The Fuzzification of Classical Structures: A General View
Keywords:
Fuzzy real line, fuzzy topological spaces, fuzzy metric spaces, fuzzy topological vector spaces, fuzzy normed linear spaces, fuzzy F-space.Abstract
The aim of this survey article, dedicated to the 50th anniversary of Zadeh’s pioneering paper "Fuzzy Sets" (1965), is to offer a unitary view to some important spaces in fuzzy mathematics: fuzzy real line, fuzzy topological spaces, fuzzy metric spaces, fuzzy topological vector spaces, fuzzy normed linear spaces. We believe that this paper will be a support for future research in this field.References
Alegre, C., Romaguera, S. (2010); Characterizations of fuzzy metrizable topological vector spaces and their asymmetric generalization in terms of fuzzy (quasi-)norms, Fuzzy Sets and Systems, 161: 2182-2192.
Bag, T., Samanta, S.K. (2003); Finite dimensional fuzzy normed linear spaces, J. of Fuzzy Mathematics, 11(3): 687-705.
Bag, T., Samanta, S.K. (2005); Fuzzy bounded linear operators, Fuzzy Sets and Systems, 151: 513-547. http://dx.doi.org/10.1016/j.fss.2004.05.004
Bag, T., Samanta, S.K. (2008); A comparative study of fuzzy norms on a linear space, Fuzzy Sets and Systems, 159: 670-684. http://dx.doi.org/10.1016/j.fss.2007.09.011
Bag, T., Samanta, S.K. (2008); Fuzzy bounded linear operators in Felbin's type fuzzy normed linear spaces, Fuzzy Sets and Systems, 159: 685-707. http://dx.doi.org/10.1016/j.fss.2007.09.006
Beg, I. (1999); Continuity of fuzzy multifunctions, J. of Applied Mathematics and Stochastic Analysis, 12(1): 17-22. http://dx.doi.org/10.1155/S1048953399000027
Beg, I. (2001); Linear fuzzy multivalued operators, J. of Fuzzy Mathematics, 9(1): 127-137.
Beg, I. (2001); Vector-valued fuzzy multifunctions, J. of Applied Mathematics and Stochas- tic Analysis, 14(3): 275-282. http://dx.doi.org/10.1155/S1048953301000235
Chalco-Cano, Y., Rojas-Medar, M.A., Osuna-Gómez, R. (2004); s-Convex fuzzy processes, Computers and Mathematics with Applications 47: 1411-1418. http://dx.doi.org/10.1016/S0898-1221(04)90133-2
Chang, C.L. (1968); Fuzzy topological spaces, J. Math. Anal. Appl., 24: 182-190. http://dx.doi.org/10.1016/0022-247X(68)90057-7
Cheng, S.C., Mordenson, J.L. (1994); Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc., 86: 429-436.
Cho, Y.J., Grabiec, M., Radu, V. (2006); On nonsymmetric topological and probabilistic structures, Nova Science Publishers, Inc., New York.
Das, R.K., Mandal, B. (2002); Fuzzy real line structure and metric space, Indian J. pure appl. Math., 33(4): 565-571.
Dubois, D., Prade, H. (1980); Fuzzy Sets and Systems: Theory and Applications, Academic Press, Inc.
Dubois, D., Prade, H. (2005); Fuzzy elements in a fuzzy set, Proc. of the 10th Intl. Fuzzy Systems Association (IFSA) Congres, Beijing, China.
Eklund, P., Gahler, W. (1988); Basic notions for fuzzy topology, Fuzzy Sets and Systems 26: 333-356. http://dx.doi.org/10.1016/0165-0114(88)90127-3
Felbin, C. (1992); Finite dimensional fuzzy normed liniar space, Fuzzy Sets and Systems, 48: 239-248. http://dx.doi.org/10.1016/0165-0114(92)90338-5
George, A., Veeramani, P. (1994); On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64: 395-399. http://dx.doi.org/10.1016/0165-0114(94)90162-7
Goguen, J.A. (1973); The fuzzy Tychonoff theorem, J. of Math. Analysis and Applications, 43: 734-742. http://dx.doi.org/10.1016/0022-247X(73)90288-6
Goleţ, I. (2010); On generalized fuzzy normed spaces and coincidence point theorems, Fuzzy Sets and Systems, 161: 1138-1144. http://dx.doi.org/10.1016/j.fss.2009.10.004
Gregori, V., Miana, J.-J., Morillas, S. (2012); Some questions in fuzzy metric spaces, Fuzzy Sets and Systems, 204: 71-85. http://dx.doi.org/10.1016/j.fss.2011.12.008
Gregori, V., Romaguera, S. (2000); Some properties of fuzzy metric space, Fuzzy Sets and Systems, 115: 485-489. http://dx.doi.org/10.1016/S0165-0114(98)00281-4
Itoh, M., Cho, M. (1998); Fuzzy bounded operators, Fuzzy Sets and Systems, 93: 353-362. http://dx.doi.org/10.1016/S0165-0114(96)00198-4
Jain, R. (1976); Tolerance analysis using fuzzy sets, Int. J. Syst, 7(12): 1393-1401. http://dx.doi.org/10.1080/00207727608942013
Janfada, M., Baghani, H., Baghani, O. (2011); On Felbin's-type fuzzy normed linear spaces and fuzzy bounded operators, Iranian Journal of Fuzzy Systems, 8(5): 117-130.
