2015
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Cover vol. 12, no. 6, December 2015
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The generation of fuzzy sets and the~construction of~characterizing functions of~fuzzy data
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Measurement results contain different kinds of uncertainty. Besides systematic errors andrandom errors individual measurement results are also subject to another type of uncertainty,socalled emph{fuzziness}. It turns out that special fuzzy subsets of the set of real numbers $RR$are useful to model fuzziness of measurement results. These fuzzy subsets $x^*$ are called emph{fuzzy numbers}. The membership functions of fuzzy numbers have to be determined. In the paper firsta characterization of membership function is given, and after that methods to obtainspecial membership functions of fuzzy numbers, socalled emph{characterizing functions} describingmeasurement results are treated.
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1
16


L.
Kovarova
Faculty of Mathematics and Physics, Charles Univer
sity in Prague, Czech Republic
Faculty of Mathematics and Physics, Charles
Czech Republic


R.
Viertl
Faculty of Mathematics and Geoinformation, Vienna University of Tech
nology, Austria
Faculty of Mathematics and Geoinformation,
Austria
Characterizing function
Fuzzy data
Generating families
Measurement results
Vectorcharacterizing function
[[1] A. Ferrero, M. Prioli and S. Salicone, Conditional randomfuzzy variables representing mea##surement results, IEEE Transactions on Instrumentation and Measurement, 64(5) (2015),##11701178.##[2] S. Jain and M. Khare, Construction of fuzzy membership functions for urban vehicular ex##haust emissions modeling, Environmental monitoring and assessment, 167(14) (2010), 691##[3] G. Klir and B. Yuan, Fuzzy sets and fuzzy logic { theory and applications, Prentice Hall,##Upper Saddle River, 1995.##[4] A. SanchoRoyo, and J. L. Verdegay, Methods for the construction of membership functions,##International Journal of Intelligent Systems, 14(12) (1999), 12131230.##[5] R. Viertl, Fuzzy models for precision measurements, Mathematics and Computers in Simu##lation, 79(4) (2008), 847878.##[6] R. Viertl, Statistical methods for fuzzy data, John Wiley & Sons, Chichester, 2011.##[7] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338353.##[8] AX. Zhu, L. Yang, B. Li, C. Qin, T. Pei and B. Liu, Construction of membership functions##for predictive soil mapping under fuzzy logic, Geoderma, 155(3) (2010), 164174.##]
Double Fuzzy ImplicationsBased Restriction Inference Algorithm
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The main condition of the differently implicational inferencealgorithm is reconsidered from a contrary direction, which motivatesa new fuzzy inference strategy, called the double fuzzyimplicationsbased restriction inference algorithm. New restrictioninference principle is proposed, which improves the principle of thefull implication restriction inference algorithm. Furthermore,focusing on the new algorithm, we analyze the basic property of itssolution, and then obtain its optimal solutions aiming at theproblems of fuzzy modus ponens (FMP) as well as fuzzy modus tollens(FMT). Lastly, comparing with the full implication restrictioninference algorithm, the new algorithm can make the inferencecloser, and generate more, better specific inference algorithms.
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17
40


Yiming
Tang
School of Computer and Information, Hefei University of Technol
ogy, Hefei 230009, China
School of Computer and Information, Hefei
China
tym608@163.com


Xuezhi
Yang
School of Computer and Information, Hefei University of Technology,
Hefei 230009, China
School of Computer and Information, Hefei
China
xzyang@hfut.edu.cn


Xiaoping
Liu
School of Computer and Information, Hefei University of Technology,
Hefei 230009, China
School of Computer and Information, Hefei
China
lxp@hfut.edu.cn


