Distributive and Lower-Modular Elements of the Lattice of Monoid Varieties

Abstract

In the lattice of semigroup varieties,the set of all neutral elements is finite,the set of all distributive elements is countably infinite,and the set of all lower-modular elements is uncountably infinite.It was established in 2018 thatthe lattice of monoid varietiescontains exactly three neutral elements.This article shows thatneutrality,distributivity,and lower-modularitycoincide in the lattice of monoid varieties.Thus,there exists only three varietiesthat are distributive and lower-modular elements of this lattice. © 2022, Pleiades Publishing, Ltd.