Characterization of Cross varieties of $J$-trivial monoids

TL;DR

This paper characterizes when a variety of J-trivial monoids is a Cross variety, identifying a specific list of 14 almost Cross subvarieties that determine this property.

Contribution

It provides a complete characterization of Cross varieties within J-trivial monoids by identifying a finite list of key almost Cross subvarieties.

Findings

A variety of J-trivial monoids is Cross iff it excludes 14 specific almost Cross subvarieties.

The 14 varieties listed are the complete set of almost Cross varieties for J-trivial monoids.

The characterization hinges on subvariety exclusion criteria.

Abstract

A finitely based, finitely generated variety with finitely many subvarieties is a Cross variety. In the present article, it is shown that a variety of $J$-trivial monoids is Cross if and only if it excludes as subvarieties a certain list of 14 almost Cross varieties. Consequently, the list of 14 varieties exhausts all almost Cross varieties of $J$-trivial monoids.