Partial metrics, valuations, and domain theory. (English) Zbl 0889.54018

O’Neill, S. J.

Andima, Susan (ed.) et al., Papers on general topology and applications. Papers presented at the 11th summer conference at the University of Southern Maine, Gorham, ME, USA, August 10–13, 1995. New York, NY: The New York Academy of Sciences. Ann. N. Y. Acad. Sci. 806, 304-315 (1996).

The author generalizes the partial metrics of S. G. Matthews [Partial metric topology, in: ‘Proceedings of the 8th Summer Conference on Topology and Applications’ (S. Andima et al, Eds.). Annals of the New York Academy of Sciences. 728, 183-197, New York.] by allowing negative values as distances. The dual \(p^*\) of a partial metric \(p\) is then defined by \(p^*(x,y)=p(x,y)-p(x,x)-p(y,y)\), the associated metric \(d\) by \(d(x,y)=p(x,y)+p^*(x,y)\). The main part of the paper is concerned with partial metrics defined on valuation spaces. A valuation space is a meet semilattice \((S,\sqsubseteq)\) with suprema for pairs of bounded elements (consistent semilattices) which is supplied with a real-valued function \(\mu\) which is strictly monotone and satisfies the modularity law \(\mu(x)+\mu(y)=\mu(x\sqcap y)+\mu(x\sqcup y)\) for bounded \(\{x,y\}\). On these spaces, a partial metric is defined by \(p(x,y)=-\mu(x\sqcap y)\). If this partial metric space is complete and the underlying meet semilattice has suprema for all nonempty bounded subsets (i.e. is a complete semilattice) then the valuation \(\mu\) is necessarily Scott-continuous and the partial metric topology coincides with the Scott topology. If the associated metric space is compact, then directed completeness of the meet semilattice follows automatically and in addition the metric topology coincides with the Lawson topology. Finally, the author singles out the special case of \(\omega\)-prime-algebraic Scott-domains for which a suitable valuation can be defined in terms of the order-theoretic structure.
For the entire collection see [Zbl 0879.00049].

Reviewer: Ph.Sünderhauf (London)

MSC:

54E35	Metric spaces, metrizability
54E55	Bitopologies
06B30	Topological lattices
06B35	Continuous lattices and posets, applications

Keywords:

partial metric; valuation; bitopology; domain