Upper-modular and related elements of the lattice of commutative semigroup varieties

TL;DR

This paper characterizes specific elements within the lattice of all commutative semigroup varieties, revealing equivalences among upper-modular, codistributive, and costandard properties, especially in the nil-case.

Contribution

It provides a complete classification of upper-modular, codistributive, and costandard elements in the lattice of commutative semigroup varieties, establishing their equivalences.

Findings

Upper-modular and codistributive elements are equivalent in this lattice.

In the nil-case, upper-modular, codistributive, and costandard properties coincide.

The paper offers a complete description of these elements within the lattice.

Abstract

We completely determine upper-modular, codistributive and costandard elements in the lattice of all commutative semigroup varieties. In particular, we prove that the properties of being upper-modular and codistributive elements in the mentioned lattice are equivalent. Moreover, in the nil-case the properties of being elements of all three types turn out to be equivalent.