Varieties and Covarieties of Languages (Extended Abstract)

Because of the isomorphism (X × A) → X ∼= X → (A → X), the transition structure of a deterministic automaton with state set X and with inputs from an alphabet A can be viewed both as an algebra and as a coalgebra. This algebra-coalgebra duality goes back to Arbib and Manes, who formulated it as a duality between reachability and observability, and is ultimately based on Kalman’s duality in systems theory between controllability and observability. Recently, it was used to give a new proof of Brzozowski’s minimization algorithm for deterministic automata. Here we will use the algebra-coalgebra duality of automata as a common perspective for the study of both varieties and covarieties, which are classes of automata and languages defined by equations and coequations, respectively. We make a first connection with Eilenberg’s definition of varieties of languages, which is based on the classical, algebraic notion of varieties of (transition) monoids.