Cross varieties of aperiodic monoids with central idempotents

Abstract

Let $\mathbf{A}$ denote the class of all aperiodic monoids with central idempotents. A description of all Cross subvarieties of $\mathbf{A}$, based on identities that they satisfy and monoids that they cannot contain, is given. The two limit subvarieties of $\mathbf{A}$, published by Marcel Jackson in 2005, turn out to be the only finitely generated, almost Cross subvarieties of $\mathbf{A}$. It follows that it is decidable in quartic time if a finite monoid in $\mathbf{A}$ generates a Cross variety.