Special elements of the lattice of epigroup varieties
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.
Keywords
Cite
@article{arxiv.1408.0356,
title = {Special elements of the lattice of epigroup varieties},
author = {V. Yu. Shaprynskii and D. V. Skokov and B. M. Vernikov},
journal= {arXiv preprint arXiv:1408.0356},
year = {2016}
}
Comments
In comparison with the previous version, we slightly optimize the proof of Theorem 1.1, eliminate a few typos and add Question 11.4