Fuzzy ee-regular spaces and strongly ee-irresolute mappings

The aim of this paper is to introduce fuzzy ($e$, almost) $e^{*}$-regular spaces in $check{S}$ostak's fuzzy topological spaces. Using the $r$-fuzzy $e$-closed sets, we define $r$-($r$-$theta$-, $r$-$etheta$-) $e$-cluster points and their properties. Moreover, we investigate the relations among $r$-($r$-$theta$-, $r$-$etheta$-) $e$-cluster points, $r$-fuzzy ($e$, almost) $e^{*}$-regular spaces and their functions

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

A Deep Study of Fuzzy Implications

This thesis contributes a deep study on the extensions of the IMPLY operator in classical binary logic to fuzzy logic, which are called fuzzy implications. After the introduction in Chapter 1 and basic notations about the fuzzy logic operators In Chapter 2 we first characterize In Chapter 3 S- and R- implications and then extensively investigate under which conditions QL-implications satisfy the thirteen fuzzy implication axioms. In Chapter 4 we develop the complete interrelationships between the eight supplementary axioms FI6-FI13 for fuzzy implications satisfying the five basic axioms FI1-FI15. We prove all the dependencies between the eight fuzzy implication axioms, and provide for each independent case a counter-example. The counter-examples provided in this chapter can be used in the applications that need different fuzzy implications satisfying different fuzzy implication axioms. In Chapter 5 we study proper S-, R- and QL-implications for an iterative boolean-like scheme of reasoning from classical binary logic in the frame of fuzzy logic. Namely, repeating antecedents $n$ times, the reasoning result will remain the same. To determine the proper S-, R- and QL-implications we get a full solution of the functional equation $I(x,y)=I(x,I(x,y))$, for all $x$, $y\in[0,1]$. In Chapter 6 we study for the most important t-norms, t-conorms and S-implications their robustness against different perturbations in a fuzzy rule-based system. We define and compare for these fuzzy logical operators the robustness measures against bounded unknown and uniform distributed perturbations respectively. In Chapter 7 we use a fuzzy implication $I$ to define a fuzzy $I$-adjunction in $\mathcal{F}(\mathbb{R}^{n})$. And then we study the conditions under which a fuzzy dilation which is defined from a conjunction $\mathcal{C}$ on the unit interval and a fuzzy erosion which is defined from a fuzzy implication $I^{'}$ to form a fuzzy $I$-adjunction. These conditions are essential in order that the fuzzification of the morphological operations of dilation, erosion, opening and closing obey similar properties as their algebraic counterparts. We find out that the adjointness between the conjunction $\mathcal{C}$ on the unit interval and the implication $I$ or the implication $I^{'}$ play important roles in such conditions

Random variational-like inclusion and random proximal operator equation for random fuzzy mappings in Banach spaces

In this paper, we introduce and study a random variational-like inclusion and its corresponding random proximal operator equation for random fuzzy mappings. It is established that the random variational-like inclusion problem for random fuzzy mappings is equivalent to a random fixed point problem. We also establish a relationship between random variational-like inclusion and random proximal operator equation for random fuzzy mappings. This equivalence is used to define an iterative algorithm for solving random proximal operator equation for random fuzzy mappings. Through an example, we show that the random Wardrop equilibrium problem is a special case of the random variational-like inclusion problem for random fuzzy mappings

The fuzzy over-relaxed proximal point iterative scheme for generalized variational inclusion with fuzzy mappings

This paper deals with the introduction of a fuzzy over-relaxed proximal point iterative scheme based on H(-, -)-cocoercivity framework for solving a generalized variational inclusion problem with fuzzy mappings. The resolvent operator technique is used to approximate the solution of generalized variational inclusion problem with fuzzy mappings and convergence of the iterative sequences generated by the iterative scheme is discussed. Our results can be treated as refinement of many previously-known results

General ω-hyperstructures and certain applications of those

The aim of this paper is to investigate general hyperstructures construction of which is based on ideas of A. D. Nezhad and R. S. Hashemi. Concept of general hyperstructures considered by the above mentioned authors is generalized on the case of hyperstructures with hyperoperations of countable arity. Speci cations of treated concepts to examples from various elds of the mathematical sturctures theory are also included.

Synthetic Quantum Systems

So far proposed quantum computers use fragile and environmentally sensitive natural quantum systems. Here we explore the new notion that synthetic quantum systems suitable for quantum computation may be fabricated from smart nanostructures using topological excitations of a stochastic neural-type network that can mimic natural quantum systems. These developments are a technological application of process physics which is an information theory of reality in which space and quantum phenomena are emergent, and so indicates the deep origins of quantum phenomena. Analogous complex stochastic dynamical systems have recently been proposed within neurobiology to deal with the emergent complexity of biosystems, particularly the biodynamics of higher brain function. The reasons for analogous discoveries in fundamental physics and neurobiology are discussed.Comment: 16 pages, Latex, 1 eps figure fil