Modular and lower-modular elements of lattices of semigroup varieties

TL;DR

This paper characterizes modular and lower-modular elements in the lattice of commutative semigroup varieties, providing classifications and new proofs for these algebraic structures.

Contribution

It offers a complete classification of modular and lower-modular elements in the lattice of commutative semigroup varieties, extending previous results and introducing new proof techniques.

Findings

Modular elements are either the entire variety COM, nil-varieties, or joins of nil-varieties with semilattices.

Nil-varieties that are modular can be characterized by 0-reduced and substitutive identities.

All lower-modular elements are also modular in the lattice of commutative semigroup varieties.

Abstract

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 COM of all commutative semigroups or a nil-variety or the join of a nil-variety with the variety of semilattices. Second, we prove that if a commutative nil-variety is a modular element of Com then it may be given within COM 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 Jezek, 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.