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)).
A. Kurz and A. Palmigiano, “Coalgebras and Modal Expansions of Logics,” Electronic Notes in Theoretical Computer Science, vol. 106, pp. 243–259, Dec. 2004. DOI: 10.1016/j.entcs.2004.05.010
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