Abstract
In 1982, Trillas introduced the notion of indistinguishability operator with the main aim of fuzzifying the crisp notion of equivalence relation. In the study of such a class of operators, an outstanding property must be stressed. Concretely, there exists a duality relationship between indistinguishability operators and metrics. The aforesaid relationship was deeply studied by several authors that introduced a few techniques to generate metrics from indistinguishability operators and vice versa. In the last years, a new generalization of the metric notion has been introduced in the literature with the purpose of developing mathematical tools for quantitative models in computer science and artificial intelligence. The aforesaid generalized metrics are known as relaxed metrics. The main purpose of the present paper is to explore the possibility of making explicit a duality relationship between indistinguishability operators and relaxed metrics in such a way that the aforementioned classical techniques to generate both concepts, one from the other, can be extended to the new framework.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alghamdi MA, Shahza N, Valero O (2013) Fixed point theorems in generalized metric spaces with applications to computer science. Fixed Point Theory A. 2013:118
Bezdek JC, Harris JD (1978) Fuzzy partitions and relations: an axiomatic basis for clustering. Fuzzy Sets Syst 256:111–127
Bukatin M, Kopperman R, Matthews SG (2014) Some corollaries of the correspondence between partial metric and multivalued equalities. Fuzzy Sets Syst 256:57–72
De Baets B, Mesiar R (1997) Pseudo-metrics and \(T\)-equivalences. Fuzzy Math 5:471–481
De Baets B, Mesiar R (2002) Metrics and \(T\)-equalities. J Math Anal Appl 267:531–547
Demirci M (2011) The order-theoretic duality and relations between partial metrics and local equalities. Fuzzy Sets Syst 192:45–57
Fourman MP, Scott DS (1979) Sheaves and logic, applications of sheaves. In: Fourman M et al (eds) Lecture notes in mathematics, vol 753. Springer, Berlin, pp 302–401
Gottwald S (1992) On t-norms which are related to distances of fuzzy sets. BUSEFAL 50:25–30
Heckmann R (1999) Approximation of metric spaces by partial metric spaces. Appl Categ Struct 7:71–83
Hitzler P, Seda AK (2010) Generalized distance functions in the theory of computation. Comput J 53:443–464
Hitzler P, Seda A (2011) Mathematical aspects of logic programming semantics. CRC Press, Boca Raton
Höhle U (1992) \(M\)-valued sets and sheaves over integral, commutative cl-monoids. In: Rodabaugh SE et al (eds) Applications of category theory to fuzzy subsets. Kluwer Academic Publishers, Dordrecht, pp 33–72
Höhle U (1993) Fuzzy equalities and indistinguishability. In: Proceedings of EUFIT’93, Aachen, vol 1, pp 358–363
Höhle U (1998) Many-valued equalities, singletons and fuzzy partitions. Soft Comput 2:134–140
Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer, Dordrecht
Matthews SG (1994) Partial metric topology. Ann N Y Acad Sci 728:183–197
Ovchinnikov S (1984) Representation of transitive fuzzy relations. In: Skala H, Termini S, Trillas E (eds) Aspects of Vageness. Reidel, Dordrecht, pp 105–118
Recasens J (2010) Indistinguishability operators: modelling fuzzy equalities and fuzzy equivalence relations. Springer, Berlin
Romaguera S, Valero O (2009) A quantitative computational model for complete partial metric spaces via formal balls. Math Struct Comput Sci 19:541–563
Romaguera S, Valero O (2012) Complete partial metric spaces have partially metrizable computational models. Int J Comput Math 89:284–290
Shahza N, Valero O (2013) On 0-complete partial metric spaces and quantitative fixed point techniques in denotational semantics. Abstr Appl Anal 2013:11 Article ID 985095
Trillas E (1982) Assaig sobre les relacions d’indistingibilitat. In: Proceedings of primer congrés català de lògica matemàtica, Barcelona, pp 51–59
Valverde L (1985) On the structure of F-indistinguishability operators. Fuzzy Sets Syst 17:313–328
Zadeh LA (1971) Similarity relations and fuzzy orderings. Inf Sci 3:177–200
Acknowledgements
The authors are very grateful to the reviewers for their valuable suggestions which have helped to improve the content of the paper. This work was partially supported by the Spanish Ministry of Economy and Competitiveness under Grants DPI2017-86372-C3-3-R, TIN2016-81731-REDT (LODISCO II) and AEI/FEDER, UE funds, by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project ref. PROCOE/4/2017 (Direcció General d’Innovació i Recerca, Govern de les Illes Balears), and by project ROBINS. The latter has received research funding from the EU H2020 framework under GA 779776. 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
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Human and animal rights
This article does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Communicated by A. Di Nola.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fuster-Parra, P., Martín, J., Miñana, JJ. et al. A study on the relationship between relaxed metrics and indistinguishability operators. Soft Comput 23, 6785–6795 (2019). https://doi.org/10.1007/s00500-018-03675-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-018-03675-9