The Hausdorff fuzzy quasi-metric

Removing the condition of symmetry in the notion of a fuzzy (pseudo)metric, in Kramosil and Michalek's sense, one has the notion of a fuzzy quasi-(pseudo-)metric. Then for each fuzzy quasi-pseudo-metric on a set X we construct a fuzzy quasi-pseudo-metric on the collection of all nonempty subsets of X, called the Hausdorff fuzzy quasi-pseudo-metric. We investigate several properties of this structure and present several illustrative examples as well as an application to the domain of words. The notion of Hausdorff fuzzy quasi-pseudo-metric when quasi-pseudo-metric fuzziness is considered in the sense of George and Veeramani is also discussed.Supported by the Plan Nacional I+D+i and FEDER, under Grant MTM2006-14925-C02-01.Rodríguez López, J.; Romaguera Bonilla, S.; Sánchez Álvarez, JM. (2010). The Hausdorff fuzzy quasi-metric. Fuzzy Sets and Systems. 161:1078-1096. https://doi.org/10.1016/j.fss.2009.09.019S1078109616

Convergence of fuzzy sets with respect to the supremum metric

We characterize the convergence of fuzzy sets in the supremum metric given by the supremum of the Hausdorff distances of the alpha-cuts of the fuzzy sets. We do it by dividing this metric into its lower and upper quasi-pseudometric parts. This characterization is given in the more general context with no assumption on the fuzzy sets. Furthermore, motivated from the theory of Convex Analysis, we also provide some results about the behavior of the convergence in the supremum metric with respect to maximizers. (C) 2014 Elsevier B.V. All rights reserved.The second and third authors thank the support of the Ministry of Economy and Competitiveness of Spain under grant MTM2012-37894-C02-01.Pedraza Aguilera, T.; Rodríguez López, J.; Romaguera Bonilla, S. (2014). Convergence of fuzzy sets with respect to the supremum metric. Fuzzy Sets and Systems. 245:83-100. https://doi.org/10.1016/j.fss.2014.03.005S8310024

A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics

[EN] In 1994, Matthews introduced the notion of partial metric and established a duality relationship between partial metrics and quasi-metrics defined on a set X. In this paper, we adapt such a relationship to the fuzzy context, in the sense of George and Veeramani, by establishing a duality relationship between fuzzy quasi-metrics and fuzzy partial metrics on a set X, defined using the residuum operator of a continuous t-norm *. Concretely, we provide a method to construct a fuzzy quasi-metric from a fuzzy partial one. Subsequently, we introduce the notion of fuzzy weighted quasi-metric and obtain a way to construct a fuzzy partial metric from a fuzzy weighted quasi-metric. Such constructions are restricted to the case in which the continuous t-norm * is Archimedean and we show that such a restriction cannot be deleted. Moreover, in both cases, the topology is preserved, i.e., the topology of the fuzzy quasi-metric obtained coincides with the topology of the fuzzy partial metric from which it is constructed and vice versa. Besides, different examples to illustrate the exposed theory are provided, which, in addition, show the consistence of our constructions comparing it with the classical duality relationship.Juan-Jose Minana acknowledges financial support from FEDER/Ministerio de Ciencia, Innovacion y Universidades-Agencia Estatal de Investigacion/Proyecto PGC2018-095709-B-C21, and by Spanish Ministry of Economy and Competitiveness under contract DPI2017-86372-C3-3-R (AEI, FEDER, UE). This work is also partially supported by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project PROCOE/4/2017 (Direccio General d'Innovacio 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 program 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.Gregori Gregori, V.; Miñana, J.; Miravet, D. (2020). A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics. Mathematics. 8(9):1-16. https://doi.org/10.3390/math809157511689MATTHEWS, S. G. (1994). Partial Metric Topology. Annals of the New York Academy of Sciences, 728(1 General Topol), 183-197. doi:10.1111/j.1749-6632.1994.tb44144.xGeorge, A., & Veeramani, P. (1994). On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64(3), 395-399. doi:10.1016/0165-0114(94)90162-7Roldán-López-de-Hierro, A.-F., Karapınar, E., & Manro, S. (2014). Some new fixed point theorems in fuzzy metric spaces. Journal of Intelligent & Fuzzy Systems, 27(5), 2257-2264. doi:10.3233/ifs-141189Gregori, V., & Miñana, J.-J. (2016). On fuzzy ψ -contractive sequences and fixed point theorems. Fuzzy Sets and Systems, 300, 93-101. doi:10.1016/j.fss.2015.12.010Gregori, V., Miñana, J.-J., Morillas, S., & Sapena, A. (2016). 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. doi:10.1007/s13398-015-0272-0Gutiérrez García, J., Rodríguez-López, J., & Romaguera, S. (2018). On fuzzy uniformities induced by a fuzzy metric space. Fuzzy Sets and Systems, 330, 52-78. doi:10.1016/j.fss.2017.05.001Beg, I., Gopal, D., Došenović, T., … Rakić, D. (2018). α-type fuzzy H-contractive mappings in fuzzy metric spaces. Fixed Point Theory, 19(2), 463-474. doi:10.24193/fpt-ro.2018.2.37Gregori, V., Miñana, J.-J., & Miravet, D. (2018). Fuzzy partial metric spaces. International Journal of General Systems, 48(3), 260-279. doi:10.1080/03081079.2018.1552687Zheng, D., & Wang, P. (2019). Meir–Keeler theorems in fuzzy metric spaces. Fuzzy Sets and Systems, 370, 120-128. doi:10.1016/j.fss.2018.08.014Romaguera, S., & Tirado, P. (2020). Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results. Mathematics, 8(2), 273. doi:10.3390/math8020273Wu, X., & Chen, G. (2020). Answering an open question in fuzzy metric spaces. Fuzzy Sets and Systems, 390, 188-191. doi:10.1016/j.fss.2019.12.006Camarena, J.-G., Gregori, V., Morillas, S., & Sapena, A. (2008). Fast detection and removal of impulsive noise using peer groups and fuzzy metrics. Journal of Visual Communication and Image Representation, 19(1), 20-29. doi:10.1016/j.jvcir.2007.04.003Camarena, J.-G., Gregori, V., Morillas, S., & Sapena, A. (2010). Two-step fuzzy logic-based method for impulse noise detection in colour images. Pattern Recognition Letters, 31(13), 1842-1849. doi:10.1016/j.patrec.2010.01.008Gregori, V., Miñana, J.-J., & Morillas, S. (2012). Some questions in fuzzy metric spaces. Fuzzy Sets and Systems, 204, 71-85. doi:10.1016/j.fss.2011.12.008Morillas, S., Gregori, V., Peris-Fajarnés, G., & Latorre, P. (2005). A fast impulsive noise color image filter using fuzzy metrics. Real-Time Imaging, 11(5-6), 417-428. doi:10.1016/j.rti.2005.06.007Gregori, V., & Romaguera, S. (2004). Fuzzy quasi-metric spaces. Applied General Topology, 5(1), 129. doi:10.4995/agt.2004.2001Park, J. H. (2004). Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals, 22(5), 1039-1046. doi:10.1016/j.chaos.2004.02.051Rodrı́guez-López, J., & Romaguera, S. (2004). The Hausdorff fuzzy metric on compact sets. Fuzzy Sets and Systems, 147(2), 273-283. doi:10.1016/j.fss.2003.09.007Schweizer, B., & Sklar, A. (1960). Statistical metric spaces. Pacific Journal of Mathematics, 10(1), 313-334. doi:10.2140/pjm.1960.10.313Sapena Piera, A. (2001). A contribution to the study of fuzzy metric spaces. Applied General Topology, 2(1), 63. doi:10.4995/agt.2001.3016Miñana, J.-J., & Valero, O. (2020). On Matthews’ Relationship Between Quasi-Metrics and Partial Metrics: An Aggregation Perspective. Results in Mathematics, 75(2). doi:10.1007/s00025-020-1173-xKarapınar, E., Erhan, İ. M., & Öztürk, A. (2013). Fixed point theorems on quasi-partial metric spaces. Mathematical and Computer Modelling, 57(9-10), 2442-2448. doi:10.1016/j.mcm.2012.06.036Künzi, H.-P. A., Pajoohesh, H., & Schellekens, M. P. (2006). Partial quasi-metrics. Theoretical Computer Science, 365(3), 237-246. doi:10.1016/j.tcs.2006.07.05

