44 research outputs found

    Characterizing a class of completable fuzzy metric spaces

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    In this paper we give a characterization of the class of completable strong (non-Archimedean) fuzzy metric spaces, in the sense of George and Veeramani.Gregori Gregori, V.; Miñana, JJ.; Morillas, S.; Sapena Piera, A. (2016). Characterizing a class of completable fuzzy metric spaces. Topology and its Applications. 203:3-11. doi:10.1016/j.topol.2015.12.070S31120

    Completable fuzzy metric spaces

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    [EN] In the context of fuzzy metrics in the sense of George and Veeramani, we introduce the concept of stratified fuzzy metric. Many well-known fuzzy metrics are stratified. We prove that stratified strong fuzzy metric spaces (X, M, *) are completable, under the assumption that * is integral (positive). In particular, stratified fuzzy ultrametric spaces are completable. (C) 2017 Elsevier B.V. All rights reserved.Valentin Gregori acknowledges the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2015-64373-P.Gregori Gregori, V.; Miñana, J.; Sapena Piera, A. (2017). Completable fuzzy metric spaces. Topology and its Applications. 225:103-111. https://doi.org/10.1016/j.topol.2017.04.016S10311122

    On Principal Fuzzy Metric Spaces

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    [EN] In this paper, we deal with the notion of fuzzy metric space (X, M, *), or simply X, due to George and Veeramani. It is well known that such fuzzy metric spaces, in general, are not completable and also that there exist p-Cauchy sequences which are not Cauchy. We prove that if every p-Cauchy sequence in X is Cauchy, then X is principal, and we observe that the converse is false, in general. Hence, we introduce and study a stronger concept than principal, called strongly principal. Moreover, X is called weak p-complete if every p-Cauchy sequence is p-convergent. We prove that if X is strongly principal (or weak p-complete principal), then the family of p-Cauchy sequences agrees with the family of Cauchy sequences. Among other results related to completeness, we prove that every strongly principal fuzzy metric space where M is strong with respect to an integral (positive) t-norm * admits completion.Samuel Morillas acknowledges financial support from Ministerio de Ciencia e Innovacion of Spain under grant PID2019-107790RB-C22 funded by MCIN/AEI/10.13039/501100011033. JuanJose Minana acknowledges financial support from Proyecto PGC2018-095709-B-C21 financiado por MCIN/AEI/10.13039/501100011033 y FEDER "Una manera de hacer Europa" and from project BUGWRIGHT2. This last project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 871260. Also acknowledge support of Generalitat Valenciana under grant CIAICO/2021/137. 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.Gregori Gregori, V.; Miñana, J.; Morillas, S.; Sapena Piera, A. (2022). On Principal Fuzzy Metric Spaces. Mathematics. 10(16):1-10. https://doi.org/10.3390/math10162860110101

    On completable fuzzy metric spaces

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    In this paper we construct a non-completable fuzzy metric space in the sense of George and Veeramani which allows to answer an open question related to continuity on the real parameter t. In addition, the constructed space is not strong (non-Archimedean).Juan Jose Minana acknowledges the support of Conselleria de Educacion, Formacion y Empleo (Programa Vali+d para investigadores en formacion) of Generalitat Valenciana, Spain and the support of Universitat Politecnica de Valencia under Grant PAID-06-12 SP20120471.Gregori Gregori, V.; Miñana, J.; Morillas, S. (2015). On completable fuzzy metric spaces. Fuzzy Sets and Systems. 267:133-139. https://doi.org/10.1016/j.fss.2014.07.009S13313926

