Modular and lower-modular elements of lattices of semigroup varieties

The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{COM}$\end{document} of all commutative semigroups or a nilvariety or the join of a nilvariety with the variety of semilattices. Second, we prove that if a commutative nilvariety is a modular element of Com then it may be given within \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{COM}$\end{document} by 0-reduced and substitutive identities only. Third, we completely classify all lower-modular elements of Com. As a corollary, we prove that an element of Com is modular whenever it is lower-modular. All these results are precise analogues of results concerning modular and lower-modular elements of the lattice of all semigroup varieties obtained earlier by Ježek, McKenzie, Vernikov, and the author. As an application of a technique developed in this paper, we provide new proofs of the ‘prototypes’ of the first and the third our results.