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Upper-modular elements of the lattice of semigroup varieties. II

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Abstract

A semigroup variety is called a variety of degree ≤2 if all its nilsemigroups are semigroups with zero multiplication, and a variety of degree >2 otherwise. We completely determine all semigroup varieties of degree >2 that are upper-modular elements of the lattice of all semigroup varieties and find quite a strong necessary condition for semigroup varieties of degree ≤2 to have the same property.

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References

  1. J. A. Gerhard, “Semigroups with an idempotent power. II. The lattice of equational subclasses of [(xy)2 = xy],” Semigroup Forum, 14, 375–388 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  2. E. A. Golubov and M. V. Sapir, “Residually small varieties of semigroups,” Sov. Math. (Iz. VUZ), 26, No. 11, 25–36 (1982).

    MATH  MathSciNet  Google Scholar 

  3. J. Ježek and R. N. McKenzie, “Definability in the lattice of equational theories of semigroups,” Semigroup Forum, 46, 199–245 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  4. L. Polák, “On varieties of completely regular semigroups. I,” Semigroup Forum, 32, 97–123 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  5. V. V. Rasin, “Varieties of orthodox Cliffordian semigroups,” Sov. Math. (Iz. VUZ), 26, No. 11, 107–110 (1982).

    MATH  MathSciNet  Google Scholar 

  6. L. N. Shevrin, B. M. Vernikov, and M. V. Volkov, “Lattices of varieties of semigroups,” Izv. Vyssh. Uchebn. Zaved., Mat., to appear.

  7. B. M. Vernikov, “Lower-modular elements of the lattice of semigroup varieties,” Semigroup Forum, 75, 554–566 (2007).

    Article  MathSciNet  Google Scholar 

  8. B. M. Vernikov, “On modular elements of the lattice of semigroup varieties,” Comment. Math. Univ. Carolin., 48, 595–606 (2007).

    MATH  MathSciNet  Google Scholar 

  9. B. M. Vernikov, “Lower-modular elements of the lattice of semigroup varieties. II,” Acta Sci. Math. (Szeged), to appear.

  10. B. M. Vernikov, “Upper-modular elements of the lattice of semigroup varieties,” Algebra Universalis, to appear.

  11. B. M. Vernikov and M. V. Volkov, “Lattices of nilpotent semigroup varieties,” in: L. N. Shevrin, ed., Algebraic Systems and Their Varieties [in Russian], Ural State Univ., Sverdlovsk (1988), pp. 53–65.

    Google Scholar 

  12. B. M. Vernikov and M. V. Volkov, “Modular elements of the lattice of semigroup varieties. II,” Contrib. Gen. Algebra, 17, 173–190 (2006).

    MathSciNet  Google Scholar 

  13. M. V. Volkov, “Commutative semigroup varieties with distributive subvariety lattices,” Contrib. Gen. Algebra, 7, 351–359 (1991).

    Google Scholar 

  14. M. V. Volkov, “Semigroup varieties with modular subvariety lattice. III,” Russ. Math. (Iz. VUZ), 36, No. 8, 18–25 (1992).

    Google Scholar 

  15. M. V. Volkov, “Semigroup varieties with modular subvariety lattice,” Russ. Acad. Sci. Dokl. Math., 46, 274–278 (1993).

    Google Scholar 

  16. M. V. Volkov, “Modular elements of the lattice of semigroup varieties,” Contrib. Gen. Algebra, 16, 275–288 (2005).

    Google Scholar 

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Authors and Affiliations

  1. Ural State University, Ekaterinburg, Russia

    B. M. Vernikov

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  1. B. M. Vernikov
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Correspondence to B. M. Vernikov.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 14, No. 7, pp. 43–51, 2008.

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Vernikov, B.M. Upper-modular elements of the lattice of semigroup varieties. II. J Math Sci 164, 182–187 (2010). https://doi.org/10.1007/s10958-009-9718-2

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