    Cauchyness and convergence in fuzzy metric spaces

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    [EN] In this paper we survey some concepts of convergence and Cauchyness appeared separately in the context of fuzzy metric spaces in the sense of George and Veeramani. For each convergence (Cauchyness) concept we find a compatible Cauchyness (convergence) concept. We also study the relationship among them and the relationship with compactness and completeness (defined in a natural sense for each one of the Cauchy concepts). In particular, we prove that compactness implies p-completeness.Almanzor Sapena acknowledges the support of Ministry of Economy and Competitiveness of Spain under grant TEC2013-45492-R. Valentín Gregori acknowledges the support of Ministry of Economy and Competitiveness of Spain under grant MTM 2012-37894-C02-01.Gregori Gregori, V.; Miñana, J.; Morillas, S.; Sapena Piera, A. (2017). Cauchyness and convergence in fuzzy metric spaces. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 111(1):25-37. https://doi.org/10.1007/s13398-015-0272-0S25371111Alaca, C., Turkoglu, D., Yildiz, C.: Fixed points in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 29, 1073–1078 (2006)Edalat, A., Heckmann, R.: A computational model for metric spaces. Theor. Comput. Sci. 193, 53–73 (1998)Engelking, R.: General topology. PWN-Polish Sci. Publ, Warsawa (1977)Fang, J.X.: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 46(1), 107–113 (1992)George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64, 395–399 (1994)George, A., Veeramani, P.: Some theorems in fuzzy metric spaces. J. Fuzzy Math. 3, 933–940 (1995)George, A., Veeramani, P.: On some results of analysis for fuzzy metric spaces. Fuzzy Sets Syst. 90, 365–368 (1997)Grabiec, M.: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 27, 385–389 (1989)Gregori, V., Romaguera, S.: Some properties of fuzzy metric spaces. Fuzzy Sets Syst. 115, 485–489 (2000)Gregori, V., Romaguera, S.: On completion of fuzzy metric spaces. Fuzzy Sets Syst. 130, 399–404 (2002)Gregori, V., Romaguera, S.: Characterizing completable fuzzy metric spaces. Fuzzy Sets Syst. 144, 411–420 (2004)Gregori, V., López-Crevillén, A., Morillas, S., Sapena, A.: On convergence in fuzzy metric spaces. Topol. Appl. 156, 3002–3006 (2009)Gregori, V., Miñana, J.J.: Some concepts realted to continuity in fuzzy metric spaces. In: Proceedings of the conference in applied topology WiAT’13, pp. 85–91 (2013)Gregori, V., Miñana, J.-J., Sapena, A.: On Banach contraction principles in fuzzy metric spaces (2015, submitted)Gregori, V., Miñana, J.-J.: std-Convergence in fuzzy metric spaces. Fuzzy Sets Syst. 267, 140–143 (2015)Gregori, V., Miñana, J.-J.: Strong convergence in fuzzy metric spaces Filomat (2015, accepted)Gregori, V., Miñana, J.-J., Morillas, S.: Some questions in fuzzy metric spaces. Fuzzy Sets Syst. 204, 71–85 (2012)Gregori, V., Miñana, J.-J., Morillas, S.: A note on convergence in fuzzy metric spaces. Iran. J. Fuzzy Syst. 11(4), 75–85 (2014)Gregori, V., Morillas, S., Sapena, A.: On a class of completable fuzzy metric spaces. Fuzzy Sets Syst. 161, 2193–2205 (2010)Gregori, V., Morillas, S., Sapena, A.: Examples of fuzzy metric spaces and applications. Fuzzy Sets Syst. 170, 95–111 (2011)Kramosil, I., Michalek, J.: Fuzzy metric and statistical metric spaces. Kybernetika 11, 326–334 (1975)Mihet, D.: On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets Syst. 158, 915–921 (2007)Mihet, D.: Fuzzy φ\varphi φ -contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets Syst. 159, 739–744 (2008)Mihet, D.: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 144, 431–439 (2004)Mishra, S.N., Sharma, N., Singh, S.L.: Common fixed points of maps on fuzzy metric spaces Internat. J. Math. Math. Sci. 17(2), 253–258 (1994)Morillas, S., Sapena, A.: On Cauchy sequences in fuzzy metric spaces. In: Proceedings of the conference in applied topology (WiAT’13), pp. 101–108 (2013)Ricarte, L.A., Romaguera, S.: A domain-theoretic approach to fuzzy metric spaces. Topol. Appl. 163, 149–159 (2014)Sherwood, H.: On the completion of probabilistic metric spaces. Z.Wahrschein-lichkeitstheorie verw. Geb. 6, 62–64 (1966)Sherwood, H.: Complete Probabilistic Metric Spaces. Z. Wahrschein-lichkeitstheorie verw. Geb. 20, 117–128 (1971)Tirado, P.: On compactness and G-completeness in fuzzy metric spaces. Iran. J. Fuzzy Syst. 9(4), 151–158 (2012)Tirado, P.: Contraction mappings in fuzzy quasi-metric spaces and [0,1]-fuzzy posets. Fixed Point Theory 13(1), 273–283 (2012)Vasuki, R., Veeramani, P.: Fixed point theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets Syst. 135(3), 415–417 (2003)Veeramani, P.: Best approximation in fuzzy metric spaces. J. Fuzzy Math. 9, 75–80 (2001

    BICOMPLETABLE STANDARD FUZZY QUASI-METRIC SPACE

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    In this paper we introduce the definition of standard fuzzy quasi-metric space then we discuss several properties after we give an example to illustrate this notion. Then we showed the existence of a standard fuzzy quasi-metric space which is not bicompletable. Here we prove that every bicompletable standard qussi-metric space admits a unique [ up to F-isometic ] bicompletion

    A Banach contraction principle in fuzzy metric spaces defined by means of t-conorms