Juan
Yang
School of Computer and Information, Hefei University of Technology,
Hefei 230009, China
School of Computer and Information, Hefei
China
uzzy inference
Fuzzy system
Compositional rule of inference (CRI) algorithm
Full implication inference algorithm
Fuzzy implication
[[1] M. Baczynski and B. Jayaram, Fuzzy implications (Studies in Fuzziness and Soft Computing,##Vol. 231), Springer, Berlin Heidelberg, 2008.##[2] S. S. Dai, D. W. Pei and S. M. Wang, Perturbation of fuzzy sets and fuzzy reasoning based##on normalized Minkowski distances, Fuzzy Sets and Systems, 189 (2012), 6373.##[3] D. Dubois and H. Prade, Fuzzy sets in approximate reasoning, Part 1: Inference with possi##bility distributions, Fuzzy Sets and Systems, 40 (1991), 143202.##[4] J. Fodor, On contrapositive symmetry of implications in fuzzy logic, First European Congress##on Fuzzy and Intelligent Technologies, Aachen, (1993), 1342{1348.##[5] J. Fodor and M. Roubens, Fuzzy Preference Modeling and Multicriteria Decision Support,##Kluwer Academic Publishers, Dordrecht, 1994.##[6] S. Gottwald, A treatise on manyvalued logics, Research Studies Press, Baldock, 2001.##[7] P. Hajek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.##[8] J. Hou, F. You and H. X. Li, Fuzzy systems constructed by triple I algorithm and their##response ability, Progress in Natural Science, 15 (2005), 2937.##[9] B. Jayaram, On the law of importation (x ^ y) ! z (x ! (y ! z)) in fuzzy logic, IEEE##Transactions on Fuzzy Systems, 16 (2008), 130144.##[10] E. P. Klement, R. Mesiar and E. Pap, Triangular Norms, Kluwer Academic Publishers,##Dordrecht, 2000.##[11] H. X. Li, Probability representations of fuzzy systems, Science in China. Series F, 49 (2006),##[12] H. X. Li, F. You and J. Y. Peng, Fuzzy controllers based on some fuzzy implication operators##and their response functions, Progress in Natural Science, 14 (2004), 1520.##[13] H. W. Liu, Fully implicational methods for approximate reasoning based on intervalvalued##fuzzy sets, Journal of Systems Engineering and Electronics, 21 (2010), 224232.##[14] H. W. Liu and G. J. Wang, Continuity of triple I methods based on several implications,##Computers and Mathematics with Applications, 56 (2008), 20792087.##[15] H. W. Liu and C. Li, Fully implicational methods for intervalvalued fuzzy reasoning##with multiantecedent rules, International Journal of Computational Intelligence Systems,##4 (2011), 929945.##[16] M. Mas, M. Monserrat, J. Torrens and E. Trillas, A survey on fuzzy implication functions,##IEEE Transactions on Fuzzy Systems, 15 (2007), 11071121.##[17] D. W. Pei, R0 implication: characteristics and applications, Fuzzy Sets and Systems, 131##(2002), 297302.##[18] D. W. Pei, On the strict logic foundation of fuzzy reasoning, Soft Computing, 8 (2004),##[19] D. W. Pei, Unied full implication algorithms of fuzzy reasoning, Information Sciences, 178##(2008), 520530.##[20] D. W. Pei, Formalization of implication based fuzzy reasoning method, International Journal##of Approximate Reasoning, 53 (2012), 837846. ##[21] J. Y. Peng, Fully implicational triple I restriction algorithm for fuzzy reasoning based on##some familiar implication operators, Progress in Natural Science, 15 (2005), 539546.##[22] H. Seki and M. Mizumoto, SIRMs connected fuzzy inference method adopting emphasis and##suppression, Fuzzy Sets and Systems, 215 (2013), 112126.##[23] S. J. Song, C. B. Feng and C. X. Wu, Theory of restriction degree of triple I method with##total inference rules of fuzzy reasoning, Progress in Natural Science, 11 (2001), 5866.##[24] S. J. Song, C. B. Feng and E. S. Lee, Triple I method of fuzzy reasoning, Computers and##Mathematics with Applications, 44 (2002), 15671579.##[25] Y. M. Tang and X. P. Liu, Dierently implicational universal triple I method of (1, 2, 2)##type, Computers and Mathematics with Applications, 59 (2010), 19651984.##[26] Y. M. Tang, F. J. Ren and Y. X. Chen, Reversibility of FMTuniversal triple I method##based on IL operator, American Journal of Engineering and Technology Research, 11 (2011),##27632766.##[27] Y. M. Tang, F. J. Ren and Y. X. Chen, Universal triple I method and its application to textual##emotion polarity recognition, In: Third International Conference on Quantitative Logic and##Soft Computing, Xi'an, China, (2012), 189196.##[28] L. X. Wang, A course in fuzzy systems and control, PrenticeHall, Englewood Clis, NJ,##[29] G. J. Wang, Fully implicational triple I method for fuzzy reasoning, Science in China. Series##E, 29 (1999), 4353.##[30] G. J. Wang and L. Fu, Unied forms of triple I method, Computers and Mathematics with##Applications, 49 (2005), 923932.##[31] G. J. Wang and H. J. Zhou, Introduction to Mathematical Logic and Resolution Principle,##Science Press, Beijing and Alpha Science International Limited, Oxford, U.K., 2009.##[32] L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision##processes, IEEE Transactions on Systems Man and Cybernetics, 3 (1973), 28{44.##[33] J. C. Zhang and X. Y. Yang, Some properties of fuzzy reasoning in propositional fuzzy logic##systems, Information Sciences, 180 (2010), 46614671.##]
Power harmonic aggregation operator with trapezoidal intuitionistic fuzzy numbers for solving MAGDM problems
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Trapezoidal intuitionistic fuzzy numbers (TrIFNs) express abundant and flexible information in a suitable manner and are very useful to depict the decision information in the procedure of decision making. In this paper, some new aggregation operators, such as, trapezoidal intuitionistic fuzzy weighted power harmonic mean (TrIFWPHM) operator, trapezoidal intuitionistic fuzzy ordered weighted power harmonic mean (TrIFOWPHM) operator, trapezoidal intuitionistic fuzzy induced ordered weighted power harmonic mean (TrIFIOWPHM) operator and trapezoidal intuitionistic fuzzy hybrid power harmonic mean (TrIFhPHM) operator are introduced to aggregate the decision information. The desirable properties of these operators are presented in detail. A prominent characteristic of these operators is that, the aggregated value by using these operators is also a TrIFN. It is observed that the proposed TrIFWPHM operator is the generalization of trapezoidal intuitionistic fuzzy weighted harmonic mean (TrIFWHM) operator, trapezoidal intuitionistic fuzzy weighted arithmetic mean (TrIFWAM) operator, trapezoidal intuitionistic fuzzy weighted geometric mean (TrIFWGM) operator and trapezoidal intuitionistic fuzzy weighted quadratic mean (TrIFWQM) operator, {it i.e.,} we can easily reduce the TrIFWPHM operator to TrIFWHM, TrIFWGM, TrIFWAM and TrIFWQM operators, depending upon the decision situation. Further, we develop an approach to multiattribute group decision making (MAGDM) problem on the basis of the proposed aggregation operators. Finally, the effectiveness and applicability of our proposed MAGDM model, as well as comparison analysis with other approaches are illustrated with a practical example.
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74


