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Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e., classes of finite algebras closed under finite products, subalgebras and quotients. In this article, Reiterman’s theorem is generalized to finite Eilenberg-Moore algebras for a monad T on a category  D: we prove that a class of finiteT -algebras is a pseudovariety iff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinite monadT ^ associated to the monadT , which gives a categorical view of the construction of the space of profinite terms.

Reiterman’s Theorem on Finite Algebras for a Monad