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Partial metrics and co-continuous valuations. (English) Zbl 0945.06006
Nivat, Maurice (ed.), Foundations of software science and computation structures. 1st international conference, FoSSaCS ’98. Held as part of the joint European conferences on Theory and practice of software, ETAPS ’98, Lisbon, Portugal, March 28 - April 4, 1998. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1378, 125-139 (1998).
Summary: The existence of deep connections between partial metrics and valuations is well known in domain theory. However, the treatment of non-algebraic continuous Scott domains has been not quite satisfactory so far. In this paper we return to the continuous normalized valuations $\mu$ on the systems of open sets and introduce notions of co-continuity ($\{U_i, i\in I\}$ is a filtered system of open sets $\Rightarrow \mu\left(\text{Int}\left(\bigcap_{i\in I}U_i\right)\right)= \inf_{i\in I} \mu(U_i)$) and strong non-degeneracy ($U\subset V$ are open sets $\Rightarrow \mu(U)< \mu(V)$) for such valuations. We call the resulting class of valuations CC-valuations. The first central result of this paper is a construction of CC-valuations for Scott topologies on all continuous dcpo’s with countable bases. This is a surprising result because neither co-continuous nor strongly non-degenerate valuations are usually possible for ordinary Hausdorff topologies. Another central result is a new construction of partial metrics. Given a continuous Scott domain $A$ and a CC-valuation $\mu$ on the system of Scott open subsets of $A$, we construct a continuous partial metric on $A$ which yields the Scott topology as $u(x,y)= \mu(A\setminus (C_x\cap C_y))- \mu(I_x\cap I_y)$, where $C_x= \{y\in A\mid y\sqsubseteq x\}$ and $I_x= \{y\in A\mid\{x,y\}$ is unbounded}. This construction covers important cases based on the real line and makes it possible to obtain an induced metric on $\text{Total}(A)$ without the unpleasant restrictions known from earlier work. For the entire collection see [Zbl 0889.00031].

06B35Continuous lattices and posets, applications