Satyajit
Das
Department of Mathematics, Indian Institute of Technology Patna,
India
Department of Mathematics, Indian Institute
India
satyajitnit.das@gmail.com


Debashree
Guha
Department of Mathematics, Indian Institute of Technology Patna,
India
Department of Mathematics, Indian Institute
India
debashree@iitp.ac.in
Intuitionistic fuzzy number
Power mean
Harmonic mean
Ranking
Multiattribute group decision making
[[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87{96.##[2] K. T. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and##Systems, 31(3) (1989), 343{349.##[3] K. T. Atanassov, G. Pasi and R. Yager, Intuitionistic fuzzy interpretations of multicriteria##multiperson and multimeasurement tool decision making, International Journal of Systems##Science, 36(14) (2005), 859{868.##[4] B. F. Baird, Managerial decisions under uncertainty: An introduction to the analysis of##decision making, John Wiley and Sons, 1989.##[5] A. P. G. Beliakov and T. Calvo, Aggregation functions: A guide for practitioners, Springer##Verlag, Berlin Heidelberg, 2007. ##[6] F. E. Boran, S. Genc, M. Kurt and D. Akay, A multicriteria intuitionistic fuzzy group##decision making for supplier selection with topsis method, Expert Systems with Applications,##36(8) (2009), 11363{11368.##[7] P. Burillo, H. Bustince and V. Mohedano, Some denitions of intuitionistic fuzzy number.##rst properties, In: Proceedings of the 1st Workshop on Fuzzy Based Expert Systems, (1994),##[8] M. Casanovas and J. M. Merigo, Fuzzy aggregation operators in decision making with##dempstershafer belief structure, Expert Systems with Applications, 39(8) (2012), 7138{##[9] S.M. Chen and J.M. Tan, Handling multicriteria fuzzy decisionmaking problems based on##vague set theory, Fuzzy Sets and Systems, 67(2) (1994), 163{172.##[10] H. Chen, C. Liu and Z. Sheng, Induced ordered weighted harmonic averaging (IOWHA) oper##ator and its application to combination forecasting method, Chinese Journal of Management##Science, 12(5) (2004), 35{40.##[11] Z. Chen and W. Yang, A new multiple attribute group decision making method in intuition##istic fuzzy setting, Applied Mathematical Modelling, 35(9) (2011), 4424{4437.##[12] F. Chiclana, F. Herrera and E. HerreraViedma, The ordered weighted geometric operator:##Properties and application in mcdm problems, In: Technologies for Constructing Intelligent##Systems, Springer, (2002), 173{183.##[13] G. Choquet, Theory of capacities, In: Annales de linstitut Fourier, 5 (1954), 131{295.##[14] S. Das and D. Guha, Ranking of intuitionistic fuzzy number by centroid point, Journal of##Industrial and Intelligent Information, 1(2) (2013), 107{110.##[15] S. Das, B. Dutta and D. Guha, Weight computation of criteria in a decisionmaking problem##by knowledge measure with intuitionistic fuzzy set and intervalvalued intuitionistic fuzzy set,##Soft Computing, DOI 10.1007/s0050001518133.##[16] S. K. De, R. Biswas and A. R. Roy, An application of intuitionistic fuzzy sets in medical##diagnosis, Fuzzy Sets and Systems, 117(2) (2001), 209{213.##[17] B. Farhadinia, An ecient similarity measure for intuitionistic fuzzy sets, Soft Computing,##18(1) (2014), 85{94.##[18] M. Grabisch and C. Labreuche, A decade of application of the choquet and sugeno integrals##in multicriteria decision aid, Annals of Operations Research, 175(1) (2010), 247{286.##[19] P. Grzegorzewski, Distances and orderings in a family of intuitionistic fuzzy numbers, In:##EUSFLAT Conference, Citeseer, (2003), 223{227.##[20] S. Gupta and V. Kapoor, Fundamentals of mathematical statistics: A modern approach,##Sultan Chand and Sons, 1982.##[21] W. Jianqiang and Z. Zhong, Aggregation operators on intuitionistic trapezoidal fuzzy number##and its application to multicriteria decision making problems, Journal of Systems Engineer##ing and Electronics, 20(2) (2009), 321{326.##[22] A. Kharal, Homeopathic drug selection using intuitionistic fuzzy sets, Homeopathy, 98(1)##(2009), 35{39.##[23] D.F. Li, A note on using intuitionistic fuzzy sets for faulttree analysis on printed circuit##board assembly, Microelectronics Reliability, 48(10) (2008), 17{41.##[24] D.F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application##to madm problems, Computers and Mathematics with Applications, 60(6) (2010), 1557{1570.##[25] A. Matuszak, Dierencesarithmeticgeometricharmonicmeans, The Economist at##large;http://economistatlarge.com/nance/appliednance/dierencesarithmeticgeometric##harmonicmeans.##[26] D. Meng and Z. Pei, On weighted unbalanced linguistic aggregation operators in group deci##sion making, Information Sciences, 223 (2013), 31{41.##[27] F. Meng and X. Chen, Intervalvalued intuitionistic fuzzy multicriteria group decision mak##ing based on cross entropy and 2additive measures, Soft Computing, 19(7) (2015), 2071##[28] F. Meng, X. Chen and Q. Zhang, Multiattribute decision analysis under a linguistic hesitant##fuzzy environment, Information Sciences, 267 (2014), 287{305. ##[29] F. Meng and X. Chen, Entropy and similarity measure of atanassovs intuitionistic fuzzy sets##and their application to pattern recognition based on fuzzy measures, Pattern Analysis and##Applications, DOI 10.1007/s10044{014{0378{6.##[30] F. Meng, X. Chen and Q. Zhang, Some uncertain generalized shapley aggregation operators##for multiattribute group decision making, Journal of Intelligent and Fuzzy Systems, DOI:##10.3233/IFS131069.##[31] G. A. Papakostas, A. G. Hatzimichailidis and V. G. Kaburlasos, Distance and similarity##measures between intuitionistic fuzzy sets: A comparative analysis from a pattern recognition##point of view, Pattern Recognition Letters, 34(14) (2013), 1609{1622.##[32] J. H. Park, J. M. Park and Y. C. Kwun, 2tuple linguistic harmonic operators and their##applications in group decision making, KnowledgeBased Systems, 44 (2013), 10{19.##[33] M.H. Shu, C.H. Cheng and J.R. Chang, Using intuitionistic fuzzy sets for faulttree analysis##on printed circuit board assembly, Microelectronics Reliability, 46(12) (2006), 2139{2148.