Semigroup Forum

, Volume 89, Issue 1, pp 41–51 | Cite as

Unary enhancements of inherently non-finitely based semigroups

  • K. Auinger
  • I. Dolinka
  • T. V. Pervukhina
  • M. V. Volkov
RESEARCH ARTICLE

Abstract

This paper is a follow up of an article published in 2012 by three of the authors, more precisely, of a part of that article dealing with inherently nonfinitely based involutory semigroups. We exhibit a simple condition under which a finite involutory semigroup whose semigroup reduct is inherently nonfinitely based is also inherently nonfinitely based as a unary semigroup. As applications, we get already known as well as new examples of inherently nonfinitely based involutory semigroups. We also show that for finite regular semigroups, our condition is not only sufficient but also necessary for the property of being inherently nonfinitely based to persist. This leads to an algorithmic description of regular inherently nonfinitely based involutory semigroups.

Keywords

Involutory semigroup Inherently nonfinitely based semigroup Twisted semilattice Twisted Brandt monoid Regular semigroup Matrix semigroup 

Notes

Acknowledgement

The authors are indebted to an anonymous referee for several inspiring remarks that have led to an improved presentation of Sect. 4.

References

  1. 1.
    Auinger, K., Dolinka, I., Volkov, M.V.: Matrix identities involving multiplication and transposition. J. Eur. Math. Soc. 14(3), 937–969 (2012) MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, Berlin (1981) MATHCrossRefGoogle Scholar
  3. 3.
    Dolinka, I.: On identities of finite involution semigroups. Semigroup Forum 80(1), 105–120 (2010) MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Fajtlowicz, S.: Equationally complete semigroups with involution. Algebra Univers. 1(1), 355–358 (1972) MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Goldberg, I.A., Volkov, M.V.: Identities of semigroups of triangular matrices over a finite field. Mat. Zametki 73(4), 502–510 (2003) in Russian; Engl. translation Math. Notes 73(4), 474–481 (2003) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs, vol. 12. Clarendon Press, Oxford (1995) MATHGoogle Scholar
  7. 7.
    Jackson, M., Volkov, M.V.: The algebra of adjacency patterns: Rees matrix semigroups with reversion. In: Fields of Logic and Computation. Lecture Notes in Computer Science, vol. 6300, pp. 414–443. Springer, Berlin (2010) CrossRefGoogle Scholar
  8. 8.
    Lawrence, J., Willard, R.: On finitely based groups and nonfinitely based quasivarieties. J. Algebra 203(1), 1–11 (2008) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lidl, R., Niederreiter, H.: Finite Fields. Addison-Wesley, Cambridge (1997) Google Scholar
  10. 10.
    Nambooripad, K.S.S., Pastijn, F.J.C.M.: Regular involution semigroups. In: Semigroups, Szeged, 1981. Colloquia Mathematica Societatis János Bolyai, vol. 39, pp. 199–249. North-Holland, Amsterdam (1985) Google Scholar
  11. 11.
    Pin, J.-É.: Varieties of Formal Languages. Plenum, New York (1986) MATHCrossRefGoogle Scholar
  12. 12.
    Sapir, M.V.: Problems of Burnside type and the finite basis property in varieties of semigroups. Izv. Akad. Nauk SSSR, Ser. Mat. 51(2), 319–340 (1987) in Russian; Engl. translation Math. USSR-Izv. 30(2), 295–314 (1988) Google Scholar
  13. 13.
    Sapir, M.V.: Inherently nonfinitely based finite semigroups. Mat. Sb. 133(2), 154–166 (1987) in Russian; Engl. translation Math. USSR–Sb. 61(1), 155–166 (1988) Google Scholar
  14. 14.
    Sapir, M.V.: Identities of finite inverse semigroups. Int. J. Algebra Comput. 3(1), 115–124 (1993) MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Serre, J.-P.: Cours d’Arithmetique. Presses Universitaires de France, Paris (1980) Google Scholar
  16. 16.
    Volkov, M.V.: The finite basis problem for finite semigroups. Sci. Math. Jpn. 53(1), 171–199 (2001) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • K. Auinger
    • 1
  • I. Dolinka
    • 2
  • T. V. Pervukhina
    • 3
  • M. V. Volkov
    • 3
  1. 1.Fakultät für MathematikUniversität WienWienAustria
  2. 2.Department of Mathematics and InformaticsUniversity of Novi SadNovi SadSerbia
  3. 3.Institute of Mathematics and Computer ScienceUral Federal UniversityEkaterinburgRussia

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