On Yager and Hamacher t-Norms and Fuzzy Metric Spaces

Recently, Gregori et al. have discussed (Fuzzy Sets Syst 2011;161:2193 2205) the so-called strong fuzzy metrics when looking for a class of completable fuzzy metric spaces in the sense of George and Veeramani and state the question of finding a non-strong fuzzy metric space for a continuous t-norm different from the minimum. Later on, Gutíerrez-García and Romaguera solved this question (Fuzzy Sets Syst 2011;162:91 93) by means of two examples for the product and the Lukasiewicz t-norm, respectively. In this direction, they posed to find further examples of nonstrong fuzzy metrics for continuous t-norms that are greater than the product but different from minimum. In this paper, we found an example of this kind. On the other hand, Tirado established (Fixed Point Theory 2012;13:273 283) a fixed-point theorem in fuzzy metric spaces, which was successfully used to prove the existence and uniqueness of solution for the recurrence equation associated with the probabilistic divide and conquer algorithms. Here, we generalize this result by using a class of continuous t-norms known as ω-Yager t-norms.The second author acknowledges the support of the Ministry of Economy and Competitiveness of Spain under grant MTM2012-37894-C02-01 and the support of Universitat Politecnica de Valencia under grant PAID-06-12-SP20120471.Castro Company, F.; Tirado Peláez, P. (2014). On Yager and Hamacher t-Norms and Fuzzy Metric Spaces. International Journal of Intelligent Systems. 29:1173-1180. https://doi.org/10.1002/int.21688S1173118029Sherwood, H. (1966). On the completion of probabilistic metric spaces. Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 6(1), 62-64. doi:10.1007/bf00531809Gregori, V. (2002). On completion of fuzzy metric spaces. Fuzzy Sets and Systems, 130(3), 399-404. doi:10.1016/s0165-0114(02)00115-xGregori, V., Morillas, S., & Sapena, A. (2010). On a class of completable fuzzy metric spaces. Fuzzy Sets and Systems, 161(16), 2193-2205. doi:10.1016/j.fss.2010.03.013Gutiérrez García, J., & Romaguera, S. (2011). Examples of non-strong fuzzy metrics. Fuzzy Sets and Systems, 162(1), 91-93. doi:10.1016/j.fss.2010.09.017Yager, R. R. (1980). On a general class of fuzzy connectives. Fuzzy Sets and Systems, 4(3), 235-242. doi:10.1016/0165-0114(80)90013-5Castro-Company, F., & Tirado, P. (2012). Some classes of t-norms and fuzzy metric spaces. doi:10.1063/1.4756272George, 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)90162-7Hadžić, O., & Pap, E. (2001). Fixed Point Theory in Probabilistic Metric Spaces. doi:10.1007/978-94-017-1560-7Klement, E. P., Mesiar, R., & Pap, E. (2000). Triangular Norms. Trends in Logic. doi:10.1007/978-94-015-9540-

Cauchyness and convergence in fuzzy metric spaces

[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 $$\varphi$$ 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

The depth and the attracting centre for a continuous map on a fuzzy metric interval

