Topological Relational Quantales

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 concrete 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 quantales (see [13] for reference) into quantales of relations, so that joins are represented as unions, non-commutative products as compositions of relations, and involutions as taking inverses. One of the most significant conditions for our representation theorem is that the quantaleQ is join-generated by its functional elements, i.e. those a ∈ Q s.t. a∗ ·a ≤ e. This condition is analogous to that given by Jónsson and Tarski in [9] for representability of relation algebras, and indeed our methodology is similar in that is based on notions derived from the theory of canonical extensions. This connection with the theory of canonical extensions is an element of novelty, for, as far as we know, the theorems of representation for quantales that appear in the literature rely on techniques inspired only to functional analysis and the theory of C∗-algebras. On the other hand, it is worth remarking that the theory of canonical extensions such as it is presented in [3, 6, 7, 8] cannot be straightforwardly applied to the context of quantales, because the infinite joins would be destroyed when passing to Qσ. So we work in a setting of meet-dense extensions of quantales that preserve all joins, where we consider only the π-extended operations. Indeed, we assume that Q can be embedded into a P(X) so that ∨ Q is the union in P(X) and Q is meet-dense in P(X)1. A special class of representable quantales is then formed by those topological relational quantales that are join-generated by {graph(f) ⊆ R | f : A→ X is a continuous map and A ⊆ X is open in X}. Let X be a topological space and R ⊆ X ×X an equivalence relation which is lowerIf Q is also a frame, namely if the finite meet induced by the complete join-semilattice order distributes over arbitrary joins, then this condition is equivalent to saying that Q, seen as a frame, is spatial and corresponds to a T1 topology.