The correspondence between partial metrics and semivaluations

Abstract

Partial metrics, or the equivalent weightable quasi-metrics, have been introduced in Matthews (Proc. 8th Summer Conf. on General Topology and Applications; Ann. New York Acad. Sci. 728 (1994) 183) as part of the study of the denotational semantics of data flow networks (Theoret. Comput. Sci. 151 (1995) 195). The interest in valuations in connection to Domain Theory derives from e.g. Jones and Plotkin (LICS ’89, IEEE Computer Society Press, Silver Spring, MD, 1998, pp. 186–195), Jones (Ph.D. Thesis, University of Edinburgh, 1989), Edalat (LICS’94, IEEE Computer Society Press, Silver Spring, MD, 1994) and Heckmann (Fund. Inform. 24(3) (1995) 259). Connections between partial metrics and valuations have been discussed in the literature, e.g. O'Neill (in: S. Andima et al. (Eds.), Proc. 11th Summer Conf. on General Topology and Applications; Ann. New York Acad. Sci. 806 (1997) 304), Bukatin and Scott (in: S. Adian, A. Nerode (Eds.), Logical Foundations of Computer Science, Lecture Notes in Computer Science, Vol. 1234, Springer, Berlin, 1997, pp. 33–43) and Bukatin and Shorina (in: M. Nivat (Ed.), Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science, Vol. 1378, Springer, Berlin, 1998, pp. 125–139). 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 (Kúnzi (Bolyai Soc. Math. Stud. 4 (1993) 303). 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 Smyth (in: G.M. Reed, A.W. Roscoe, R.F. Wachter (Ed.), Topology and Category Theory in Computer Science, Oxford University Press, Oxford, 1991, pp. 207–229), the Baire quasi-metric spaces of (Theoret. Comput. Sci. 151 (1995) 195), the complexity spaces of Schellekens (in: Proc. MFPS 11, Electronic Notes in Theoretical Computer Science, Vol. I, Elsevier, Amsterdam, 1995, pp. 211–232) and the interval domain (Proc. Twelfth Ann. IEEE Symp. on Logic in Computer Science, IEEE Press, New York, 1997, pp. 248–257). 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.