Kaleva, O., Seikkala, S. (1984); On fuzzy metric spaces, Fuzzy Sets and Systems, 12: 215- 229. http://dx.doi.org/10.1016/0165-0114(84)90069-1
Katsaras, A.K., Liu, D.B. (1977); Fuzzy vector spaces and fuzzy topological vector spaces, Journal of Mathematical Analysis and Applications, 58: 135-146. http://dx.doi.org/10.1016/0022-247X(77)90233-5
Katsaras, A.K. (1981); Fuzzy topological vector spaces I, Fuzzy Sets and Systems, 6: 85-95. http://dx.doi.org/10.1016/0165-0114(81)90082-8
Katsaras, A.K. (1984); Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12:143-154. http://dx.doi.org/10.1016/0165-0114(84)90034-4
Kerre, K.K. (2011); A tribute to Zadeh's extension principle, Scientia Iranica D, 18(3): 593-595. http://dx.doi.org/10.1016/j.scient.2011.04.011
Krishna, S.V., Sarma, K.M. (1991); Fuzzy topological vector spaces - topological generation and normability. Fuzzy Sets and Systems, 41: 89-99. http://dx.doi.org/10.1016/0165-0114(91)90159-N
Kramosil, I., Michálek, J. (1975); Fuzzy metric and statistical metric spaces, Kybernetica, 11: 326-334.
Lowen, R. (1976); Fuzzy topological spaces and fuzzy compactness, Journal of Mathemat- ical Analysis and Applications, 56: 621-633. http://dx.doi.org/10.1016/0022-247X(76)90029-9
Lowen, R. (1977); Initial and final topologies and the fuzzy Tychonoff theorem, J. of Math. Analysis and Applications, 58: 11-21. http://dx.doi.org/10.1016/0022-247X(77)90223-2
Lupiá-ez, F.G. (2001); On Mich alek's fuzzy topological spaces, Kybernetika, 37(2): 159- 163.
Matloka, M. (2000); Convex fuzzy processes, Fuzzy Sets and Systems, 110: 109-114. http://dx.doi.org/10.1016/S0165-0114(98)00053-0
Michálek, J. (1975); Fuzzy topologies, Kybernetika, 11(5): 345-354.
Mizumoto, M., Tanaka, J. (1979); Some properties of fuzzy numbers, in M.M. Gupta et all, Editors, Advances in Fuzzy Set Theory and Applications (North-Holland, New York), 153-164.
Nădăban, S., Dzitac, I. (2014); Atomic Decompositions of Fuzzy Normed Linear Spaces for Wavelet Applications, Informatica, 25(4): 643-662. http://dx.doi.org/10.15388/Informatica.2014.33
Nădăban, S. (2014); Fuzzy pseudo-norms and fuzzy F-spaces, Fuzzy Sets and Systems, doi: 10.1016/j.fss.2014.12.010. http://dx.doi.org/10.1016/j.fss.2014.12.010
Nădăban, S., Dzitac, I. (2014);Special types of fuzzy relations, Procedia Computer Science, 31: 552-557. http://dx.doi.org/10.1016/j.procs.2014.05.301
Nădăban, S. (2015); Fuzzy euclidean normed spaces for data mining applications, International J. of Computers Communications & Control, 10(1): 70-77. http://dx.doi.org/10.15837/ijccc.2015.1.1564
Nădăban, S. (2015); Some fundamental properties of fuzzy linear relations between vector spaces, Filomat (in press).
Nădăban, S., Bînzar, T., Pater, F., Ţerei, C., Hoară, S. (2015); Katsaras's type fuzzy norm under triangular norms, Theory and Applications of Mathematics & Computer Science, 5(2): 148-157.
Nădăban, S. (2015); Fuzzy continuous mappings in fuzzy normed linear spaces, International J. of Computers Communications & Control, 10(6): 835-843. http://dx.doi.org/10.15837/ijccc.2015.6.2074
Papageorgiou, N.S. (1985); Fuzzy topology and fuzzy multifunctions, J. of Mathematical Analysis and Applications, 109: 397-425. http://dx.doi.org/10.1016/0022-247X(85)90159-3
Qiu, D., Yang, F., Shu, L. (2010); On convex fuzzy processes and their generalizations, International J. of Fuzzy Systems, 12: 268-273.
