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, double-struck A sign), ie sets X equipped with an algebra double-struck A sign of subsets of X, allows for the definition of powerspaces V (and hence transition systems (X, double-struck A sign) → V(X, double-struck A sign)).