Distributive and lower-modular elements of the lattice of monoid varieties

TL;DR

This paper investigates the properties of the lattice of monoid varieties, revealing that neutrality, distributivity, and lower-modularity are equivalent and identifying the exact number of such elements.

Contribution

It proves that neutrality, distributivity, and lower-modularity coincide in the lattice of monoid varieties, identifying exactly three such elements.

Findings

Neutral, distributive, and lower-modular elements coincide in the lattice.

There are exactly three such elements in the lattice.

The properties of these elements are characterized and counted.

Abstract

The sets of all neutral, distributive and lower-modular elements of the lattice of semigroup varieties are finite, countably infinite and uncountably infinite, respectively. In 2018, we established that there are precisely three neutral elements of the lattice of monoid varieties. In the present work, it is shown that the neutrality, distributivity and lower-modularity coincide in the lattice of monoid varieties. Thus, there are precisely three distributive and lower-modular elements of this lattice.