Profinite Monads, Profinite Equations, and Reiterman’s Theorem

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 paper Reiterman’s theorem is generalised to finite Eilenberg-Moore algebras for a monad \(\mathbf {T}\) on a variety \(\mathscr {D}\) of (ordered) algebras: a class of finite \(\mathbf {T}\)-algebras is a pseudovariety iff it is presentable by profinite (in-)equations. As an application, quasivarieties of finite algebras are shown to be presentable by profinite implications. Other examples include finite ordered algebras, finite categories, finite \(\infty \)-monoids, etc.