On the lattice of overcommutative varieties of monoids

Abstract

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.

5 pages

Cited by in corpus (4)

• The lattice of varieties of monoids
• Varieties of monoids with complex lattices of subvarieties
• Special elements of the lattice of monoid varieties
• Cancellable elements of the lattice of monoid varieties