Standard fuzzy uniform structures based on continuous t-norms

This paper deals with fuzzy uniform structures previously introduced by the authors [Fuzzy uniform structures and continuous t-norms, Fuzzy Sets Syst. 161 (2009) 1011-1021]. Our approach involves a covariant functor psi from the category of fuzzy uniform spaces and fuzzy uniformly continuous mappings (in our sense) to the category of uniform spaces and uniformly continuous mappings. We show that psi is well-behaved with respect to some significant fuzzy uniform concepts. and its behavior provides a method to introduce notions of tine fuzzy uniform structure and Stone-tech fuzzy compactification in this context. Our method also applies to obtain fuzzy versions of some classical results on topological algebra and hyperspaces. The case of quasi-uniform structures is also analyzed. (C) 2011 Elsevier B.V. All rights reserved.This research is supported by the Plan Nacional I+D+i, under Grants MTM2009-12872-C02-01 and MTM2009-12872-C02-02.Gutiérrez, J.; Romaguera Bonilla, S.; Sanchis, M. (2012). Standard fuzzy uniform structures based on continuous t-norms. Fuzzy Sets and Systems. 195:75-89. doi:10.1016/j.fss.2011.10.008S758919

Hyperspace of a fuzzy quasi-uniform space

[EN] The aim of this paper is to present a fuzzy counterpart method of constructing the Hausdorff quasi-uniformity of a crisp quasi-uniformity. This process, based on previous works due to Morsi [25] and Georgescu [9], allows to extend probabilistic and Hutton [0, 1]-quasi-uniformities on a set X to its power set. In this way, we obtain an endofunctor for each one of the categories of those objects. We will demonstrate the commutativity of these endofunctors with Lowen and Katsaras' functors. Furthermore, we will prove the compatibility of our construction with the Hausdorff fuzzy quasi-pseudometric introduced in [33].The second author is supported by the grant MTM2015-64373-P (MINECO/FEDER, UE). The authors are grateful to the reviewers for useful comments which have improved the first version of the paperPedraza Aguilera, T.; Rodríguez López, J. (2020). Hyperspace of a fuzzy quasi-uniform space. Iranian Journal of Fuzzy Systems. 17(2):97-114. https://doi.org/10.22111/IJFS.2020.5222S9711417