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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:
 54E35 Metric spaces, metrizability 54E15 Uniform structures and generalizations 68Q55 Semantics 06A12 Semilattices 06B35 Continuous lattices and posets, applications
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##### References:
 [1] Birkhoff, G.: Lattice theory. AMS colloquium publications 25 (1984) · Zbl 0063.00402 [2] Bonsangue, M. M.; Van Breugel, F.; Rutten, J. J. M.M.: Generalized metric spacescompletion, topology, and powerdomains via the yoneda embedding. Theoret. comp. Sci. 193, 1-51 (1998) · Zbl 0997.54042 [3] Bukatin, M. A.; Scott, J. S.: Towards computing distances between programs via Scott domains. Lecture notes in computer science 1234, 33-43 (1997) · Zbl 0889.68099 [4] Bukatin, M. A.; Shorina, S. Y.: Partial metrics and co-continuous valuations. Lecture notes in computer science 1378, 125-139 (1998) · Zbl 0945.06006 [5] Crawley, P.; Dilworth, R. P.: Algebraic theory of lattices. (1973) · Zbl 0494.06001 [6] A. Edalat, Domain theory and integration, LICS’94, IEEE Computer Society Press, Silver Spring, MD, 1994. [7] A. Edalat, M.H. Escardo, P.J. Potts, Semantics of Exact Real Arithmetic, Proc. Twelfth Ann. IEEE Symp. on Logic in Computer Science, IEEE Press, New York, 1997, pp. 248--257. [8] Fletcher, P.; Lindgren, W.: Quasi-uniform spaces. (1982) · Zbl 0501.54018 [9] Gunter, C. A.: Semantics of programming languages. (1992) · Zbl 0823.68059 [10] Heckmann, R.: Lower bag domains. Fund. inform. 24, No. 3, 259-281 (1995) · Zbl 0831.68060 [11] Heckmann, R.: Approximation of metric spaces by partial metric spaces. Appl. categorical struct. 7, 71-83 (1999) · Zbl 0993.54029 [12] C. Jones, Probabilistic non-determinism, Ph.D. Thesis, University of Edinburgh, 1989. [13] C. Jones, G. Plotkin, A probabilistic powerdomain of evaluations, in: LICS’89, IEEE Computer Society Press, Silver Spring, MD, 1998, pp. 186--195. · Zbl 0716.06003 [14] H.P. Künzi, Nonsymmetric topology, in: Proc. Szekszárd Conf., Bolyai Soc. Math. Stud. 4 (1993) 303--338. [15] H.P. Künzi, V. Vajner, Weighted quasi-metrics, in: Proc. 8th Summer Conf. on General Topology and Applications; Ann. New York Acad. Sci. 728 (1994) 64--77. · Zbl 0915.54023 [16] S.G. Matthews, Partial metric topology, in: Proc. 8th Summer Conf. on General Topology and Applications; Ann. New York Acad. Sci. 728 (1994) 183--197. · Zbl 0911.54025 [17] Matthews, S. G.: An extensional treatment of lazy data flow deadlock. Theoret. comput. Sci. 151, 195-205 (1995) · Zbl 0872.68110 [18] S.J. O’Neill, Partial metrics, valuations and domain theory, in: S. Andima et al. (Eds.), Proc. 11th Summer Conf. General Topology and Applications; New York. Ann. New York Acad. Sci. 806 (1997) 304--315. [19] S.J. O’Neill, A fundamental study into the theory and application of the partial metric spaces, Ph.D. Thesis, University of Warwick, 1998. [20] Romaguera, S.; Schellekens, M.: Quasi-metric properties of complexity spaces. Topology appl. 98, 311-322 (1999) · Zbl 0941.54028 [21] M.P. Schellekens, The Smyth completion: a common foundation for denotational semantics and complexity analysis, in: Proc. MFPS 11, Electronic Notes in Theoretical Computer Science, Vol. I, Elsevier, Amsterdam, 1995, pp. 211--232. · Zbl 0910.68135 [22] M.P. Schellekens, On upper weightable spaces, in: Proc. 11th Summer Conf. General Topology and Applications; Ann. New York Acad. Sci. 806 (1996) 348--363. · Zbl 0884.54016 [23] M.B. Smyth, Quasi-uniformities: Reconciling Domains with Metric Spaces, Lecture Notes in Computer Science, Vol. 298, Springer, Berlin, 1987, pp. 236--253. [24] Smyth, M. B.: Totally bounded spaces and compact ordered spaces as domains of computation. Topology and category theory in computer science, 207-229 (1991) [25] Weber, H.: Uniform lattices ia generalization of topological Riesz spaces and topological Boolean rings. Ann. mat. Pura appl. 160, 347-370 (1991) · Zbl 0790.06006 [26] Weber, H.: Uniform lattices iiorder continuity and exhaustivity. Ann. mat. Pura appl. (IV) 165, 133-158 (1993) · Zbl 0799.06014