Saadati, R., Vaezpour, S.M. (2005); Some results on fuzzy Banach spaces, J. Appl. Math.
Saadati, R., Park, J.H. (2006); Intuitionistic fuzzy Euclidean normed spaces, Communica- tions in Mathematical Analysis, 1(2): 85-90.
Saadati, R., Park, J.H. (2006); On the intuitionistic fuzzy topological spaces, Chaos, Soli- tons and Fractals, 27: 331-344. http://dx.doi.org/10.1016/j.chaos.2005.03.019
Sadeqi, I., Kia, F.S. (2009); Fuzzy normed linear space and its topological structure, Chaos, Solitons and Fractals, 40: 2576-2589. http://dx.doi.org/10.1016/j.chaos.2007.10.051
Schweizer, B., Sklar, A. (1960); Statistical metric spaces, Pacific J. Math., 10: 314-334.
Shostak, A.P. (1985); On a fuzzy topological structure, Suppl. Rend. Circ. Mat. Palermo, Ser. II, 11: 89-103.
Shostak, A.P. (1989); Two decades of fuzzy topology: basic ideas, notions, and results, Russian Math. Surveys, 44(6): 125-186. http://dx.doi.org/10.1070/RM1989v044n06ABEH002295
Syau, Y.-R., Low, C.-Y., Wu, T.-H. (2002); A note on convex fuzzy processes, Applied Mathematics Letters 15: 193-196. http://dx.doi.org/10.1016/S0893-9659(01)00117-3
Tsiporkova-Hristoskova, E., De Baets, B., Kerre, E. (1997); A fuzzy inclusion based approach to upper inverse images under fuzzy multivalued mappings, Fuzzy Sets and Systems, 85: 93-108. http://dx.doi.org/10.1016/0165-0114(95)00345-2
Tsiporkova, E., De Baets, B., Kerre, E. (1998); Continuity of fuzzy multivalued mapping, Fuzzy Sets and Systems, 94: 335-348. http://dx.doi.org/10.1016/S0165-0114(96)00263-1
Warren, R.H. (1978); Neighborhoods, bases and continuity in fuzzy topological space, Rocky Mountain Journal of Mathematics, 8(3): 459-470. http://dx.doi.org/10.1216/RMJ-1978-8-3-459
Xiao, J.Z., Zhu, X.H. (2002); On linearly topological structures and property of fuzzy normed linear space, Fuzzy Sets and Systems, 125(2): 153-161. http://dx.doi.org/10.1016/S0165-0114(00)00136-6
Xiao, J.Z., Zhu, X.H. (2003); Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems, 133(3): 389-399. http://dx.doi.org/10.1016/S0165-0114(02)00274-9
Zadeh, L.A. (1965); Fuzzy Sets, Informations and Control, 8: 338-353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X
Zadeh, L.A. (1975); The concept of a linguistic variable and its application to approximate reasoning, parts 1,2 and 3, Inf. Sciences 8: 199-249, 8: 301-357, 9: 43-80.
Zadeh, L.A. (2011); My Life and Work - A Retrospective View, Special Issue on Fuzzy Set Theory and Applications, Dedicated to the 90th Birthday of prof. Lotfi A. Zadeh in Aplied and Computational Mathematics, 10(1):4-9, 2011.
Zadeh, L.A. (2011); Foreword, Special Issue on Fuzzy Sets and Systems, Dedicated to the 90th Birthday of Prof. Lotfi A. Zadeh in International J. of Computers Communications& Control, 6(3): 385-386.
Zadeh, L.A., Tufis, D., Filip, F.G., Dzitac I. (2008); From Natural Language to Soft Com- puting: New Paradigms in Artificial Intelligence , Ed. House of Romanian Academy.
Zimmermann, H.-J. (2001); Fuzzy set theory - and its applications, 4th Edition, Kluwer Academic Publisher.
Published
Issue
Section
License
ONLINE OPEN ACCES: Acces to full text of each article and each issue are allowed for free in respect of Attribution-NonCommercial 4.0 International (CC BY-NC 4.0.
You are free to:
-Share: copy and redistribute the material in any medium or format;
-Adapt: remix, transform, and build upon the material.
The licensor cannot revoke these freedoms as long as you follow the license terms.
DISCLAIMER: The author(s) of each article appearing in International Journal of Computers Communications & Control is/are solely responsible for the content thereof; the publication of an article shall not constitute or be deemed to constitute any representation by the Editors or Agora University Press that the data presented therein are original, correct or sufficient to support the conclusions reached or that the experiment design or methodology is adequate.