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A characterization of partial metrizability: Domains are quantifiable. (English) Zbl 1043.54011
Following {\it S. G. Matthews} [Ann. N.Y. Acad. Sci. 728, 183--197 (1994; Zbl 0911.54025)] a quasi-metric space $(X, d)$ is called weightable if there exists a mapping $w: X\to [0,\infty)$ such that $d(x, y) +w(x)= d(y, x)+w(y)$ for each pair $x,y\in X$. It is known that every weightable quasi-metric space is quasi-developable. However, it is an open problem to characterize those quasi-uniform spaces $(X, {\Cal U})$ for which there exists a weightable quasi-metric $d$ an $X$ such that $(X,{\Cal U})= (X,{\Cal U}_d)$. Therefore the following theorem is interesting. Theorem 1. Let $(X, {\Cal U})$ be a quasi-uniform join semilattice, i.e., a quasi-uniform space such that (i) ($X,\le_{\Cal U})$ is a partially ordered set such that for each pair $x,y\in X$ there exists a supremum $x\vee y$, where $\le_{\Cal U}= \bigcap\{U\mid U\in{\Cal U}\}$, and (ii) the mapping $(x, y)\mapsto x\vee y$ is quasi-uniformly continuous. Then there exists a weightable invariant quasi-metric $d$ an $X$ such that $(X, {\Cal U})= (X, {\Cal U}_d)$ if and only if there exists a $Q$-join co-valuation on $(X,{\Cal U})$. It is shown that this theorem is strong enough to solve a major problem from domain theory. Theorem 2. Every domain (i.e., every directed complete partially ordered set $(P, \sqsubseteq)$ with a countable basis) is quantifiable (i.e., there exists a weightable quasi-metric $d$ an $P$ such that the topology induced by $d$ coincides with the Scott-topology of $P$ and the associated preorder $\le_d$ coincides with $\sqsubseteq$).

MSC:
54E15Uniform structures and generalizations
54E35Metric spaces, metrizability
06A12Semilattices
06B35Continuous lattices and posets, applications
WorldCat.org
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References:
[1] G. Birkhoff, Lattice Theory, Vol. 25, AMS Colloquium Publications, Providence, RI, 1984.
[2] Bonsangue, M. M.; Van Breugel, F.; Rutten, J. J. M.M.: Generalized metric spacescompletion, topology, and powerdomains via the yoneda embedding. Theoret. comput. 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 (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, Lecture Notes in Computer Science ’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, 1997, pp. 248--257.
[8] Flagg, B.; Kopperman, R.: The asymmetric topology of computer science. Lecture notes in computer science 802, 544-553 (1993)
[9] Flagg, R. C.; Kopperman, R.: Continuity spacesreconciling domains and metric spaces. Theoret. comput. Sci. 177, No. 1, 111-138 (1997) · Zbl 0901.68109
[10] R. Flagg, P. Sünderhauf, K. Wagner, A logical approach to quantitative domain theory, Preprint, 1996.
[11] Flagg, R. C.; Sünderhauf, P.: The essence of ideal completion in quantitative form. Theoret. comput. Sci. 278, 141-158 (2002) · Zbl 1002.68091
[12] Fletcher, P.; Lindgren, W.: Quasi-uniform spaces. (1982) · Zbl 0501.54018
[13] Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S.: A compendium of continuous lattices. (1980) · Zbl 0452.06001
[14] Gunter, C. A.: Semantics of programming languages. (1992) · Zbl 0823.68059
[15] Heckmann, R.: Lower bag domains. Fund. inform. 24, No. 3, 259-281 (1995) · Zbl 0831.68060
[16] R. Heckmann, Spaces of valuations, in: Proc. 11th Summer Conf. on General Topology and Applications, Annals of the New York Academy Science, Vol. 806, 1996, pp. 174--200. · Zbl 0905.54006
[17] Heckmann, R.: Approximation of metric spaces by partial metric spaces. Appl. categor. Struct. 7, 71-83 (1999) · Zbl 0993.54029
[18] C. Jones, G. Plotkin, A probabilistic powerdomain of evaluations, in: Lecture Notes in Computer Sciences ’89, IEEE Computer Society Press, Silver Spring, MD, 1989, pp. 186--195.