##[34] E. Szmidt and J. Kacprzyk, Intuitionistic fuzzy sets in some medical applications, in: Com##putational Intelligence, Theory and Applications, Springer, (2001), 148{151.##[35] S.P. Wan and J.Y. Dong, Method of intuitionistic trapezoidal fuzzy number for multi##attribute group decision, Control and Decision, 25(5) (2010), 773{776.##[36] S.P. Wan, Multiattribute decision making method based on intervalvalued intuitionistic##trapezoidal fuzzy number, Control and Decision, 26(6) (2011), 857{861.##[37] S.Wan, Method based on fractional programming for intervalvalued intuitionistic trapezoidal##fuzzy number multiattribute decision making, Control and Decision, 27(3) (2012), 455{458.##[38] S.P. Wan, Q.Y. Wang and J.Y. Dong, The extended VIKOR method for multiattribute##group decision making with triangular intuitionistic fuzzy numbers, KnowledgeBased Sys##tems, 52 (2013), 65{77.##[39] S.P. Wan, D.F. Li and Z.F. Rui, Possibility mean, variance and covariance of triangular##intuitionistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 24(4) (2013), 847{##[40] S.P. Wan, Multiattribute decision making method based on possibility variance coecient of##triangular intuitionistic fuzzy numbers, International Journal of Uncertainty, Fuzziness and##KnowledgeBased Systems, 21(2) (2013), 223{243.##[41] S.P. Wan and D.F. Li, Fuzzy linmap approach to heterogeneous MADM considering com##parisons of alternatives with hesitation degrees, Omega, 41(6) (2013), 925{940.##[42] S.P. Wan, Power average operators of trapezoidal intuitionistic fuzzy numbers and appli##cation to multiattribute group decision making, Applied Mathematical Modelling, 37(6)##(2013), 4112{4126.##[43] S.P. Wan and D.F. Li, Atanassovs intuitionistic fuzzy programming method for heteroge##neous multiattribute group decision making with atanassovs intuitionistic fuzzy truth degrees,##IEEE Tran. On Fuzzy Systems, 22(2) (2014), 300{312.##[44] S.Wan and J. Dong, A possibility degree method for intervalvalued intuitionistic fuzzy multi##attribute group decision making, Journal of Computer and System Sciences, 80(1) (2014),##[45] S.P. Wan and J.Y. Dong, Possibility method for triangular intuitionistic fuzzy multi##attribute group decision making with incomplete weight information, International Journal##of Computational Intelligence Systems, 7(1) (2014), 65{79.##[46] S.P. Wan and J.Y. Dong, Power geometric operators of trapezoidal intuitionistic fuzzy##numbers and application to multiattribute group decision making, Applied Soft Computing,##29 (2015), 153{168.##[47] J.Q. Wang, R. Nie, H.Y. Zhang and X.H. Chen, New operators on triangular intuitionistic##fuzzy numbers and their applications in system fault analysis, Information Sciences, 251##(2013), 79{95.##[48] G. Wei, Some induced geometric aggregation operators with intuitionistic fuzzy information##and their application to group decision making, Applied Soft Computing, 10(2) (2010), 423{##[49] G. Wei, Some arithmetic aggregation operators with intuitionistic trapezoidal fuzzy numbers##and their application to group decision making, Journal of Computers, 5(3) (2010), 345{351.##[50] J. Wu and Q.W. Cao, Same families of geometric aggregation operators with intuitionistic##trapezoidal fuzzy numbers, Applied Mathematical Modelling, 37(1) (2013), 318{327.##[51] Z. Xu and Q. Da, The ordered weighted geometric averaging operators, International Journal##of Intelligent Systems, 17(7) (2002), 709{716.##[52] Z. Xu and J. Chen, An interactive method for fuzzy multiple attribute group decision making,##Information Sciences, 177(1) (2007), 248{263.##[53] Z. Xu, Intuitionistic fuzzy aggregation operators, IEEE Tran. on Fuzzy Systems, 15(6) (2007),##1179{1187.##[54] Z. Xu, Fuzzy harmonic mean operators, International Journal of Intelligent Systems, 24(2)##(2009), 152{172.##[55] Z. Xu and R. R. Yager, Powergeometric operators and their use in group decision making,##IEEE Tran. on Fuzzy Systems, 18(1) (2010), 94{105.##[56] Y. Xu and H. Wang, The induced generalized aggregation operators for intuitionistic fuzzy##sets and their application in group decision making, Applied Soft Computing, 12(3) (2012),##1168{1179.##[57] R. R. Yager, Connectives and quantiers in fuzzy sets, Fuzzy Sets and Systems, 40(1) (1991),##[58] R. R. Yager, On a semantics for neural networks based on fuzzy quantiers, International##Journal of Intelligent Systems, 7(8) (1992), 765{786.##[59] R. R. Yager and D. Filev, Fuzzy logic controllers with ##exible structures, In: Proceedings##of the 2nd International Conference on Fuzzy Sets and Neural Networks, Tizuka, (1992),##[60] R. R. Yager and J. Kacprzyk, The ordered weighted averaging operators:Theory and appli##cations, Kluwer, Norwell, MA, 1997.##[61] R. R. Yager, The power average operator, IEEE Tran. on Systems and Humans, Man and##Cybernetics, 31(6) (2001), 724{731.##[62] R. R. Yager, Generalized OWA aggregation operators, Fuzzy Optimization and Decision Mak##ing, 3(1) (2004), 93{107.##[63] R. R. Yager, On generalized bonferroni mean operators for multicriteria aggregation, Inter##national Journal of Approximate Reasoning, 50(8) (2009), 1279{1286.##[64] J. Ye, Multicriteria group decisionmaking method using vector similarity measures for trape##zoidal intuitionistic fuzzy numbers, Group Decision and Negotiation, 21(4) (2012), 519{530.##[65] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338{353.##]
A Comparative Study of Fuzzy Inner Product Spaces
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In the present paper, we investigate a connection between two fuzzy inner product one of which arises from Felbin's fuzzy norm and the other is based on Bag and Samanta's fuzzy norm. Also we show that, considering a fuzzy inner product space, how one can construct another kind of fuzzy inner product on this space.
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75
93


