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On Matthews’ Relationship Between Quasi-Metrics and Partial Metrics: An Aggregation Perspective


Borsík and Doboš studied the problem of how to merge a family of metric spaces into a single one through a function. They called such functions metric preserving and provided a characterization of them in terms of the so-called triangle triplets. Since then, different papers have extended their study to the case of generalized metric spaces. Concretely, Mayor and Valero (Inf Sci 180:803–812, 2010) provided two characterizations of those functions, called quasi-metric aggregation functions, that allows us to merge a collection of quasi-metric spaces into a new one. In Massanet and Valero (in: Sainz-Palmero et al (eds) Proceedings of the 16th Spanish conference on fuzzy technology and fuzzy logic, European Society for Fuzzy Logic and Techonology, Valladolid, 2012) gave a characterization of the functions, called partial metric aggregation function, that are useful for merging a collection of partial metric spaces into single one as final output. Inspired by the preceding work, Martín et al. (in: Bustince et al (eds) Aggregation functions in theory and in practice. Advances in intelligent systems and computing, vol 228, Springer, Berlin, 2013) addressed the problem of constructing metrics from quasi-metrics, in a general way, using a class of functions that they called metric generating functions. In particular, they solved the posed problem providing a characterization of such functions and, thus, all ways under which a metric can be induced from a quasi-metric from an aggregation viewpoint. Following this idea, we propose the same problem in the framework of partial metric spaces. So, we characterize those functions that are able to generate a quasi-metric from a partial metric, and conversely, in such a way that Matthews’ relationship between both type of generalized metrics is retrieved as a particular case. Moreover, we study if both, the partial order and the topology induced by a partial metric or a quasi-metric, respectively, are preserved by the new method in the spirit of Matthews. Furthermore, we discuss the relationship between the new functions and those families introduced in the literature, i.e., metric preserving functions, quasi-metric aggregation functions, partial metric aggregation functions and metric generating functions.

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  1. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. Springer, Heidelberg (1999)

    Book  Google Scholar 

  2. Alghamdi, M.A., Shahzad, N., Valero, O.: Fixed point theorems in generalized metric spaces with applications to computer science. Fixed Point Theory Appl. 2013, 118 (2013)

    Article  MathSciNet  Google Scholar 

  3. Borsík, J., Doboš, J.: On a product of metric spaces. Mathematica Slovaka 31, 193–205 (1981)

    MATH  Google Scholar 

  4. Deza, M.M., Deza, E.: Encyclopedia of Distances. Springer, Berlin (2009)

    Book  Google Scholar 

  5. Doboš, J.: Metric Preserving Functions. Štroffek, Košice (1998)

    MATH  Google Scholar 

  6. García-Raffi, L.M., Romaguera, S., Sánchez-Pérez, E.A.: The supremum asymmetric norm on sequence algebras: a general framework to measure complexity spaces. Electron. Notes Theor. Comput. Sci. 74, 39–50 (2003)

    Article  Google Scholar 

  7. García-Raffi, L.M., Romaguera, S., Sánchez-Pérez, E.A.: Sequence spaces and asymmetric norms in the theory of computational complexity. Math. Comput. Modell. 36, 1–11 (2002)

    Article  MathSciNet  Google Scholar 

  8. Goubault-Larrecq, J.: Non-Hausdorff Topology and Domain Theory. Cambridge University Press, New York (2013)

    Book  Google Scholar 

  9. Haghi, R.H., Rezapour, Sh, Shahzad, N.: Be careful on partial metric fixed point results. Topol. Its Appl. 160(3), 450–454 (2013)

    Article  MathSciNet  Google Scholar 

  10. Hitzler, P., Seda, A.K.: Mathematical Aspects of Logic Programming Semantics. CRC Press, Boca Raton (2010)

    MATH  Google Scholar 

  11. Martín, J., Mayor, G., Valero, O.: On the symmetrization of quasi-metrics: an aggregation perspective. In: Bustince, H., et al. (eds.) Aggregation Functions in Theory and in Practice. Advances in Intelligent Systems and Computing, vol. 228, pp. 319–331. Springer, Berlin (2013)

    MATH  Google Scholar 

  12. Martín, J., Mayor, G., Valero, O.: On quasi-metric aggregation functions and fixed point theorems. Fuzzy Sets Syst. 228, 88–104 (2013)

