Abstract
Let denote the class of aperiodic monoids with central idempotents. A subvariety of that is not contained in any finitely generated subvariety of is said to be inherently non-finitely generated. A characterization of inherently non-finitely generated subvarieties of , 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 , the inclusion of which is both necessary and sufficient for a subvariety of to be inherently non-finitely generated. Further, it is decidable in polynomial time if a finite set of identities defines an inherently nonfinitely generated subvariety of .
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