The correspondence between partial metrics and semivaluations

Partial metrics, or the equivalent weightable quasi-metrics, have been introduced in [Mat94] as part of the study of the denotational semantics of data flow networks (cf. also [Mat95]). The interest in valuations in connection to Domain Theory derives from e.g. [JP89], [Jon89], [Eda94] and [Hec95]. Connections between partial metrics and valuations have been discussed in the literature, e.g. [O’N97], [BS97] and [BSh98]. In each case partial metrics are generated from strictly increasing valuations. We analyze the precise relationship between these two notions. It is well known that characterizations of partial metrics in general are hard to obtain, as witnessed by the open characterization problems in the survey paper Nonsymmetric Topology ([Kün93]). Our approach to obtaining such a characterization involves the isolation of a “mathematically nice” class of spaces, which is sufficiently large to incorporate the quantitative domain theoretic examples involving partial metric spaces. For these purposes we focus on the class of quasi-metric semilattices. These structures, as will be illustrated, arise naturally in Quantitative Domain Theory and include in particular the class of totally bounded Scott domains discussed in [Smy91], the Baire quasi-metric spaces of [Mat95], the complexity spaces of [Sch95] and the interval domain ([EEP97]). We introduce the notion of a semivaluation, which generalizes the fruitful notion of a valuation on a lattice to the context of semilattices and establish a correspondence between partial metric semilattices and semivaluation spaces. AMS Subject Classification: 54E15, 54E35, 06A12, 06B35