Abstract
In the lattice of semigroup varieties, the set of all neutral elements is finite, the set of all distributive elements is countably infinite, and the set of all lower-modular elements is uncountably infinite. It was established in 2018 that the lattice of monoid varieties contains exactly three neutral elements. This article shows that neutrality, distributivity, and lower-modularity coincide in the lattice of monoid varieties. Thus, there exists only three varieties that are distributive and lower-modular elements of this lattice.
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References
Grätzer G., General Lattice Theory, Birkhäuser, Basel (1978).
Vernikov B.M., “Special elements in lattices of semigroup varieties,” Acta Sci. Math. (Szeged), vol. 81, no. 1, 79–109 (2015).
Gusev S.V., “Special elements of the lattice of monoid varieties,” Algebra Univ., vol. 97, no. 2, Paper no. 29. 12 pp. (2018).
Gusev S.V., “Standard elements of the lattice of monoid varieties,” Algebra Logic, vol. 59, no. 6, 415–422 (2021).
Gusev S.V. and Lee E.W.H., “Cancellable elements of the lattice of monoid varieties,” Acta Math. Hungar., vol. 165, no. 1, 156–168 (2021).
Shaprynski V.Yu., Skokov D.V., and Vernikov B.M., “Cancellable elements of the lattices of varieties of semigroups and epigroups,” Comm. Algebra, vol. 47, no. 11, 4697–4712 (2019).
Burris S. and Sankappanavar H.P., A Course in Universal Algebra, Springer, New York (1981).
Gusev S.V. and Vernikov B.M., “Chain varieties of monoids,” Dissertationes Math., vol. 534, 1–73 (2018).
Wismath S.L., “The lattice of varieties and pseudovarieties of band monoids,” Semigroup Forum, vol. 33, no. 1, 187–198 (2019).
Lee E.W.H., “Varieties generated by 2-testable monoids,” Studia Sci. Math. Hungar., vol. 49, 366–389 (2012).
Sapir O.B., “Limit varieties of J-trivial monoids,” Semigroup Forum, vol. 101, no. 1, 236–260 (2021).
Acknowledgment
The author is grateful to the referee for a series of useful remarks.
Funding
The author was supported by the Ministry of Science and Higher Education of the Russian Federation (Project FEUZ–2020–0016).
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 6, pp. 1248–1255. https://doi.org/10.33048/smzh.2022.63.606
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Gusev, S.V. Distributive and Lower-Modular Elements of the Lattice of Monoid Varieties. Sib Math J 63, 1069–1074 (2022). https://doi.org/10.1134/S0037446622060064
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DOI: https://doi.org/10.1134/S0037446622060064