Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

On Varieties Generated by Minimal Complex Semigroups

  • Published:
Order Aims and scope Submit manuscript

Abstract

A semigroup is complex if it generates a variety the subvariety lattice of which contains an isomorphic copy of every finite lattice. It is known that a complex semigroup has at least four elements and that up to isomorphism and anti-isomorphism, there are four complex semigroups of order four. Subvarieties of the varieties generated by two of these four minimal complex semigroups have previously been described. To complete the study, we describe subvarieties of the varieties generated by the remaining two semigroups.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  2. Birjukov, A.P.: Varieties of idempotent semigroups. Algebra i Log. 9, 255–273 (1970) (Russian)

    MathSciNet  Google Scholar 

  3. Edmunds, C.C.: Varieties generated by semigroups of order four. Semigroup Forum 21, 67–81 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fennemore, C.F.: All varieties of bands, i, ii. Math. Nachr. 48, 237–262 (1971)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Gerhard, J.A., Petrich, M.: All varieties of regular orthogroups. Semigroup Forum 31, 311–351 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  7. Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon, Oxford (1995)

    MATH  Google Scholar 

  8. Jackson, M.: Finite semigroups whose varieties have uncountably many subvarieties. J. Algebra 228, 512–535 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lee, E.W.H.: Identity bases for some non-exact varieties. Semigroup Forum 68, 445–457 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lee, E.W.H.: Subvarieties of the variety generated by the five-element Brandt semigroup. Int. J. Algebra Comput. 16, 417–441 (2006)

    Article  MATH  Google Scholar 

  11. Lee, E.W.H.: Minimal semigroups generating varieties with complex subvariety lattices. Int. J. Algebra Comput. 17, 1553–1572 (2007)

    Article  MATH  Google Scholar 

  12. Lee, E.W.H.: On identity bases of exclusion varieties for monoids. Commun. Algebra 35, 2275–2280 (2007)

    Article  MATH  Google Scholar 

  13. Oates, S., Powell, M.B.L: Identical relations in finite groups. J. Algebra 1, 11–39 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  14. Perkins, P.: Bases for equational theories of semigroups. J. Algebra 11, 298–314 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  15. Petrich, M., Reilly, N.R.: Completely Regular Semigroups. Wiley, New York (1999)

    MATH  Google Scholar 

  16. Pollák, G.: On the consequences of permutation identities. Acta Sci. Math. (Szeged) 34, 323–333 (1973)

    MATH  MathSciNet  Google Scholar 

  17. Pollák, G.: On two classes of hereditarily finitely based semigroup identities. Semigroup Forum 25, 9–33 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  18. Vernikov, B.M., Volkov, M.V.: The structure of lattices of nilsemigroup varieties. Izv. Ural. Gos. Univ. Mat. Mekh. No. 3(18), 34–52 (2000) (Russian)

    MathSciNet  Google Scholar 

  19. Vernikov, B.M., Volkov, M.V.: Commuting fully invariant congruences on free semigroups. Contrib. Gen. Algebra 12 (Vienna, 1999), 391–417, Heyn, Klagenfurt (2000)

    MathSciNet  Google Scholar 

  20. Volkov, M.V.: György Pollák’s work on theory of semigroup varieties: its significance and its influence so far. Acta Sci. Math. (Szeged) 68, 875–894 (2002)

    MathSciNet  Google Scholar 

  21. Zhang, W.T., Luo, Y.F.: The subvariety lattice of the join of two semigroup varieties. Acta Math. Sin. Engl. Ser. (in press)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Wen Ting Zhang & Yan Feng Luo

Authors
  1. Wen Ting Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Feng Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yan Feng Luo.

Additional information

This research was partially supported by the National Natural Science Foundation of China (No.10571077) and the Natural Science Foundation of Gansu Province (No.3ZS052-A25-017)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, W.T., Luo, Y.F. On Varieties Generated by Minimal Complex Semigroups . Order 25, 243–266 (2008). https://doi.org/10.1007/s11083-008-9092-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-008-9092-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation