The Fuzzification of Classical Structures: A General View

Authors

  • Ioan Dzitac Agora University of Oradea & Aurel Vlaicu University of Arad

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.

Author Biography

Ioan Dzitac, Agora University of Oradea & Aurel Vlaicu University of Arad

Associate Editor in Chief of IJCCC

Rector of Agora University

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

2015-10-03

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