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 finitealgebras. Two well–known theorems that characterize varieties and pseudovarieties of alge-bras play an important role here: Birkhoff’s theorem and Birkhoff’s theorem for finite alge-bras, the latter also known as Reiterman’s theorem. We prove, under mild assumptions, acategorical version of Birkhoff’s theorem for (finite) algebras to establish a one–to–one cor-respondence 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, weget a one–to–one correspondence between (pseudo)equational T–theories and their dual,(pseudo)coequational B–theories. Particular instances of (pseudo)coequational B-theorieshave been already studied in language theory under the name of “varieties of languages” toestablish Eilenberg–type correspondences. All in all, we get a one–to–one correspondencebetween (pseudo)varieties of T–algebras and (pseudo)coequational B–theories, which willbe shown to be exactly the nature of Eilenberg–type correspondences.