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On the Lattice of Subvarieties of the Wreath Product of the Variety of Semilattices and the Variety of Semigroups with Zero Multiplication

Abstract

It is known that the monoid wreath product of any two semigroup varieties that are atoms in the lattice of all semigroup varieties may have a finite as well as an infinite lattice of subvarieties. If this lattice is finite, then as a rule it has at most eleven elements. This was proved in a paper of the author in 2007. The exclusion is the monoid wreath product Sl w N 2 of the variety of semilattices and the variety of semigroups with zero multiplication. The number of elements of the lattice L(Sl w N 2) of subvarieties of Sl w N 2 is still unknown. In our paper, we show that the lattice L(Sl w N 2) contains no less than 33 elements. In addition, we give some exponential upper bound of the cardinality of this lattice.

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References

  1. 1.

    M. A. Arbib, ed., Algebraic Theory of Machines, Languages and Semigroups, Academic Press, New York (1968).

    MATH  Google Scholar 

  2. 2.

    A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. 1, Amer. Math. Soc., Providence, Rhode Island (1964).

  3. 3.

    S. Eilenberg, Automata, Languages and Machines, Vol. B, Academic Press, New York (1976).

    MATH  Google Scholar 

  4. 4.

    T. Evans, “The lattice of semigroup varieties,” Semigroup Forum, 2, 1–43 (1971).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

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

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Ju. G. Košelev, “Varieties preserved under wreath products,” Semigroup Forum, 12, 95–107 (1976).

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Yu. G. Koshelev, “Associativity of multiplication of semigroup varieties,” in: Int. Algebraic Conf. in Memory of A. I. Mal’cev. Collection of Theses on Theory of Models and Algebraic Systems, Novosibirsk (1989), p. 63.

  8. 8.

    G. Lallement, Semigroups and Combinatorial Applications, Wiley, New York (1979).

    MATH  Google Scholar 

  9. 9.

    A. I. Mal’cev, “On multiplying of classes of algebraic systems,” Sib. Mat. Zh., 8, No. 2, 346–365 (1967).

    Google Scholar 

  10. 10.

    H. Neumann, Varieties of Groups, Springer, New York (1967).

    Book  MATH  Google Scholar 

  11. 11.

    M. V. Sapir, “Inherently nonfinitely based finite semigroups,” Mat. Sb., 133, No. 2, 154–166 (1987).

    MATH  Google Scholar 

  12. 12.

    M. V. Sapir, “On Cross semigroup varieties and related questions,” Semigroup Forum, 42, 345–364 (1991).

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    M. V. Sapir and E. V. Sukhanov, “On varieties of periodic semigroups,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 4, 48–55 (1981).

  14. 14.

    L. N. Shevrin and E. V. Sukhanov, “Structural aspects of the theory of semigroup varieties,” Sov. Math. (Iz. VUZ), 33, No. 6, 1–34 (1989).

    MATH  Google Scholar 

  15. 15.

    L. N. Shevrin, B. M. Vernikov, and M. V. Volkov, “Lattices of semigroup varieties,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 3–36 (2009).

  16. 16.

    L. N. Shevrin and M. V. Volkov, “Identities of semigroups,” Sov. Math. (Iz. VUZ), 29, No. 11, 1–64 (1989).

    MATH  Google Scholar 

  17. 17.

    L. A. Skornjakov, “Regularity of the wreath product of monoids,” Semigroup Forum, 18, No. 1, 83–86 (1979).

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    B. Tilson, “Categories as algebra: an essential ingredient in the theory of monoids,” J. Pure Appl. Algebra, 48, No. 1-2, 83–198 (1987).

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    A. V. Tishchenko, “On different definitions of the wreath product of semigroup varieties,” Fundam. Prikl. Mat., 2, No. 2, 233–249 (1996).

    MathSciNet  MATH  Google Scholar 

  20. 20.

    A. V. Tishchenko, “Wreath product of varieties and semi-Archimedean varieties of semigroups,” Trans. Moscow Math. Soc., 203–222 (1996).

  21. 21.

    A. V. Tishchenko, “The wreath product of the atoms of the lattice of semigroup varieties,” Russ. Math. Surv., 53, 870–871 (1998).

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    A. V. Tishchenko, “The ordered monoid of semigroup varieties under wreath product,” Fundam. Prikl. Matem., 5, No. 1, 283–305 (1999).

    MathSciNet  MATH  Google Scholar 

  23. 23.

    A. V. Tishchenko, “The wreath product of the atoms of the lattice of semigroup varieties,” Trans. Moscow Math. Soc., 93–118 (2007).

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Correspondence to A. V. Tishchenko.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 6, pp. 191–212, 2014.

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Tishchenko, A.V. On the Lattice of Subvarieties of the Wreath Product of the Variety of Semilattices and the Variety of Semigroups with Zero Multiplication. J Math Sci 221, 436–451 (2017). https://doi.org/10.1007/s10958-017-3236-4

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