The correspondence between partial metrics and semivaluations
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 \(\forall x,y,z\in X,\) \(f(x \sqcap z)\leq f(x \sqcap y)+ f(y \sqcap z)-f(y).\)
scientific article; zbMATH DE number 2081676
scientific article