[EN] Let I be a fuzzy metric interval and f be a continuous map from I to I. Denote by R(f), Ω(f) and ω(x, f) the set of recurrent points of f, the set of non-wandering points of f and the set of ω- limit points of x under f, respectively. Write ω(f) = ∪x∈Iω(x, f), ωn+1(f) = ∪x∈ωn(f)ω(x, f) and Ωn+1(f) = Ω(f|Ωn(f)) for any positive integer n. In this paper, we show that Ω2(f) = R(f) and the depth of f is at most 2, and ω3(f) = ω2(f) and the depth of the attracting centre of f is at most 2.Project supported by NNSF of China (11761011, 71862003) and NSF of Guangxi (2018GXNSFAA294010) and SF of Guangxi University of Finance and Economics (2019QNB10).Sun, T.; Li, L.; Su, G.; Han, C.; Xia, G. (2020). The depth and the attracting centre for a continuous map on a fuzzy metric interval. Applied General Topology. 21(2):285-294. https://doi.org/10.4995/agt.2020.13126OJS285294212A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Sys. 64 (1994), 395-399. https://doi.org/10.1016/0165-0114(94)90162-7M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Sys. 27 (1989), 385-389. https://doi.org/10.1016/0165-0114(88)90064-4V. Gregori and J. J. Miñana, Some remarks on fuzzy contractive mappings, Fuzzy Sets Sys. 251 (2014), 101-103. https://doi.org/10.1016/j.fss.2014.01.002V. Gregori and J. J. Miñana, On fuzzy Ψ-contractive sequences and fixed point theorems, Fuzzy Sets Sys. 300 (2016), 93-101. https://doi.org/10.1016/j.fss.2015.12.010V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets Sys. 125 (2002), 245-252. https://doi.org/10.1016/S0165-0114(00)00088-9X. Hu, Z. Mo and Y. Zhen, On compactnesses of fuzzy metric spaces (Chinese), J. Sichuan Norm. Univer. (Natur. Sei.) 32 (2009), 184-187.I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975), 336-344.C. Li and Y. Zhang, On connectedness of the Hausdorff fuzzy metric spaces, Italian J. Pure Appl. Math. 42 (2019), 458-466.D. Mihet, Fuzzy Ψ-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets Sys. 159 (2008), 739-744. https://doi.org/10.1016/j.fss.2007.07.006D. Mihet, A note on fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets Sys. 251 (2014), 83-91. https://doi.org/10.1016/j.fss.2014.04.010J. Rodríguez-López and S. Romaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets Sys. 147 (2004), 273-283. https://doi.org/10.1016/j.fss.2003.09.007B. Schweizer and A. Sklar, Statistical metrics paces, Pacif. J. Math. 10 (1960), 385-389. https://doi.org/10.2140/pjm.1960.10.313Y. Shen, D. Qiu and W. Chen, Fixed point theorems in fuzzy metric spaces, Appl. Math. Letters 25 (2012), 138-141. https://doi.org/10.1016/j.aml.2011.08.002D. Wardowski, Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets Sys. 222 (2013), 108-114. https://doi.org/10.1016/j.fss.2013.01.012D. Zheng and P. Wang, On probabilistic Ψ-contractions in Menger probabilistic metric spaces, Fuzzy Sets Sys. 350 (2018), 107-110. https://doi.org/10.1016/j.fss.2018.02.011D. Zheng and P. Wang, Meir-Keeler theorems in fuzzy metric spaces, Fuzzy Sets Sys. 370 (2019), 120-128. https://doi.org/10.1016/j.fss.2018.08.01

