The order-theoretic duality and relations between partial metrics and local equalities

For a GL-monoid L, Höhle introduced L-valued sets to formalize the mathematical theory of identity and existence in monoidal logic. For the purpose of analyzing deadlock situations in data flow networks, the notion of partial metric space was proposed by Matthews. Later on a lattice-theoretic generalization of (pseudo) partial metric spaces, extending the range of (pseudo) partial metrics to a value lattice V, were studied under the name of V-(pseudo) pmetric spaces. Referring to a GL-monoid L and a dual GL-monoid V, we show that L-valued sets are order-theoretically dual to V-pseudopmetric spaces, and apply this duality to the representation of V-pseudopmetric spaces and the determination of their various categorical properties. In addition to this, the present paper provides not only some new results about the representation of L-valued sets and their categories, but also some non-trivial categorical connections between L-valued sets and V-pseudopmetric spaces.