Geodesic
Inherently non-finitely generated varieties of aperiodic monoids with central idempotents
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 26, Tome 423 (2014), pp. 166-182

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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 $\mathscr A$. 
@article{ZNSL_2014_423_a8,
     author = {Edmond W. H. Lee},
     title = {Inherently non-finitely generated varieties of aperiodic monoids with central idempotents},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {166--182},
     publisher = {mathdoc},
     volume = {423},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_423_a8/}
}
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Edmond W. H. Lee. Inherently non-finitely generated varieties of aperiodic monoids with central idempotents. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 26, Tome 423 (2014), pp. 166-182. http://geodesic.mathdoc.fr/item/ZNSL_2014_423_a8/