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Special elements of the lattice of monoid varieties

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Abstract

We completely classify all neutral and costandard elements in the lattice \(\mathbb {MON}\) of all monoid varieties. Further, we prove that an arbitrary upper-modular element of \(\mathbb {MON}\) except the variety of all monoids is either a completely regular or a commutative variety. Finally, we verify that all commutative varieties of monoids are codistributive elements of \(\mathbb {MON}\). Thus, the problems of describing codistributive or upper-modular elements of \(\mathbb {MON}\) are completely reduced to the completely regular case.

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Acknowledgements

The author is sincerely grateful to Professor B.M. Vernikov for his great assistance in the improvement of the initial version of the manuscript and to Professor M.V. Volkov for helpful discussions.

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Authors and Affiliations

  1. Institute of Natural Sciences and Mathematics, Ural Federal University, Lenina str. 51, 620000, Ekaterinburg, Russia

    Sergey V. Gusev

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  1. Sergey V. Gusev
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Correspondence to Sergey V. Gusev.

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Presented by M. Jackson.

The work is supported by the Russian Foundation for Basic Research (Grant 17-01-00551) and by the Ministry of Education and Science of the Russian Federation (project 1.6018.2017/8.9).

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Gusev, S.V. Special elements of the lattice of monoid varieties. Algebra Univers. 79, 29 (2018). https://doi.org/10.1007/s00012-018-0513-0

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