Cancellable elements of the lattice of monoid varieties

TL;DR

This paper characterizes the cancellable elements in the lattice of monoid varieties, identifying exactly five such elements through structural analysis and establishing necessary conditions for modularity.

Contribution

It provides the first example of a monoid variety with a modular but non-distributive subvariety lattice and fully describes all cancellable elements in the lattice of monoid varieties.

Findings

Identified five cancellable elements in the lattice of monoid varieties.

Constructed the first example of a monoid variety with modular but non-distributive subvariety lattice.

Established a necessary condition for the modularity of elements in the lattice.

Abstract

The set of all cancellable elements of the lattice of semigroup varieties has recently been shown to be countably infinite. But the description of all cancellable elements of the lattice $\mathbb{MON}$ of monoid varieties remains unknown. This problem is addressed in the present article. The first example of a monoid variety with modular but non-distributive subvariety lattice is first exhibited. Then a necessary condition of the modularity of an element in $\mathbb{MON}$ is established. These results play a crucial role in the complete description of all cancellable elements of the lattice $\mathbb{MON}$. It turns out that there are precisely five such elements.