[19] C. Jones, Probabilistic non-determinism, Ph.D. Thesis, University of Edinburgh, 1989.
[20] Jones, N.: Computability and complexity from a programming perspective, foundations of computing series. (1997) · Zbl 0940.68050
[21] H.P. Künzi, Nonsymmetric topology, in: Proc. Szekszárd Conf., Bolyai Soc. Math. Studies, Vol. 4, 1993, pp. 303--338.
[22] Künzi, H. P.; Schellekens, M. P.: On the yoneda completion of a quasi-metric space. Theoret. comput. Sci. 278, 159-194 (2002) · Zbl 1025.54014
[23] 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
[24] J.D. Lawson, Valuations on continuous lattices, in: R.E. Hoffman (Ed.), Continuous Lattices and Related Topics, Mathematik Arbeitspapiere, Vol. 27, Universität Bremen.
[25] S.G. Matthews, Partial metric topology, in: Proc. 8th Summer Conf. on General Topology and Applications, Ann. New York Acad. Sci., Vol. 728, 1994, pp. 183--197. · Zbl 0911.54025
[26] Matthews, S. G.: An extensional treatment of lazy data flow deadlock. Theoret. comput. Sci. 151, 195-205 (1995) · Zbl 0872.68110
[27] Nachbin, L.: Topology and order. Van nostrand mathematical studies 4 (1965) · Zbl 0131.37903
[28] O’neill, S. J.: Partial metrics, valuations and domain theory. Annals of the New York Academy of sciences 806, 304-315 (1997)
[29] S.J. O’Neill, A fundamental study into the theory and application of the partial metric spaces, Ph.D. Thesis, University of Warwick, 1998.
[30] G. Plotkin, course notes, 1983.
[31] Romaguera, S.; Schellekens, M.: Quasi-metric properties of complexity spaces. Topol. appl. 98, 311-322 (1999) · Zbl 0941.54028
[32] S. Romaguera, M. Schellekens, Duality and quasi-normability for complexity spaces, Appl. General Topology 3 (2002). · Zbl 1022.54018
[33] S. Romaguera, M. Schellekens, A characterization of norm-weightable Riesz spaces, preprint. · Zbl 1270.46004
[34] 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
[35] M.P. Schellekens, On upper weightable spaces, in: Proc. 11th Summer Conf. on General Topology and Applications. Annals of the New York Academy Science, Vol. 806, 1996, pp. 348--363. · Zbl 0884.54016
[36] M.P. Schellekens, Complexity Spaces Revisited, extended abstract, in: Proc. 8th Prague Topology Conf., Topology Atlas, 1996.
[37] Schellekens, M. P.: Complexity spaceslifting & directedness. Topology proc. 22, 403-425 (1999)
[38] M.P. Schellekens, The correspondence between partial metrics and semivaluations, Theoret. Comput. Sci., to appear. · Zbl 1052.54026
[39] M.P. Schellekens, Extendible spaces, Appl. General Topology 3 (2002). · Zbl 1038.68078
[40] Seda, A. K.: Quasi-metrics and fixed points in computing. Bull. EATCS 60, 154-163 (1996)
[41] M.B. Smyth, Quasi-uniformities: reconciling domains with metric spaces, Lecture Notes in Computer Science, Vol. 298, Springer, Berlin, 1987, pp. 236--253.
[42] Smyth, M. B.: Totally bounded spaces and compact ordered spaces as domains of computation. Topology and category theory in computer science, 207-229 (1991)
[43] Smyth, M. B.: Completeness of quasi-uniform and syntopological spaces. J. London math. Soc. 49, 385-400 (1994) · Zbl 0798.54036
[44] P. Sünderhauf, The Smyth-completion of a quasi-uniform space, in: M. Droste, Y. Gurevich (Eds.), Semantics of Programming Languages and Model Theory, Algebra, Logic and Applications, Vol. 5, Gordon and Breach Science Publ., London, 1993, pp. 189--212. · Zbl 0799.54023
[45] Sünderhauf, P.: Quasi-uniform completeness in terms of Cauchy nets. Acta math. Hungar. 69, 47-54 (1995) · Zbl 0845.54016
[46] 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
[47] Weber, H.: Uniform lattices iiorder continuity and exhaustivity. Ann. mat. Pura appl. (IV) 165, 133-158 (1993) · Zbl 0799.06014