An Eilenberg–like Theorem for Algebras on a Monad

An Eilenberg–like theorem is shown for algebras on a given monad. The main idea is to explore the approach given by Bojańczyk that defines, for a given monad T on a category D, pseudovarieties of T–algebras as classes of finite T–algebras closed under homomorphic images, subalgebras, and finite products. To define pseudovarieties of recognizable languages, which is the other main concept for an Eilenberg–like theorem, we use a category C that is dual to D and a recent duality result between Eilenberg–Moore categories of algebras and coalgebras by Salamanca, Bonsangue, and Rot. Using this duality, we define the concept of a pseudovariety of recognizable languages based on the category C. With this new approach, we can study different kinds of pseudovarieties of algebras as well as different kinds of pseudovarieties of recognizable languages to (re)discover some existing and new Eilenberg–like theorems. By dropping finiteness conditions we also derive an Eilenberg–like theorem for varieties of T–algebras, that is, classes of T–algebras closed under homomorphic images, subalgebras, and (not necessarily finite) products.