A Characterization of Strong Completeness in Fuzzy Metric Spaces

[EN] Here, we deal with the concept of fuzzy metric space(X,M,*), due to George and Veeramani. Based on the fuzzy diameter for a subset ofX, we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory.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 was 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 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.Gregori Gregori, V.; Miñana, J.; Roig, B.; Sapena Piera, A. (2020). A Characterization of Strong Completeness in Fuzzy Metric Spaces. Mathematics. 8(6):1-11. https://doi.org/10.3390/math8060861S11186Menger, K. (1942). Statistical Metrics. Proceedings of the National Academy of Sciences, 28(12), 535-537. doi:10.1073/pnas.28.12.535George, 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)90162-7Gregori, V., & Romaguera, S. (2000). Some properties of fuzzy metric spaces. Fuzzy Sets and Systems, 115(3), 485-489. doi:10.1016/s0165-0114(98)00281-4Gregori, V. (2002). On completion of fuzzy metric spaces. Fuzzy Sets and Systems, 130(3), 399-404. doi:10.1016/s0165-0114(02)00115-xAtanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87-96. doi:10.1016/s0165-0114(86)80034-3Gregori, V., Romaguera, S., & Veeramani, P. (2006). A note on intuitionistic fuzzy metric spaces☆. Chaos, Solitons & Fractals, 28(4), 902-905. doi:10.1016/j.chaos.2005.08.113Gregori, V., & Sapena, A. (2018). Remarks to «on strong intuitionistic fuzzy metrics». Journal of Nonlinear Sciences and Applications, 11(02), 316-322. doi:10.22436/jnsa.011.02.12Abu-Donia, H. M., Atia, H. A., & Khater, O. M. A. (2020). Common fixed point theorems in intuitionistic fuzzy metric spaces and intuitionistic (ϕ,ψ)-contractive mappings. Journal of Nonlinear Sciences and Applications, 13(06), 323-329. doi:10.22436/jnsa.013.06.03Gregori, 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.2015.12.010Miheţ, D. (2007). On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems, 158(8), 915-921. doi:10.1016/j.fss.2006.11.012Wardowski, D. (2013). Fuzzy contractive mappings and fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 222, 108-114. doi:10.1016/j.fss.2013.01.012Gregori, 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/s13398-015-0272-0Gregori, V., & Miñana, J.-J. (2017). Strong convergence in fuzzy metric spaces. Filomat, 31(6), 1619-1625. doi:10.2298/fil1706619gGrabiec, M. (1988). Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 27(3), 385-389. doi:10.1016/0165-0114(88)90064-4George, A., & Veeramani, P. (1997). On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems, 90(3), 365-368. doi:10.1016/s0165-0114(96)00207-2Miheţ, D. (2008). Fuzzy -contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 159(6), 739-744. doi:10.1016/j.fss.2007.07.006Vasuki, R., & Veeramani, P. (2003). Fixed point theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets and Systems, 135(3), 415-417. doi:10.1016/s0165-0114(02)00132-xGregori, V., & Romaguera, S. (2004). Characterizing completable fuzzy metric spaces. Fuzzy Sets and Systems, 144(3), 411-420. doi:10.1016/s0165-0114(03)00161-1Gregori, 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.2011.12.008Ricarte, L. A., & Romaguera, S. (2014). A domain-theoretic approach to fuzzy metric spaces. Topology and its Applications, 163, 149-159. doi:10.1016/j.topol.2013.10.014Gregori, V., López-Crevillén, A., Morillas, S., & Sapena, A. (2009). On convergence in fuzzy metric spaces. Topology and its Applications, 156(18), 3002-3006. doi:10.1016/j.topol.2008.12.043Sherwood, H. (1966). On the completion of probabilistic metric spaces. Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 6(1), 62-64. doi:10.1007/bf00531809Shukla, S., Gopal, D., & Sintunavarat, W. (2018). A new class of fuzzy contractive mappings and fixed point theorems. Fuzzy Sets and Systems, 350, 85-94. doi:10.1016/j.fss.2018.02.010Beg, 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.2018.2.37Zheng, D., & Wang, P. (2019). Meir–Keeler theorems in fuzzy metric spaces. Fuzzy Sets and Systems, 370, 120-128. doi:10.1016/j.fss.2018.08.014Rakić, D., Došenović, T., Mitrović, Z. D., de la Sen, M., & Radenović, S. (2020). Some Fixed Point Theorems of Ćirić Type in Fuzzy Metric Spaces. Mathematics, 8(2), 297. doi:10.3390/math802029

