Unveiling Eilenberg-type Correspondences: Birkhoff's Theorem for (finite) Algebras + Duality

The purpose of the present paper is to show that: Eilenberg-type correspondences = Birkhoff's theorem for (finite) algebras + duality. We consider algebras for a monad T on a category D and we study (pseudo)varieties of T-algebras. Pseudovarieties of algebras are also known in the literature as varieties of finite algebras. Two well-known theorems that characterize varieties and pseudovarieties of algebras play an important role here: Birkhoff's theorem and Birkhoff's theorem for finite algebras, the latter also known as Reiterman's theorem. We prove, under mild assumptions, a categorical version of Birkhoff's theorem for (finite) algebras to establish a one-to-one correspondence between (pseudo)varieties of T-algebras and (pseudo)equational T-theories. Now, if C is a category that is dual to D and B is the comonad on C that is the dual of T, we get a one-to-one correspondence between (pseudo)equational T-theories and their dual, (pseudo)coequational B-theories. Particular instances of (pseudo)coequational B-theories have been already studied in language theory under the name of "varieties of languages" to establish Eilenberg-type correspondences. All in all, we get a one-to-one correspondence between (pseudo)varieties of T-algebras and (pseudo)coequational B-theories, which will be shown to be exactly the nature of Eilenberg-type correspondences.