Abstract
We relate two approaches to Quantitative Domain Theory, the partial metrics of Steve Matthews and the measurements of Keye Martin, by showing that stable partial metrics are in one-to-one correspondence to weakly modular measurements. It is shown that every ω-algebraic domain admits a partial metric whose associated measurement assigns zero precisely to the set of maximal elements, a condition which features prominently in Martin's work. A partial metric gives rise to a metric in a standard way; we study the conditions under which the resulting space is complete and show that every ω-algebraic domain admits a partial metric of this kind. We discuss a number of examples and counterexamples to locate the strength of our results more exactly.
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Waszkiewicz, P. Quantitative Continuous Domains. Applied Categorical Structures 11, 41–67 (2003). https://doi.org/10.1023/A:1023012924892
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DOI: https://doi.org/10.1023/A:1023012924892