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arXiv · 1408.0356

Special elements of the lattice of epigroup varieties

V. Yu. Shaprynskii·D. V. Skokov·B. M. Vernikov
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Abstract

We study special elements of eight types (namely, neutral, standard, costandard, distributive, codistributive, modular, lower-modular and upper-modular elements) in the lattice EPI of all epigroup varieties. Neutral, standard, costandard, distributive and lower-modular elements are completely determined. A strong necessary condition and a sufficient condition for modular elements are found. Modular elements are completely classified within the class of commutative varieties, while codistributive and upper-modular elements are completely determined within the wider class of strongly permutative varieties. It is verified that an element of EPI is costandard if and only if it is neutral; is standard if and only if it is distributive; is modular whenever it is lower-modular; is neutral if and only if it is lower-modular and upper-modular simultaneously. We found also an application of results concerning neutral and lower-modular elements of EPI for studying of definable sets of epigroup varieties.

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V. Yu. Shaprynskii, D. V. Skokov, B. M. Vernikov. 2014-08-02. Special elements of the lattice of epigroup varieties. https://arxiv.org/abs/1408.0356

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