Special elements of the lattice of epigroup varieties

TL;DR

This paper investigates special elements within the lattice of epigroup varieties, providing classifications and conditions for various element types, and explores applications to definable sets of epigroup varieties.

Contribution

It offers complete classifications of several special elements in the lattice of epigroup varieties and establishes new conditions for modular elements, advancing the structural understanding of this lattice.

Findings

Complete determination of neutral, standard, costandard, distributive, and lower-modular elements.

Conditions for modular elements, especially in commutative varieties.

Characterizations of codistributive and upper-modular elements in strongly permutative 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.