Topological Relational Quantales

Abstract We introduce and present results about a class of quantales, the topological relational quantales, that can be associated with tuples (X,R) such that X is a topological space and R is a lower-semicontinuous equivalence relation on X (see Definition 1 below). Our motivating case studies and examples of topological relational quantales include con-crete quantales such as the relational quantales of [11] and the quantale Pen introduced in [12] (see Examples 1–5 below). In this setting, we focus in particular on the question of the representability of quantales into quantales of binary relations on a set, which has already been studied by some authors in the literature: for instance, in [1] it is shown that every quantale Q can be order-embedded into a quantale of relations in such a way that the noncommutative product is represented as relational composition, but this method does not extend to involutive quantales and the join of Q is not in general represented as the union. In [10], it is shown that any involutive quantale is embeddable into a quantale of join-semilattice endomorphisms, which give back the quantales of relations when the sup lattice is a powerset. We present sufficient conditions for the representability of unital involutive quantale