Equational and implicational classes of coalgebras

AbstractIf F:Set→Set is a functor which is bounded and preserves weak generalized pullbacks then a class of F-coalgebras is a covariety, i.e., closed under H (homomorphic images), S (sub-coalgebras) and ∑ (sums), if and only if it can be defined by a set of “coequations”. Similarly, quasi-covarieties, i.e., classes closed under H and ∑, can be characterized by implications of coequations. These results are analogous to the theorems of Birkhoff and of Mal'cev in classical universal algebra