Inherently non-finitely generated varieties of aperiodic monoids with central idempotents

Let $\mathscr A$ denote the class of aperiodic monoids with central idempotents. A subvariety of $\mathscr A$ that is not contained in any finitely generated subvariety of $\mathscr A$ is said to be inherently non-finitely generated. A characterization of inherently non-finitely generated subvarieties of $\mathscr A$, based on identities that they cannot satisfy and monoids that they must contain, is given. It turns out that there exists a unique minimal inherently non-finitely generated subvariety of $\mathscr A$, the inclusion of which is both necessary and sufficient for a subvariety of $\mathscr A$ to be inherently non-finitely generated. Further, it is decidable in polynomial time if a finite set of identities defines an inherently non-finitely generated subvariety of $\mathscr A$.