Modular elements in the lattice of monoid varieties

TL;DR

This paper classifies all modular elements within the lattice of monoid varieties, providing a comprehensive understanding of their structure and properties.

Contribution

It offers the first complete classification of modular elements in the lattice of all monoid varieties, filling a gap in algebraic lattice theory.

Findings

Identified all modular elements in the lattice of monoid varieties.

Characterized the structure of these modular elements.

Enhanced understanding of the lattice's algebraic properties.

Abstract

An element $x$ of a lattice $L$ is modular if $L$ has no five-element sublattice isomorphic to the pentagon in which $x$ would correspond to the lonely midpoint. In the present work, we classify all modular elements of the lattice of all monoid varieties.