On the lattice of overcommutative varieties of monoids

We study the lattice of varieties of monoids, i. e., algebras with two operations, namely an associative binary operation and a 0-ary operation that fixes the neutral element. It was unknown so far, whether this lattice satisfies some non-trivial identity. The objective of this note is to give the negative answer to this question. Namely, we prove that any finite lattice is a homomorphic image of some sublattice of the lattice of overcommutative varieties of monoids (i.e., varieties that contain the variety of all commutative monoids). This implies that the lattice of overcommutative varieties of monoids, and therefore, the lattice of all varieties of monoids does not satisfy any non-trivial identity.