On modular and cancellable elements of the lattice of semigroup varieties

TL;DR

This paper characterizes all semigroup varieties satisfying certain permutational identities of length 3 that are modular in the lattice of all semigroup varieties, and provides an example of a modular but not cancellable element.

Contribution

It completely classifies semigroup varieties with permutational identities of length 3 that are modular, and constructs an example distinguishing modularity from cancellability.

Findings

Identified all semigroup varieties satisfying specific permutational identities of length 3 that are modular.

Provided an explicit example of a semigroup variety that is modular but not cancellable.

Enhanced understanding of the lattice structure of semigroup varieties.

Abstract

We completely determine all semigroup varieties satysfiyng a permutational identity of length 3 that are modular elements of the lattice of all semigroup varieties. Using this result, we provide an example of a semigroup variety that is a modular but not cancellable element of this lattice.