Modular elements of the lattices of varieties of semigroups and epigroups. I
Vyacheslav Yu. Shaprynskiǐ, Dmitry V. Skokov
TL;DR
This paper addresses the problem of describing modular elements in the lattices of semigroup varieties $\mathbb{SEM}$ and epigroup varieties $\mathbb{EPI}$. It develops strengthened necessary and sufficient conditions for modularity, reducing the task to nil-varieties and proving that a modular variety has a decomposition $\mathbf{V}=\mathbf{M}\vee\mathbf{N}$ with $\mathbf{M}\in\{\mathbf{T},\mathbf{SL}\}$ and $\mathbf{N}$ nil-variety; the same conclusions hold for both lattices. A central methodological advance is the reduction to G-sets and the introduction of $\mathcal{T}$-coordinated congruences, $\mathcal{T}$-codes, and stabilizers, enabling a detailed analysis of modularity via the action of groups on letter-sets and the corresponding stabilizer subgroups. The results lay groundwork for a complete classification to be presented in Part II and situate modular elements within the broader context of lattice identities and neutral joins, including relationships to joins with semilattices and comparisons to modularity in related varieties such as monoids.
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.