M.
Saheli
Department of Mathematics, ValieAsr University of Rafsanjan, Raf
sanjan, Iran
Department of Mathematics, ValieAsr University
Iran
Fuzzy norm
Fuzzy inner product
Fuzzy Hilbert space
[[1] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math.,##11(3) (2003), 687705.##[2] T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems, 151##(2005), 513547.##[3] S. C. Cheng and J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces,##Bull. Cal. Math. Soc., 86 (1994), 429436.##[4] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48 (1992),##[5] M. Goudarzi and S. M. Vaezpour, On the denition of fuzzy Hilbert spaces and its application,##J. Nonlinear Sci. Appl., 2(1) (2009), 4659.##[6] M. Goudarzia and S. M. Vaezpour, R. Saadati, On the intuitionistic fuzzy inner product##spaces, Chaos, Solitons and Fractals, 41 (2009), 11051112.##[7] A. Hasankhani, A. Nazari and M. Saheli, Some properties of fuzzy Hilbert spaces and norm##of operators, Iranian Journal of Fuzzy Systems, 7(3) (2010), 129157.##[8] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984),##[9] I. Karmosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11##(1975), 326334.##[10] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12 (1984), 143##[11] P. Mazumdar and S. K. Samanta,On fuzzy inner product spaces, The Journal of Fuzzy Math##ematics 16(2) (2008), 377392.##[12] S. Mukherjee and T. Bag,Fuzzy real inner product space and its properties, Annals of Fuzzy##Mathematics and Informatics 6(2) (2013), 377389.##[13] S. Vijayabalaji,Fuzzy strong ninner product space, International Journal of Applied Mathe##matics., 1(2) (2010), 176185.##[14] S. Vijayabalaji,Equivalent fuzzy strong ninner product space, International Journal of Open##Problems in Computer Science and Mathematics, 4(4) (2011), 2632.##[15] J. Xiao and X. Zhu, Fuzzy normed space of operators and its completeness, Fuzzy Sets and##Systems, 133 (2003), 389399.##]
Coupled common fixed point theorems for $varphi$contractions in probabilistic metric spaces and applications
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2
In this paper, we give some new coupled common fixed point theorems for probabilistic $varphi$contractions in Menger probabilistic metric spaces. As applications of the main results, we obtain some coupled common fixed point theorems in usual metric spaces and fuzzy metric spaces. The main results of this paper improvethe corresponding results given by some authors. Finally, we give one example to illustrate the main results of this paper.
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95
108


S. H.
Wang
Department of Mathematics and Physics, North China Electric Power
University, Baoding, China
Department of Mathematics and Physics, North
China


