Finiteness conditions for lattices of monoid varieties

TL;DR

This paper classifies aperiodic monoid varieties with central idempotents based on the finiteness and chain conditions of their subvariety lattices, revealing equivalences among these properties.

Contribution

It provides a complete classification of such monoid varieties regarding their subvariety lattice properties, establishing key equivalences.

Findings

Finite subvariety lattice is equivalent to ascending chain condition.

Ascending chain condition implies descending chain condition.

Classification of aperiodic monoid varieties with chain conditions.

Abstract

We classify all varieties of aperiodic monoids with central idempotents whose subvariety lattice is finite or satisfies the descending chain condition or satisfies the ascending chain condition. It turns out that for varieties in this class, the properties of having a finite subvariety lattice and a subvariety lattice satisfying the ascending chain condition are equivalent, and thus the property of having a subvariety lattice satisfying the ascending chain condition implies the one of having a subvariety lattice satisfying the descending chain condition.