Coalgebras and Modal Expansions of Logics

In this paper we construct a setting in which the question of when a logic supports a classical modal expansion can be made precise. Given a fully selfextensional logic S, we find sufficient conditions under which the Vietoris endofunctor V on S-referential algebras can be defined and we propose to define the modal expansions of S as the logic that arises from the V-coalgebras. As an example, we also show how the Vietoris endofunctor on referential algebras extends the Vietoris endofunctor on Stone spaces.From another point of view, we examine when a category of 'spaces' (X,A), ie sets X equipped with an algebra A of subsets of X, allows for the definition of powerspaces V (and hence transition systems (X,A)→V(X,A)).