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

TL;DR

This paper investigates the conditions under which semigroup and epigroup varieties are modular elements in their lattices, refining previous criteria and setting the stage for a full classification.

Contribution

It provides strengthened necessary and sufficient conditions for modularity in the lattices of semigroup and epigroup varieties, improving upon earlier criteria.

Findings

Refined criteria for modular elements in semigroup and epigroup variety lattices

Established foundational results for future complete classification

Enhanced understanding of lattice structure in algebraic varieties

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.