A construction of a fuzzy topology from a strong fuzzy metric

[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 ${\mathcal T}:2^X \to [0,1]$ induced by a fuzzy metric  $m: X\times X \times [0,\infty)$ was constructed. In this paper we extend  this construction to get the fuzzy topology ${\mathcal T}: [0,1]^X \to [0,1]$ and study some properties of this fuzzy topology.54AGrecova, S.; Sostak, A.; Uljane, I. (2016). A construction of a fuzzy topology from a strong fuzzy metric. Applied General Topology. 17(2):105-116. doi:10.4995/agt.2016.4495.SWORD105116172Chang, C. . (1968). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications, 24(1), 182-190. doi:10.1016/0022-247x(68)90057-7Goguen, J. . (1967). L-fuzzy sets. Journal of Mathematical Analysis and Applications, 18(1), 145-174. doi:10.1016/0022-247x(67)90189-8Goguen, J. . (1973). The fuzzy tychonoff theorem. Journal of Mathematical Analysis and Applications, 43(3), 734-742. doi:10.1016/0022-247x(73)90288-6George, 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)90162-7George, A., & Veeramani, P. (1997). On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems, 90(3), 365-368. doi:10.1016/s0165-0114(96)00207-2V. Gregori, A. López-Crevillén and S. Morillas, On continuity and uniform continuity in fuzzy metric spaces, Proc. Workshop Appl. Topology WiAT'09 (2009), 85-91.Gregori, V., López-Crevillén, A., Morillas, S., & Sapena, A. (2009). On convergence in fuzzy metric spaces. Topology and its Applications, 156(18), 3002-3006. doi:10.1016/j.topol.2008.12.043V. Gregori and J. Mi-ana, Some concepts related to continuity in fuzzy metric spaces, Proc. Workshop Appl. Topology WiAT'13 (2013), 85-91.Gregori, V., Morillas, S., & Sapena, A. (2010). On a class of completable fuzzy metric spaces. Fuzzy Sets and Systems, 161(16), 2193-2205. doi:10.1016/j.fss.2010.03.013Gregori, V., & Romaguera, S. (2004). Characterizing completable fuzzy metric spaces. Fuzzy Sets and Systems, 144(3), 411-420. doi:10.1016/s0165-0114(03)00161-1Höhle, U. (1980). Upper semicontinuous fuzzy sets and applications. Journal of Mathematical Analysis and Applications, 78(2), 659-673. doi:10.1016/0022-247x(80)90173-0I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975), 336-344.Kubiak, T., & Sostak, A. P. (2004). A fuzzification of the category of M-valued L-topological spaces. Applied General Topology, 5(2), 137. doi:10.4995/agt.2004.1965Lowen, R. (1976). Fuzzy topological spaces and fuzzy compactness. Journal of Mathematical Analysis and Applications, 56(3), 621-633. doi:10.1016/0022-247x(76)90029-9Lowen, R. (1977). Initial and final fuzzy topologies and the fuzzy Tychonoff theorem. Journal of Mathematical Analysis and Applications, 58(1), 11-21. doi:10.1016/0022-247x(77)90223-2Mardones-Pérez, I., & de Prada Vicente, M. A. (2015). Fuzzy pseudometric spaces vs fuzzifying structures. Fuzzy Sets and Systems, 267, 117-132. doi:10.1016/j.fss.2014.06.003Mardones-Pérez, I., & de Prada Vicente, M. A. (2012). A representation theorem for fuzzy pseudometrics. Fuzzy Sets and Systems, 195, 90-99. doi:10.1016/j.fss.2011.11.008Menger, K. (1951). Probabilistic Geometry. Proceedings of the National Academy of Sciences, 37(4), 226-229. doi:10.1073/pnas.37.4.226Miheţ, D. (2007). On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems, 158(8), 915-921. doi:10.1016/j.fss.2006.11.012Miñana, J.-J., & Šostak, A. (2016). Fuzzifying topology induced by a strong fuzzy metric. Fuzzy Sets and Systems, 300, 24-39. doi:10.1016/j.fss.2015.11.005Sapena Piera, A. (2001). A contribution to the study of fuzzy metric spaces. Applied General Topology, 2(1), 63. doi:10.4995/agt.2001.3016A. Sapena and S. Morillas, On strong fuzzy metrics, Proc. Workshop Appl. Topology WiAT'09 (2009), 135-141.Schweizer, B., & Sklar, A. (1960). Statistical metric spaces. Pacific Journal of Mathematics, 10(1), 313-334. doi:10.2140/pjm.1960.10.313A. Sostak, On a fuzzy topological structure, Suppl. Rend. Circ. Matem. Palermo, Ser II 11 (1985), 125-186.Shostak, A. P. (1989). Two decades of fuzzy topology: basic ideas, notions, and results. Russian Mathematical Surveys, 44(6), 125-186. doi:10.1070/rm1989v044n06abeh002295Šostak, A. P. (1996). Basic structures of fuzzy topology. Journal of Mathematical Sciences, 78(6), 662-701. doi:10.1007/bf02363065Ying, M. (1991). A new approach for fuzzy topology (I). Fuzzy Sets and Systems, 39(3), 303-321. doi:10.1016/0165-0114(91)90100-5Ying, M. (1992). A new approach for fuzzy topology (II). Fuzzy Sets and Systems, 47(2), 221-232. doi:10.1016/0165-0114(92)90181-3Ying, M. (1993). A new approach for fuzzy topology (III). Fuzzy Sets and Systems, 55(2), 193-207. doi:10.1016/0165-0114(93)90132-2Ying, M. (1993). Compactness in fuzzifying topology. Fuzzy Sets and Systems, 55(1), 79-92. doi:10.1016/0165-0114(93)90303-yYue, Y., & Shi, F.-G. (2010). On fuzzy pseudo-metric spaces. Fuzzy Sets and Systems, 161(8), 1105-1116. doi:10.1016/j.fss.2009.10.00

