Journal of Mathematical Sciences

, Volume 237, Issue 3, pp 410–419 | Cite as

Pseudocomplements in the Lattice of Subvarieties of a Variety of Multiplicatively Idempotent Semirings

  • E. M. VechtomovEmail author
  • A. A. Petrov


The lattice L(𝔐) of all subvarieties of the variety 𝔐 of multiplicatively idempotent semirings is studied. Some relations have been obtained. It is proved that L(𝔐) is a pseudocomplemented lattice. Pseudocomplements in the lattice L(𝔐) are described. It is shown that they form a 64-element Boolean lattice with respect to the inclusion. It is established that the lattice L(𝔐) is infinite and nonmodular.


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  1. 1.
    A. Clifford and G. Preston, The Algebraic Theory of Semigroups, Vol. II [Russian translation], Mir, Moscow (1972).Google Scholar
  2. 2.
    G. Graetzer, General Lattice Theory [Russian translation], Mir, Moscow (1982).Google Scholar
  3. 3.
    A. I. Mal’tsev, Algebraic Systems [in Russian], Nauka, Moscow (1970).Google Scholar
  4. 4.
    F. Pastijn, ”Varieties generated by ordered bands. II,” Order, 22, 129–143 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    F. Pastijn and A. Romanowska, ”Idempotent distributive semirings. I,” Acta Sci. Math., 44, 239–253 (1982).MathSciNetzbMATHGoogle Scholar
  6. 6.
    S. V. Polin , ”Minimal varieties of semirings,” Mat. Zametki, 27, No. 4, 527–537 (1980).MathSciNetzbMATHGoogle Scholar
  7. 7.
    L. N. Shevrin, B. M. Vernikov, and M. V. Volkov, ”The lattice of varieties of semigroups,” Izv. Vyssh. Uchebn. Zaved., Mat., 3, 3–36 (2009).Google Scholar
  8. 8.
    L. A. Skornyakov, Elements of the Theory of Structures [in Russian], Nauka, Moscow (1982).Google Scholar
  9. 9.
    E. M. Vechtomov and A. A. Petrov, ”Multiplicatively idempotent semirings,” Fundam. Prikl. Mat., 18, No. 4, 41–70 (2013).zbMATHGoogle Scholar
  10. 10.
    E. M. Vechtomov and A. A. Petrov, ”Multiplicatively idempotent semirings with the identity x + 2xyx = x,” Vestn. Syktyvkar. Univ. Ser. 1. Mat., Mech., Inform., 17, 44–52 (2013).Google Scholar
  11. 11.
    E. M. Vechtomov and A. A. Petrov, ”The variety of semirings generated by two-element semirings with commutative idempotent multiplication,” Chebyshevskii Sb., 15, No. 3, 12–30 (2014).MathSciNetGoogle Scholar
  12. 12.
    B. M. Vernikov and M. V. Volkov, ”Complements in lattices of varieties and quasivarieties,” Izv. Vyssh. Uchebn. Zaved., Mat., 11, 17–20 (1982).MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Vyatka State UniversityVyatkaRussia

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