Almost Cross varieties of aperiodic monoids

A variety is finitely based if it can be defined by a finite set of identities. A variety is called finitely generated if it is generated by a finite algebra. A variety is called Cross if it is finitely based, finitely generated, and has finitely many subvarieties. One approach to characterizing Cross varieties is to identify their minimal non-Cross subvarieties, which are commonly called almost Cross varieties. It follows from Zorn's lemma that the exclusion of almost Cross subvarieties is not only necessary but also sufficient for a variety to be Cross. For a long time, only two explicit examples of nongroup almost Cross varieties of monoids were known: the variety of all commutative monoids and the variety of all idempotent monoids. In 2005, Jackson discovered two more explicit examples. Since then, a number of authors have found new examples and have also described almost Cross varieties in several subclasses of aperiodic monoids (i.e., monoids all of whose subgroups are trivial). In this talk, I will present a complete description of almost Cross varieties of aperiodic monoids by giving an exhaustive finite list of such varieties.