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

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 that the lattice of monoid varieties contains exactly three neutral elements. This article shows that neutrality, distributivity, and lower-modularity coincide in the lattice of monoid varieties. Thus, there exists only three varieties that are distributive and lower-modular elements of this lattice.