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Lower-Modular Elements of the Lattice of Semigroup Varieties

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Abstract

We call a semigroup variety modular [upper-modular, lower-modular, neutral] if it is a modular [respectively upper-modular, lower-modular, neutral] element of the lattice of all semigroup varieties. It is proved that if V is a lower-modular variety then either V coincides with the variety of all semigroups or V is periodic and the greatest nil-subvariety of V may be given by 0-reduced identities only. We completely determine all commutative lower-modular varieties. In particular, it turns out that a commutative variety is lower-modular if and only if it is neutral. A number of corollaries of these results are obtained.

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  1. Department of Mathematics and Mechanics, Ural State University, 620083, Lenina 51, 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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Vernikov, B. Lower-Modular Elements of the Lattice of Semigroup Varieties. Semigroup Forum 75, 554–566 (2007). https://doi.org/10.1007/s00233-007-0719-5

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  • DOI: https://doi.org/10.1007/s00233-007-0719-5

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