Fuzzy Partial Metric Spaces

"This is an Accepted Manuscript of an article published by Taylor & Francis in International Journal of General Systems on 01 Dec 2018, available online: https://doi.org/10.1080/03081079.2018.1552687"[EN] In this paper we provide a concept of fuzzy partial metric space (X, P, ¿) as an extension to fuzzy setting in the sense of Kramosil and Michalek, of the concept of partial metric due to Matthews. This extension has been defined using the residuum operator ¿¿ associated to a continuous t-norm ¿ and without any extra condition on ¿. Similarly, it is defined the stronger concept of GV -fuzzy partial metric (fuzzy partial metric in the sense of George and Veeramani). After defining a concept of open ball in (X, P, ¿), a topology TP on X deduced from P is constructed, and it is showed that (X, TP) is a T0-space.Valentin Gregori acknowledges the support of the Ministry of Economy and Competitiveness of Spain under Grant MTM2015-64373-P (MINECO/Feder, UE). Juan Jose Minana acknowledges the partially support of the Ministry of Economy and Competitiveness of Spain under Grant TIN2016-81731-REDT (LODISCO II) and AEI/FEDER, UE funds, by the Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project ref. PROCOE/4/2017 (Direccio General d'Innovacio i Recerca, Govern de les Illes Balears), and by project ROBINS. The latter has received research funding from the European Union 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.Gregori Gregori, V.; Miñana, J.; Miravet-Fortuño, D. (2018). Fuzzy Partial Metric Spaces. International Journal of General Systems. https://doi.org/10.1080/03081079.2018.1552687SBukatin, M., Kopperman, R., & Matthews, S. (2014). Some corollaries of the correspondence between partial metrics and multivalued equalities. 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On the construction of metrics from fuzzy metrics and its application to the fixed point theory of multivalued mappings

