Small monoids generating varieties with uncountably many subvarieties

TL;DR

This paper demonstrates that a specific small monoid, of order ten, generates a variety with uncountably many subvarieties, providing new examples and answering a recent open question in algebraic theory.

Contribution

It introduces a small, finitely based monoid of order six that generates a variety with uncountably many subvarieties, the first minimal example of its kind.

Findings

Rees quotient monoid M(aabb) of order ten is of type 2^{}

A new minimal monoid of order six with uncountably many subvarieties

First finitely based monoid of minimal size with this property

Abstract

An algebra that generates a variety with uncountably many subvarieties is said to be of type $2^{\aleph_0}$. We show that the Rees quotient monoid $M(aabb)$ of order ten is of type $2^{\aleph_0}$, thereby affirmatively answering a recent question of Glasson. As a corollary, we exhibit a new example of type $2^{\aleph_0}$ monoid of order six, which turns out to be minimal and the first of its kind that is finitely based.