Advertisement

Order

, Volume 28, Issue 1, pp 139–155 | Cite as

Special Elements in the Lattice of Overcommutative Semigroup Varieties Revisited

  • Vyacheslav Yu. Shaprynskiǐ
  • Boris M. Vernikov
Article
  • 36 Downloads

Abstract

We completely determine all distributive, codistributive, standard, costandard, and neutral elements in the lattice of overcommutative semigroup varieties, thus correcting a gap in a previous paper.

Keywords

Semigroup Variety Lattice of subvarieties Overcommutative variety Distributive element Standard element Neutral element 

Mathematics Subject Classifications (2010)

Primary 20M07; Secondary 08B15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Grätzer, G.: General Lattice Theory, 2nd edn. Birkhauser, Basel (1998)MATHGoogle Scholar
  2. 2.
    Ježek, J., McKenzie, R.N.: Definability in the lattice of equational theories of semigroups. Semigroup Forum 46, 199–245 (1993)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    McKenzie, R.N., McNulty, G.F., Taylor, W.F.: Algebras. Lattices. Varieties, vol. I. Wadsworth & Brooks/Cole, Monterey (1987)Google Scholar
  4. 4.
    Shevrin, L.N., Vernikov, B.M., Volkov, M.V.: Lattices of semigroup varieties, Izv. VUZ. Matem., No. 3, pp. 3–36 (2009) [Russian; Engl. translation: Russian Math. Izv. VUZ 53(3), 1–28 (2009)]Google Scholar
  5. 5.
    Vernikov, B.M.: Special elements in the lattice of overcommutative semigroup varieties. Mat. Zametki 70, 670–678 (2001) [Russian; Engl. translation: Math. Notes 70, 608–615 (2001)]MathSciNetGoogle Scholar
  6. 6.
    Vernikov, B.M.: On modular elements of the lattice of semigroup varieties. Comment. Math. Univ. Carol. 48, 595–606 (2007)MathSciNetMATHGoogle Scholar
  7. 7.
    Vernikov, B.M.: Lower-modular elements of the lattice of semigroup varieties. Semigroup Forum 75, 554–566 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Vernikov, B.M.: Lower-modular elements of the lattice of semigroup varieties. II. Acta Sci. Math. (Szeged) 74, 539–556 (2008)MathSciNetMATHGoogle Scholar
  9. 9.
    Vernikov, B.M.: Upper-modular elements of the lattice of semigroup varieties. Algebra Univers. 59, 405–428 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Vernikov, B.M.: Upper-modular elements of the lattice of semigroup varieties. II. Fund. and Appl. Math. 14(7), 43–51 (2008) [Russian; Engl. translation: J. Math. Sci. 164, 182–187 (2010)]MathSciNetGoogle Scholar
  11. 11.
    Vernikov, B.M.: Codistributive elements of the lattice of semigroup varieties. Izv. VUZ. Matem. (in press, Russian)Google Scholar
  12. 12.
    Vernikov, B.M., Shaprynskiǐ, V.Y.: Distributive elements of the lattice of semigroup varieties. Algebra and Logic (in press, Russian)Google Scholar
  13. 13.
    Vernikov, B.M., Volkov, M.V.: Modular elements of the lattice of semigroup varieties. II. Contrib. Gen. Algebra 17, 173–190 (2006)MathSciNetGoogle Scholar
  14. 14.
    Volkov, M.V.: Young diagrams and the structure of the lattice of overcommutative semigroup varieties. In: Higgins, P.M. (ed.) Transformation Semigroups. Proc. Int. Conf. held at the Univ. Essex, University of Essex, Colchester, pp. 99–110 (1994)Google Scholar
  15. 15.
    Volkov, M.V.: Modular elements of the lattice of semigroup varieties. Contrib. Gen. Algebra 16, 275–288 (2005)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Vyacheslav Yu. Shaprynskiǐ
    • 1
  • Boris M. Vernikov
    • 1
  1. 1.Department of Mathematics and MechanicsUral State UniversityYekaterinburgRussia

Personalised recommendations