[EN] We present a procedure to construct a compatible metric from a given fuzzy metric space. We use this approach to obtain a characterization of a large class of complete fuzzy metric spaces by means of a fuzzy version of Caristi’s fixed point theorem, obtaining, in this way, partial solutions to a recent question posed in the literature. Some illustrative examples are also given.The authors thank the referees for several useful suggestions. Salvador Romaguera and Pedro Tirado acknowledge the support of the Ministry of Economy and Competitiveness of Spain, grant MTM2012-37894-C02-01.Castro Company, F.; Romaguera Bonilla, S.; Tirado Peláez, P. (2015). On the construction of metrics from fuzzy metrics and its application to the fixed point theory of multivalued mappings. Fixed Point Theory and Applications. 2015:226. https://doi.org/10.1186/s13663-015-0476-1S2015:226Kelley, JL: General Topology. Springer, New York (1955)Schweizer, B, Sklar, A: Statistical metric spaces. Pac. J. Math. 10, 314-334 (1960)Klement, E, Mesiar, R, Pap, E: Triangular Norms. Kluwer Academic, Dordrecht (2000)Hamacher, H: Über logische Verknüpfungen unscharfer Aussagen und deren zugehörige Bewertungsfunktionen. In: Progress in Cybernetics and Systems Research, pp. 276-287. Hemisphere, New York (1975)Kramosil, I, Michalek, J: Fuzzy metrics and statistical metric spaces. Kybernetika 11, 326-334 (1975)George, A, Veeramani, P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64, 395-399 (1994)Gregori, V, Romaguera, S: Some properties of fuzzy metric spaces. Fuzzy Sets Syst. 115, 485-489 (2000)Radu, V: On the triangle inequality in PM-spaces. STPA, West University of Timişoara 39 (1978)Abbas, M, Ali, B, Romaguera, S: Multivalued Caristi’s type mappings in fuzzy metric spaces and a characterization of fuzzy metric completeness. Filomat 29(6), 1217-1222 (2015)Cho, YJ, Grabiec, M, Radu, V: On Nonsymmetric Topological and Probabilistic Structures. Nova Science Publishers, New York (2006)Hadžić, O, Pap, E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht (2001)Mihet, D: A note on Hicks type contractions on generalized Menger spaces. STPA, West University of Timişoara 133 (2002)Mihet, D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 144, 431-439 (2004)Radu, V: Some fixed point theorems in PM spaces. In: Stability Problems for Stochastic Models. Lecture Notes in Mathematics, vol. 1233, pp. 125-133. Springer, Berlin (1985)Radu, V: Some remarks on the probabilistic contractions on fuzzy Menger spaces (The Eighth Intern. Conf. on Applied Mathematics and Computer Science, Cluj-Napoca, 2001). Autom. Comput. Appl. Math. 11(1), 125-131 (2002)Chauhan, S, Shatanawi, W, Kumar, S, Radenović, S: Existence and uniqueness of fixed points in modified intuitionistic fuzzy metric spaces. J. Nonlinear Sci. Appl. 7, 28-41 (2014)Hussain, N, Salimi, P, Parvaneh, V: Fixed point results for various contractions in parametric and fuzzy b-metric spaces. J. Nonlinear Sci. Appl. 8, 719-739 (2015)Mihet, D: Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. J. Nonlinear Sci. Appl. 6, 35-40 (2013)Hicks, TL: Fixed point theory in probabilistic metric spaces. Zb. Rad. Prir.-Mat. Fak. (Novi Sad) 13, 63-72 (1983)Radu, V: Some suitable metrics on fuzzy metric spaces. Fixed Point Theory 5, 323-347 (2004)O’Regan, D, Saadati, R: Nonlinear contraction theorems in probabilistic spaces. Appl. Math. Comput. 195, 86-93 (2008)Caristi, J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241-251 (1976)Kirk, WA: Caristi’s fixed-point theorem and metric convexity. Colloq. Math. 36, 81-86 (1976)Ansari, QH: Metric Spaces: Including Fixed Point Theory and Set-Valued Maps. Alpha Science, Oxford (2010

A Bibliography on Fuzzy Automata, Grammars and Lanuages

This bibliography contains references to papers on fuzzy formal languages, the generation of fuzzy languages by means of fuzzy grammars, the recognition of fuzzy languages by fuzzy automata and machines, as well as some applications of fuzzy set theory to syntactic pattern recognition, linguistics and natural language processing

(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces

We develop a kind of pregeometry consisting of a web of overlapping fuzzy lumps which interact with each other. The individual lumps are understood as certain closely entangled subgraphs (cliques) in a dynamically evolving network which, in a certain approximation, can be visualized as a time-dependent random graph. This strand of ideas is merged with another one, deriving from ideas, developed some time ago by Menger et al, that is, the concept of probabilistic- or random metric spaces, representing a natural extension of the metrical continuum into a more microscopic regime. It is our general goal to find a better adapted geometric environment for the description of microphysics. In this sense one may it also view as a dynamical randomisation of the causal-set framework developed by e.g. Sorkin et al. In doing this we incorporate, as a perhaps new aspect, various concepts from fuzzy set theory.Comment: 25 pages, Latex, no figures, some references added, some minor changes added relating to previous wor