Skip to main content
SpringerLink
Log in

Dualities for quasi-varieties of bands

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

We complete the characterization of finite bands admitting a natural duality, by showing that every finite normal band admits a natural duality. In particular we show that every finite normal band is finitely related.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Aichinger, E., Mayr, P., McKenzie, R.: On the number of finite algebraic structures. To appear in J. Eur. Math. Soc.

  2. Banaschewski, B.: Projective covers in categories of topological spaces and topological algebras. In: General Topology and Its Relation to Modern Analysis and Algebra, Proc. Conf., Kanpur, 1968, pp. 63–91. Academia, Prague (1971)

    Google Scholar 

  3. Biryukov, A.P.: Varieties of idempotent semigroups. Algebra Log. 9, 153–164 (1970)

    Article  Google Scholar 

  4. Clark, D.M., Davey, B.A.: Natural Dualities for the Working Algebraist. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  5. Davey, B.A., Jackson, M., Pitkethly, J., Szabó, Cs.: Finite degree: algebras in general and semigroups in particular. Semigroup Forum 83, 89–110 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Davey, B.A., Knox, B.J.: Regularising natural dualities. Acta Math. Univ. Comen. 68, 295–318 (1999)

    MATH  MathSciNet  Google Scholar 

  7. Davey, B.A., Knox, B.J.: From rectangular bands to k-primal algebras. Semigroup Forum 64, 29–54 (2002)

    Article  MathSciNet  Google Scholar 

  8. Davey, B.A., Quackenbush, R.W.: Natural duality for dihedral varieties. J. Aust. Math. Soc. A 61, 216–228 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Davey, B.A., Werner, H.: Dualities and equivalences for varieties of algebras. In: Contributions to Lattice Theory, Szeged, 1980. Colloquia Mathematica Societatis János Bolyai, vol. 33, pp. 101–275. North-Holland, Amsterdam (1983)

    Google Scholar 

  10. Davey, B.A., Willard, R.: The dualisability of a quasi-variety is independent of the generating algebra. Algebra Univers. 45, 103–106 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dolinka, I.: Finite regular bands are finitely related. Bull. Aust. Math. Soc. 87, 1–9 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fennemore, C.: All varieties of bands. Math. Nachr. 48, 237–262 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gerhard, J.A.: The lattice of equational classes of idempotent semigroups. J. Algebra 15, 195–224 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  14. Golubov, E.A., Sapir, M.V.: Varieties of finitely approximable semigroups. Sov. Math. Dokl. 20(4), 828–832 (1979)

    Google Scholar 

  15. Hobby, D.: Nondualisable semigroups. Bull. Aust. Math. Soc. 65, 491–502 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hofmann, K.H., Mislove, M., Stralka, A.: The Pontryagin Duality of Compact 0-Dimensional Semi-Lattices and Its Applications. Lecture Notes in Mathematics, vol. 396. Springer, Berlin (1974)

    Google Scholar 

  17. Howie, J.M.: Fundamentals of Semigroups Theory. Oxford University Press, Oxford (1995)

    Google Scholar 

  18. Jackson, M.: Dualisability of finite semigroups. Int. J. Algebra Comput. 13, 481–497 (2003)

    Article  MATH  Google Scholar 

  19. Kimura, N.: Note on idempotent semigroups, IV. Proc. Jpn. Acad. 34, 121–123 (1958)

    Article  MATH  Google Scholar 

  20. Mayr, P.: On finitely related semigroups. Semigroup Forum 86, 613–633 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. McKenzie, R.: Residually small varieties of semigroups. Algebra Univers. 13, 171–201 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  22. McLean, D.: Idempotent semigroups. Am. Math. Mon. 61, 110–113 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  23. Quackenbush, R.W., Szabó, Cs.: Nilpotent groups are not dualisable. J. Aust. Math. Soc. 72, 173–179 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Quackenbush, R.W., Szabó, Cs.: Strong duality for metacyclic groups. J. Aust. Math. Soc. 73, 377–392 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Saramago, M.J.: Some remarks on dualisability and endodualisability. Algebra Univers. 43, 197–212 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shafaat, A.: On the structure of certain idempotent semigroups. Trans. Am. Math. Soc. 149, 371–378 (1970)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank the referee of this paper for his helpful suggestions. This paper contains a part of the author’s doctoral thesis written under supervision of Professor Brian Davey and Dr. Marcel Jackson. The author would like to thank them for their excellent guidance.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Muscat, 123, Oman

    Nadia AlDhamri

Authors
  1. Nadia AlDhamri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nadia AlDhamri.

Additional information

Communicated by Mikhail Volkov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

AlDhamri, N. Dualities for quasi-varieties of bands. Semigroup Forum 88, 417–432 (2014). https://doi.org/10.1007/s00233-013-9541-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-013-9541-4

Keywords

Search

Navigation