A. A. N.
Abdou
Department of Mathematics, King Abdulaziz University, Jeddah,
Saudi Arabia
Department of Mathematics, King Abdulaziz
Saudi Arabia


Y. J.
Cho
Department of Education Mathematics and RINS, Gyeongsang National
University, Jinju, Korean
Department of Education Mathematics and RINS,
Korea
Menger probabilistic metric space
probabilistic $varphi$contraction
coupled fixed points
[[1] T. G. Bhashkar and V. Lakshmikantham, Fixed point theorems in partially ordered metric##spaces and applications, Nonlinear Anal., 65 (2006), 1379{1393.##[2] S. S. Chang, Y. J. Cho and S. M. Kang, Nonlinear Operator Theory in Probabilistic Metric##Spaces, Nova Science Publishers, Inc., New York, 2001.##[3] B. S. Choudhury, K. Das and P. N. Dutta, A xed point result in Menger spaces using a real##function, Acta Math. Hungar., 122 (2009), 203{216.##[4] B. S. Choudhury and A. Kundu, A coupled coincidence point result in partially ordered metric##spaces for compatible mappings, Nonlinear Anal., 73 (2010), 2524{2531.##[5] L. B. Ciric, Solving the Banach xed point principle for nonlinear contractions in probabilistic##metric spaces, Nonlinear Anal., 72 (2010), 2009{2018.##[6] L. B. Ciric, R. P. Agarwal and B. Samet, Mixed monotonegeneralized contractions in par##tially ordered probabilistic metric spaces, Fixed Point Theory Appl., (2011) 2011:56.##[7] L. B. Ciric, D. Mihet and R. Saadati, Monotone generzliaed contractions in partially ordered##probabilistic metric spaces, Topology Appl., 156 (2009), 2838{2844.##[8] J. X. Fang, Fixed point theorems of local contraction mappings on Menger spaces, Appl.##Math. Mech., 12 (1991), 363{372.##[9] J. X. Fang, Common xed point theorems of compatible and weakly compatible maps in##Menger spaces, Nonlinear Anal., 71 (2009), 1833{1843.##[10] J. X. Fang, On 'contractions in probabilistic and fuzzy metric spaces, Fuzzy Sets Syst., 267##(2014), 86{99.##[11] O. Hadzic and E. Pap, Fixed Point Theory in PMSpaces, Kluwer Academic Publ., 2001.##[12] X. Q. Hu, Common coupled xed point theorems for contractive mappings in fuzzy metric##spaces, Fixed Point Theory Appl., Article ID 363716 (2011), 2011.##[13] J. Jachymski, On probabilistic 'contractions on Menger spaces, Nonlinear Anal., 73 (2010),##2199{2203.##[14] K. Karapinar, Coupled xed point theorems for nonlinear contractions in cone metric spaces,##Comput. Math. Appl., 59 (2010), 3656{3668.##[15] I. Kramosi and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11##(1975), 336{344.##[16] V. Lakahmikantham and L. B. Ciric, Coupled xed point theorems for nonlinear contractions##in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341{4349.##[17] N. V. Luong and N. X. Thuan, Coupled xed points in partially ordered metric spaces and##application, Nonlinear Anal., 74 (2011), 983{992.##[18] K. Menger, Statistical metric, Proc Natl. Acad. USA., 28 (1942), 535{537.##[19] D. O'Regan and R. Saadati, Nonlinear contraction theorems in probabilistic spaces, Appl.##Math. Comput., 195 (2008), 86{93.##[20] R. Saadati, Generalized distance and xed point theorems in partially ordered probabilistic##metric spaces, Mate. Vesnik, 65 (2013), 82{93.##[21] B. Samet, Coupled xed point theorems for a generalized MeirKeeler contraction in partially##ordered metric spaces, Nonlinear Anal., 71 (2010), 4508{4517.##[22] B. Schweizer and A. Sklar, Probabilisitc Metric Spaces, Elsevier/NorthHolland, New York,##[23] S. Sedghi, I. Altun and N. Shobec, Coupled xed point theorems for contractions in fuzzy##metric spaces, Nonlinear Anal., 72 (2010), 1298{1304. ##[24] V. M. Sehgal and A. T. BharuchaReid, Fixed points of contraction mappings on probabilistic##metric space, Math Syst. Theory, 6 (1972), 87{102.##[25] J. Z. Xiao, X. H. Zhu and Y. F. Cao, Common coupled xed point results for probabilistic##'contractions in Menger spaces, Nonlinear Anal., 74 (2011), 4589{4600.##]
The Urysohn, completely Hausdorff and completely regular axioms in $L$fuzzy topological spaces
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2
In this paper, the Urysohn, completely Hausdorff and completely regular axioms in $L$topological spaces are generalized to $L$fuzzy topological spaces. Each $L$fuzzy topological space can be regarded to be Urysohn, completely Hausdorff and completely regular tosome degree. Some properties of them are investigated. The relations among them and $T_2$ in $L$fuzzy topological spaces are discussed.
1

109
128


Chengyu
Liang
College of Science, North China University of Technology, No.5
Jinyuanzhuang Road, Shijingshan District, 100144 Beijing, P.R. China
College of Science, North China University
China
liangchengyu87@163.com


