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The correspondence between partial metrics and semivaluations. (English) Zbl 1052.54026
The author contributes to the study of the connections between (special) distance functions, namely so-called partial metrics (equivalently, weighted quasi-metrics), and valuations on ordered structures. His investigations are mainly motivated by various important examples from Theoretical Computer Science. The article introduces the notion of a semivaluation which is shown to form a natural generalization of the notion of a valuation on a lattice to the context of semilattices. The central result of the paper establishes a bijection between partial metric semilattices and semivaluation spaces in the context of quasi-metric semilattices. The results shed new light on the nature of partial metrics and allow for a simplified representation of well-known partial metric spaces, where the semivaluation involved is simply the partial metric self-distance function. If $(X,\preceq)$ is a meet semilattice then a real-valued function $f:(X,\preceq)\rightarrow [0,\infty)$ is called a meet valuation if $\forall x,y,z\in X,$ $f(x \sqcap z)\geq f(x\sqcap y)+f(y \sqcap z)-f(y)$ and $f$ is a meet co-valuation if $\forall x,y,z\in X,$ $f(x \sqcap z)\leq f(x \sqcap y)+ f(y \sqcap z)-f(y).$

MSC:
54E35Metric spaces, metrizability
54E15Uniform structures and generalizations
68Q55Semantics
06A12Semilattices
06B35Continuous lattices and posets, applications
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References:
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