Modular elements of the lattices of varieties of semigroups and epigroups. I

Abstract

This paper is the first part of a study devoted to description of modular elements in the lattices of semigroup and epigroup varieties. We provide strengthened necessary and sufficient conditions under which a semigroup or epigroup variety constitutes a modular element in its respective lattice. These results refine previously known criteria and lay the groundwork for a complete classification, to be presented in the second part of the study.