FuGui
Shi
School of Mathematics and Statistics, Beijing Institute of Technology,
5 South Zhongguancun Street, Haidian District, 100081 Beijing, P.R. China
School of Mathematics and Statistics, Beijing
China
fugushi@bit.edu.cn
$L$fuzzy topology
Urysohn axiom
Completely Hausdorff axiom
Completely regular axiom
[[1] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182–190.##[2] S. L. Chen and Z. X.Wu, Urysohn separation property in topological molecular lattices, Fuzzy##Sets Syst., 123 (2001), 177–184.##[3] P. Dwinger, Characterizations of the complete homomorphic images of a completely distributive##complete lattice I, Indagationes Mathematicae(Proceedings), 85 (1982), 403–414.##[4] J. M. Fang, H()completely Hausdorff axiom on Ltopological spaces, Fuzzy Sets Syst., 140##(2003), 475–469.##[5] J. M. Fang and Y. L. Yue, Urysohn closedness on completely distributive lattices, Fuzzy Sets##Syst., 144 (2004), 367–381.##[6] J. M. Fang and Y. L. Yue, Base and subbase in Ifuzzy topological spaces, J. Math. Res.##Exposition, 26 (2006), 89–95.##[7] G. Gierz, et al., A compendium of continuous lattices, Springer Verlag, Berlin, 1980.##[8] B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl., 50 (1975), 74–79.##[9] U. H¨ohle, Probabilistic metrization of fuzzy uniformities, Fuzzy Sets Syst., 8 (1982), 63–69.##[10] U. H¨ohle and A. P. ˘Sostak, Axiomatic foudations of fixedbasis fuzzy topology, In: U. H¨ohle,##S. E. Rodabaugh(Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory,##Handbook Series, vol.3, Kluwer Academic Publishers, Boston, Dordrecht, London, (1999),##123–173.##[11] T. Kubiak, On fuzzy topologies, Ph. D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985.##[12] H. Y. Li and F. G. Shi, Some separation axioms in Ifuzzy topological spaces, Fuzzy Sets##Syst., 159 (2008), 573–587.##[13] S. E. Rodabaugh, The Hausdorff separation axiom for fuzzy topological spaces, Topology##Appl., 11 (1980), 319–334.##[14] F. G. Shi, Pointwise uniformities and pointwise metrics on fuzzy lattices, Chinese Science##Bulletin, 42 (1997), 718–720.##[15] F. G. Shi, Pointwise uniformities in fuzzy set theory, Fuzzy Sets Syst., 98 (1998), 141146.##[16] F. G. Shi, Fuzzy pointwise complete regularity and imbedding theorem, The Journal of Fuzzy##Mathematics, 7 (1999), 305–310.##[17] F. G. Shi, A new approach to LT2, LUrysohn, and Lcompletely Hausdorff axioms, Fuzzy##Sets Syst., 157 (2006), 794–803.##[18] F. G. Shi, The Urysohn axiom and the completely Hausdorff axiom in Ltopological spaces,##Iranian Journal of Fuzzy Systems, 7(1) (2010), 33–45.##[19] F. G. Shi, (L;M)fuzzy metric spaces, Indian J. Math., 52 (2010), 231250.##[20] F. G. Shi, Regularity and normality of (L;M)fuzzy topological spaces, Fuzzy Sets Syst., 182##(2011), 37–52.##[21] A. P. ˇSostak, On a fuzzy topological structure, Suppl. Rend. Circ. Mat. PalermoSer. II, 11##(1985), 89–103.##[22] A. P. ˇSostak, Two decades of fuzzy topology: basic ideas, notions and results, Russian Math.##Surveys, 44 (1989), 125–186.##[23] M. Ying, A new approach to fuzzy topology (I), Fuzzy Sets Syst., 39 (1991), 303–321. ##[24] Y. L. Yue and J. M. Fang, Generated Ifuzzy topological spaces, Fuzzy Sets Syst., 154 (2005),##103–117.##[25] Y. L. Yue and J. M. Fang, On separation axioms in Ifuzzy topological spaces, Fuzzy Sets##Syst., 157 (2006), 780–793.##]
A generalization of the ChenWu duality into quantalevalued setting
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2
With the unit interval [0,1] as the truth value table, Chen and Wupresented the concept of possibility computation over dcpos.Indeed, every possibility computation can be considered as a[0,1]valued Scott open set on a dcpo. The aim of this paper is tostudy ChenWu's duality on quantalevalued setting. For clarity,with a commutative unital quantale $L$ as the truth value table, weintroduce a concept of fuzzy possibility computations over fuzzydcpos and then establish an equivalence between their denotationalsemantics and their logical semantics.
1

129
140


Chong
Shen
Department of Physics, Hebei University of Science and Technology,
Shijiazhuang 050018, P.R. China
Department of Physics, Hebei University of
China
shenchong0520@163.com


Shanshan
Zhang
Department of Physics, Hebei University of Science and Technol
ogy, Shijiazhuang 050018, P.R. China
Department of Physics, Hebei University of
China
zhangshan920805@163.com


Wei
Yao
Department of Physics, Hebei University of Science and Technology, Shi
jiazhuang 050018, P.R. China
Department of Physics, Hebei University of
China
22987944@qq.com


