# On the ascending and descending chain conditions in the lattice of monoid varieties

## TL;DR

This paper investigates the structure of the lattice of monoid varieties, providing an example where the join of two varieties results in a lattice with continuum cardinality that violates chain conditions.

## Contribution

It presents the first known example of monoid varieties with finite subvariety lattices whose join leads to a continuum-sized lattice violating chain conditions.

## Key findings

- Constructed an example of monoid varieties with finite subvariety lattices
- Showed the join of these varieties has a continuum-sized subvariety lattice
- Demonstrated violation of ascending and descending chain conditions

## Abstract

In this work we consider monoids as algebras with an associative binary operation and the nullary operation that fixes the identity element. We found an example of two varieties of monoids with finite subvariety lattices such that their join covers one of them and has a continuum cardinality subvariety lattice that violates the ascending chain condition and the descending chain condition.