Monoid varieties with extreme properties

TL;DR

This paper explores the diversity of subvariety lattices generated by finite monoids, revealing many can have continuum-sized lattices and providing the first example with countably infinite subvarieties.

Contribution

It demonstrates the existence of finite monoids generating monoid varieties with continuum and countably infinite subvarieties, including the first known example with the latter property.

Findings

Many finite monoids generate varieties with continuum many subvarieties.

The join of two Cross monoid varieties can have a continuum-sized subvariety lattice.

A finite monoid generating a variety with countably infinite subvarieties is explicitly constructed.

Abstract

Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids that generate monoid varieties with continuum many subvarieties; these include any finite inherently non-finitely based monoid and any monoid for which $xyxy$ is an isoterm. It follows that the join of two Cross monoid varieties can have a continuum cardinality subvariety lattice that violates the ascending chain condition. Regarding monoid varieties with countably infinitely many subvarieties, the first example of a finite monoid that generates such a variety is exhibited. A complete description of the subvariety lattice of this variety is given. This lattice has width three and contains only finitely based varieties, all except two of which are Cross.