163–172 (1951)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Grillet, P.A.: Semigroups: An Introduction to the Structure Theory, vol. 193. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  10. 10.
    Grillet, P.A.: A coherence theorem on Schützenberger groups. J. Aust. Math. Soc. 21(2), 129–143 (1976)CrossRefGoogle Scholar
  11. 11.
    Kilp, M., Knauer, U., Mikhalev, A.V.: Monoids, acts and categories. De Gruyter Expositions in Mathematics, vol. 29. Walter de Gruyter, Berlin (2000)Google Scholar
  12. 12.
    Krasner, M., Kaloujnine, L.: Produit complet des groupes de permutations et problème d’extension de groupes. III. Acta Sci. Math. 14, 69–82 (1951)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Krohn, K., Rhodes, J.: Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines. Trans. Am. Math. Soc. 116, 450–464 (1965)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Krohn, K., Rhodes, J.: Complexity of finite semigroups. Ann. Math. 88, 128–160 (1968)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, Singapore (1998)CrossRefGoogle Scholar
  16. 16.
    Liu, J., Kong, X.: Subgroups of endomorphisms of an independence algebra with finite Rank1. Int. Math. Forum 6(39), 1909–1913 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Maksimovskii, M.Y.: Bipolygons and multipolygons over semigroups. Math. Notes 87(5–6), 834–843 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Miller, D., Clifford, A.: Regular D-classes in semigroups. Trans. Am. Math. Soc. 82(1), 270–280 (1956)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Moghaddasi, G.: On injective and subdirectly irreducible S-acts over left zero semigroups. Turk. J. Math. 36(3), 359–365 (2012)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Oehmke, R.H.: The semigroup of a strongly connected automaton. Semigroup Forum 15(1), 351–356 (1977)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Paterson, A.: Groupoids, Inverse Semigroups, and Their Operator Algebras, vol. 170. Springer, Berlin (2012)zbMATHGoogle Scholar
  22. 22.
    Petrich, M.: The translational hull of a completely 0-simple semigroup. Glasg. Math. J. 9(1), 1–11 (1968)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rees, D.: On semi-groups. Math. Proc. Camb. Philos. Soc. 36(4), 387–400 (1940)CrossRefGoogle Scholar
  24. 24.
    Rhodes, J., Steinberg, B.: The q-Theory of Finite Semigroups. Springer, Berlin (2009)CrossRefGoogle Scholar
  25. 25.
    Schutzenberger, M.: D-représentation des demi-groupes. C. R. Acad. Sci. Paris 244(15), 1994–1996 (1957)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Skornjakov, L.: Regularity of the wreath product of monoids. Semigroup Forum 18(1), 83–86 (1979)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Steinberg, B.: A theory of transformation monoids: combinatorics and representation theory. Electron. J. Comb. 17(1), R164 (2010)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Sushkevich, A.: Theory of Generalized Groups. GNTI, Kharkiv-Kiev (1937)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Modal’X, UPLUniv Paris NanterreNanterreFrance

Personalised recommendations