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    [EN] Fixed point theory in fuzzy metric spaces has grown to become an intensive field of research. The difficulty of demonstrating a fixed point theorem in such kind of spaces makes the authors to demand extra conditions on the space other than completeness. In this paper, we introduce a new version of the celebrated Banach contracion principle in the context of fuzzy metric spaces. It is defined by means of t-conorms and constitutes an adaptation to the fuzzy context of the mentioned contracion principle more "faithful" than the ones already defined in the literature. In addition, such a notion allows us to prove a fixed point theorem without requiring any additional condition on the space apart from completeness. Our main result (Theorem 1) generalizes another one proved by Castro-Company and Tirado. Besides, the celebrated Banach fixed point theorem is obtained as a corollary of Theorem 1.Juan-José Miñana acknowledges 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 No 779776 and No 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. Valentín Gregori acknowledges the support of Generalitat Valenciana under grant AICO-2020-136.Gregori Gregori, V.; Miñana, J. (2021). A Banach contraction principle in fuzzy metric spaces defined by means of t-conorms. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 115(3):1-11. https://doi.org/10.1007/s13398-021-01068-6S111115

    Fuzzy quasi-metrics for the Sorgenfrey line

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    [EN] We endow the set of real numbers with a family of fuzzy quasi-metrics, in the sense of George and Veeramani, which are compatible with the Sorgenfrey topology. Although these fuzzy quasi-metrics are not deduced explicitly from a quasi-metric, they possess interesting properties related to completeness. For instance, we prove that they are balanced and complete in the sense of Doitchinov and that only one of them is right K-sequentially complete. We also observe that compatible fuzzy quasi-metrics for the Sorgenfrey line cannot be left (weakly right) K-sequentially complete.1 The first and second authors acknowledge the support of Spanish Ministry of Education and Science under Grant MTM 2009-12872-C02-01.Gregori Gregori, V.; Morillas, S.; Roig, B. (2013). Fuzzy quasi-metrics for the Sorgenfrey line. Fuzzy Sets and Systems. 222:98-107. https://doi.org/10.1016/j.fss.2012.11.001S9810722

    Some questions in fuzzy metric spaces

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    The George and Veeramani's fuzzy metric defined by M∗(x,y,t)=min{x,y}+tmax{x,y}+tM^*(x,y,t)=\frac{min\{x,y\}+t}{max\{x,y\}+t} on [0,∞[[0,\infty[ (the set of non-negative real numbers) has shown some advantages in front of classical metrics in the process of filtering images. In this paper we study from the mathematical point of view this fuzzy metric and other fuzzy metrics related to it. As a consequence of this study we introduce, throughout the paper, some questions relative to fuzzy metrics. Also, as another practical application, we show that this fuzzy metric is useful for measuring perceptual colour differences between colour samples.The authors wish to thank both the associated editors coordinating this submission and the reviewers for their insightful suggestions and comments which have been useful to increase the scientific quality and presentation of the paper. Also, the authors thank Dr. M. Melgosa, Dr. R. Huertas and Dr. L. Gomez-Robledo from the Department of Optics of University of Granada, for providing data, information and invaluable comments and suggestions. Valentin Gregori and Samuel Morillas acknowledge the support of Spanish Ministry of Education and Science under Grant MTM 2009-12872-C02-01. Samuel Morillas acknowledges the support of Research Project FIS2010-19839, Ministerio de Educacion y Ciencia (Espana) with European Regional Development Funds (ERDFs).Gregori Gregori, V.; Miñana Prats, JJ.; Morillas Gómez, S. (2012). Some questions in fuzzy metric spaces. Fuzzy Sets and Systems. 204:71-85. https://doi.org/10.1016/j.fss.2011.12.008718520

    A construction of a fuzzy topology from a strong fuzzy metric

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    [EN] After the inception of the concept of a fuzzy metric by I. Kramosil and J. Michalek, and especially after its revision by A. George and G. Veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. In most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. Recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. In particular, in the paper  (J.J. Mi\~{n}ana, A. \v{S}ostak, {\it Fuzzifying topology induced by a strong fuzzy metric}, Fuzzy Sets and Systems,  6938 DOI information: 10.1016/j.fss.2015.11.005.) a fuzzifying topology T:2X→[0,1]{\mathcal T}:2^X \to [0,1] induced by a fuzzy metric  m:X×X×[0,∞)m: X\times X \times [0,\infty) was constructed. In this paper we extend  this construction to get the fuzzy topology T:[0,1]X→[0,1]{\mathcal T}: [0,1]^X \to [0,1] and study some properties of this fuzzy topology.54AGrecova, S.; Sostak, A.; Uljane, I. 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