Preorders, Partial Semigroups, and Quantales

It is known that each powerset quantale is embeddable into some relational unital quantale whose underlying set is the powerset of some preorder. An aim of this paper is to understand the relational embedding as a relationship between quantales and preorders. For that, this paper introduces the notion of weak preorders, a functor from the category of weak preorders to the category of partial semigroups, and a functor from the category of partial semigroups to the category of quantales and lax homomorphisms. By using these two functors, this paper shows a correspondence among four classes of weak preorders (including the class of ordinary preorders), four classes of partial semigroups, and four classes of quantales. As a corollary of the correspondence, we can understand the relational embedding map as a natural transformation between functors onto certain category of quantales.