    Article  MathSciNet  Google Scholar 

  13. Massanet, S., Valero, O.: New results on metrics aggregation. In: Sainz-Palmero, G.I. et al. (eds.) Proceedings of the 16th Spanish Conference on Fuzzy Technology and Fuzzy Logic, European Society for Fuzzy Logic and Techonology, Valladolid, pp. 558–563 (2012)

  14. Matthews, S.G.: Partial metric topology. Ann. N. Y. Acad. Sci. 728, 183–197 (1994)

    Article  MathSciNet  Google Scholar 

  15. Matthews, S.G.: An extensional treatment of lazy data flow deadlock. Theor. Comput. Sci. 151, 195–205 (1995)

    Article  MathSciNet  Google Scholar 

  16. Mayor, G., Valero, O.: Aggregation of asymmetric distances in computer science. Inf. Sci. 180, 803–812 (2010)

    Article  MathSciNet  Google Scholar 

  17. Shahzad, N., Valero, O.: On \(0\)-complete partial metric spaces and quantitative fixed point techniques in Denotational Semantics. Abstr. Appl. Anal. 2013, 11, Article ID 985095

  18. Pestov, V., Stojmirović, A.: Indexing schemes for similarity search: an illustrated paradigm. Fundamenta Informaticae 70, 367–385 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Pestov, V., Stojmirović, A.: Indexing schemes for similarity search in datasets of short protein fragments. Inf. Syst. 32, 1145–1165 (2007)

    Article  Google Scholar 

  20. Romaguera, S., Sánchez-Pérez, E.A., Valero, O.: Computing complexity distances between algorithms. Kybernetika 39, 569–582 (2003)

    MathSciNet  MATH  Google Scholar 

  21. Romaguera, S., Schellekens, M.P.: Quasi-metric properties of complexity spaces. Topol. Its Appl. 98, 311–322 (1999)

    Article  MathSciNet  Google Scholar 

  22. Romaguera, S., Schellekens, M.P., Valero, O.: The complexity space of partial functions: a connection between complexity analysis and denotational semantics. Int. J. Comput. Math. 88, 1819–1829 (2011)

    Article  MathSciNet  Google Scholar 

  23. Romaguera, S., Tirado, P., Valero, O.: New results on mathematical foundations of asymptotic complexity analysis of algorithms via complexity space. Int. J. Comput. Math. 89, 1728–1741 (2012)

    Article  MathSciNet  Google Scholar 

  24. Romaguera, S., Valero, O.: Asymptotic complexity analysis and denotational semantics for recursive programs based on complexity spaces. In: Afzal, M. (ed.) Semantics Advances in Theories and Mathematical Models, vol. 1, pp. 99–120. InTech Open Science, Rijeka (2012)

    Google Scholar 

  25. Schellekens, M.P.: The Smyth completion: a common foundation for denotational semantics and complexity analysis. Electron. Notes Theor. Comput. Sci. 1, 211–232 (1995)

    Article  MathSciNet  Google Scholar 

  26. Shahzad, N., Valero, O.: Fixed point theorems in quasi-metric spaces and the specialization partial order. Fixed Point Theory 19(2), 733–750 (2018)

    Article  MathSciNet  Google Scholar 

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Correspondence to Juan-José Miñana.

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The authors acknowledge financial support from FEDER/Ministerio de Ciencia, Innovación y Universidades-Agencia Estatal de Investigación/\(_{-}\)Proyecto PGC2018-095709-B-C21. This work is also partially supported by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by Project PROCOE/4/2017 (Direcció General d’Innovació i Recerca, Govern de les Illes Balears) and by projects ROBINS and BUGWRIGHT2. These two latest projects have received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreements Nos. 779776 and 871260, respectively. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein.

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Miñana, JJ., Valero, O. On Matthews’ Relationship Between Quasi-Metrics and Partial Metrics: An Aggregation Perspective. Results Math 75, 47 (2020).

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  • Partial metric space
  • quasi-metric space
  • aggregation function
  • quasi-metric generating
  • partial metric generating

Mathematics Subject Classification

  • 06A06
  • 54A10
  • 54C30
  • 54E40
  • 54E35