# On the lattice of overcommutative varieties of monoids

## TL;DR

This paper proves that the lattice of overcommutative monoid varieties is complex enough to contain images of all finite lattices, showing it does not satisfy any non-trivial lattice identities.

## Contribution

It demonstrates that the lattice of overcommutative monoid varieties is highly complex, containing images of all finite lattices, and does not satisfy any non-trivial identities.

## Key findings

- Any finite lattice is a homomorphic image of a sublattice of the overcommutative varieties lattice.
- The lattice of overcommutative varieties does not satisfy any non-trivial lattice identities.
- The result extends to the entire lattice of all monoid varieties.

## 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.