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 Bojan´czyk 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 o

Duality of equations and coequations via contravariant adjunctions

In this paper we show duality results between categories of equations and categories of coequations. These dualities are obtained as restrictions of dualities between categories of algebras and coalgebras, which arise by lifting contravariant adjunctions on the base categories. By extending this approach to (co)algebras for (co)monads, we retrieve th

Regular Varieties of Automata and Coequations

In this paper we use a duality result between equations and coequations for automata, proved by Ballester-Bolinches, Cosme-Ll´opez, and Rutten to characterize nonempty classes of deterministic automata that are closed under products, subautomata, homomorphic images, and sums. One characterization is as classes of automata defined by regular equations and the second one is as classes of automata satisfying sets of coequations called varieties of languages. We show how our results are related to Birkhoff’s theorem for regular varieties