  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.
