Equations and Coequations for Weighted Automata

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.