Minimal monoids generating varieties with complex subvariety lattices

TL;DR

This paper demonstrates that the 6-element Brandt monoid generates the smallest finitely universal monoid variety with complex subvariety lattices, and explores joins of Cross varieties with this property.

Contribution

It identifies the minimal monoid generator for finitely universal varieties and shows that joins of certain Cross varieties can also be finitely universal.

Findings

The 6-element Brandt monoid generates a finitely universal variety.

The join of two Cross varieties can be finitely universal.

Constructs a finitely universal variety with uncountably many subvarieties.

Abstract

A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. We show that the 6-element Brandt monoid generates a finitely universal variety of monoids and, by the previous results, it is the smallest generator for a monoid variety with this property. It is also deduced that the join of two Cross varieties of monoids can be finitely universal. In particular, we exhibit a finitely universal variety of monoids with uncountably many subvarieties which is the join of two Cross varieties of monoids whose lattices of subvarieties are the 6-element and the 7-element chains, respectively.