It is unknown so far, whether the lattice of all varieties of monoids
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.Comment: 5 page
