A Birkhoo-like Axiomatizability Result for Hidden Algebra and Coalgebra

A characterization result for behaviorally deenable classes of hidden algebras shows that a class of hidden algebras is behaviorally deenable by equations if and only if it is closed under coproducts, quotients, morphisms and representative inclusions. The second part of the paper categorically generalizes this result to a framework of any category with coproducts, a nal object and an inclusion system; this is general enough to include all coalgebra categories of interest. As a technical issue, the notions of equation and satisfaction are axiomatized in order to include the diierent approaches in the literature.