Groupoid Quantales: a non \'etale setting

It is well known that if G is an \'etale topological groupoid then its topology can be recovered as the sup-lattice generated by G-sets, i.e. by the images of local bisections. This topology has a natural structure of unital involutive quantale. We present the analogous construction for any non \'etale groupoid with sober unit space G_0. We associate a canonical unital involutive quantale with any inverse semigroup of G-sets which is also a sheaf over G_0. We introduce axiomatically the class of quantales so obtained, and revert the construction mentioned above by proving a representability theorem for this class of quantales, under a natural spatiality condition

Abstract Logics as Dialgebras

AbstractThe aim of this report is to propose a line of research that studies the connections between the theory of consequence operators as developed in [1] and [4] and the theory of dialgebras. The first steps in this direction are taken in this report, namely some of the basic notions of the theory of consequence operators - such as abstract logics - are translated into notions of the theory of dialgebras, and internal characterizations of the corresponding classes of objects are presented. Moreover it is shown that the class of coalgebras that corresponds to abstract logics of empty signature is a covariety

Residuation algebras with functional duals

We employ the theory of canonical extensions to study residuation algebras whose associated relational structures are functional, i.e., for which the ternary relations associated to the expanded operations admit an interpretation as (possibly partial) functions. Providing a partial answer to a question of Gehrke, we demonstrate that functionality is not definable in the language of residuation algebras (or even residuated lattices), in the sense that no equational or quasi-equational condition in the language of residuation algebras is equivalent to the functionality of the associated relational structures. Finally, we show that the class of Boolean residuation algebras such that the atom structures of their canonical extensions are functional generates the variety of Boolean residuation algebras

Coalgebras and Modal Expansions of Logics

AbstractIn 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))