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Abstract. 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 the duality between equations and coequations for automata proved by Ballester-Bolinches, Cosme-Llópez and Rutten, and generalize it to dynamical systems.

Abstract. In this paper we use a duality result between equations and coequations for automata, proved by Ballester-Bolinches, Cosme-Llópez, 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.

Abstract. We study weighted automata from both an algebraic and a coalgebraic perspective. In particular, we consider equations and coequations for weighted automata. We prove a duality result that relates sets of equations (congruences) with (certain) subsets of coequations. As a consequence, we obtain two equivalent but complementary ways to define classes of weighted automata. We show that this duality cannot be generalized to linear congruences in general but we obtain partial results when weights are from a field.