Changcheng
Zhang
Department of Physics, Hebei University of Science and Tech
nology, Shijiazhuang 050018, P.R. China
Department of Physics, Hebei University of
China
puregenius@126.com
Fuzzy Scott topology
$L$fuzzy possibility computation
Denotational semantics
$L$fuzzy predicate transformer
$L$fuzzy logical semantics
[[1] R. Belohlavek, Fuzzy relational systems: foundations and principles, Kluwer Academic/##Plenum Publishers, New York, 2002.##[2] R. Belohlavek, Concept lattices and order in fuzzy logic, Annals of Pure and Applied Logic,##128(13) (2004), 227298.##[3] Y. X. Chen and H. Y. Wu, Domain semantics of possibility computations, Information Sciences,##178(2) (2008), 26612679.##[4] Y. X. Chen and A. Jung, An introduction to fuzzy predicate transformers, The Invited Talk##at the Third International Symposium on Domain Theory, Shaanxi Normal University, Xian,##China, 2004.##[5] P. W. Chen, H. Lai and D. Zhang, Core##ective hull of nite strong Ltopological spaces,##Fuzzy Sets and Systems, 182(1) (2011), 7992.##[6] L. Fan, A new approach to quantitative domain theory, Electronic Notes in Theroretical##Computer Science, 45 (2001), 7787.##[7] J. A. Goguen, Lfuzzy sets, Journal of Mathematial Analysis and Applications, 18(1) (1967),##[8] C. A. R. Hoare, Communicating sequential process, Communications of the ACM, 21(8)##(1978), 666677.##[9] C. Jones, Probabilistic nondeterminism, PhD thesis, Department of Computer Science, University##of Edinburgh, Edinburgh, 1990.##[10] C. Jones and G. Plotkin, A probabilistic powerdomain of evaluations, In Proceedings of the##Fourth Annual Symposium on Logic in Computer Science, (1989), 186195.##[11] H. Lai and D. Zhang, Complete and directed complete ##categories, Theoretical Computer##Science, 388(13) (2007), 125.##[12] G. D. Plotkin, A powerdomain construction, SIAM Journal on Computing, 5(3) (1976),##[13] G. D. Plotkin, A powerdomain for countable nondeterminism, In M. Nielsen and E. M.##Schmidt (editors), Automata, Languages and programming, Lecture Notes in Computer Science,##EATCS, SpringerVerlag, 140 (1982), 412428.##[14] G. D. Plotkin, Probabilistic powerdomains, In Proceedings CAAP, (1982), 271287.##[15] K. I. Rosenthal, Quantales and their applications, Longman Scientic and Technical, New##York, 1990.##[16] S. E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies,##pp. 91116, Chapter 2 in U. Hohle and S. E. Rodabaugh, eds, Mathematics of Fuzzy Sets:##Topology, and Measure Theory, The handbooks of Fuzzy Sets Series, Volume 3 (1999), Kluwer##Academic Publishers (Boston/ Dordrecht/London).##[17] S. E. Rodabaugh, Relationship of algebraic theories to powerset theories and fuzzy topo##logical theories for latticevalued mathematics, International Journal of Mathematics and##Mathematical Sciences, vol. 2007, Article ID 43645, 71 pages, 2007. doi:10.1155/2007/43645##[18] D. S. Scott, A type theoretical alternative to ISWIM, CUCH, OWHY, Theoretical Computer##Science, 121(12) (1993), 411440.##[19] D. S. Scott, Continuous lattices, In: E. Lawvere (Ed.), Toposes, Algebraic Geometry and##Logic, Lecture Notes in Mathematics, SpringerVerlag, 274 (1972), 97136.##[20] M. B. Smyth, Powerdomains, Journal of Computer and Systems Sciences, 16(1) (1978),##[21] N. SahebDjahromi, CPOs of measures for nondeterminism, Theoretical Computer Science,##12(1) (1980), 1937.##[22] R. Tix, K. Keimel and G. Plotkin, Semantic domains for combining probability and non##determinism, Electronic Notes in Theoretical Computer Science, 222 (2009), 399.##[23] K. R. Wagner, Liminf convergence in ##categories, Theoretical Computer Science, 184(12)##(1997), 61104.##[24] W. Yao and L. X. Lu, Fuzzy Galois connections on fuzzy posets, Mathmatical Logic Quarterly,##55(1) (2009), 105112. ##[25] W. Yao, Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed complete##posets, Fuzzy Sets and Systems, 161(7) (2010), 937987.##[26] W. Yao and F. G. Shi, Quantitative domains via fuzzy sets: Part II: fuzzy Scott topology on##fuzzy directed complete posets, Fuzzy Sets and Systems, 173(1) (2011), 121.##[27] W. Yao, A survey of fuzzications of frames, the PapertPapertIsbell adjunction and sobri##ety, Fuzzy Sets and Systems, 190(1) (2012), 6381.##[28] W. Yao, A categorical isomorphism between injective fuzzy T0spaces and fuzzy continuous##lattices, IEEE Transactions on Fuzzy Systems, Article in press.##[29] W. Yao, A more general truth valued table for latticevalued convergence spaces, Preprint.##[30] Q. Y. Zhang and L. Fan, Continuity in quantitative domains, Fuzzy Sets and Systems, 154(1)##(2005), 118131.##[31] Q. Y. Zhang and W. X. Xie, Sectionretractionpairs between fuzzy domains, Fuzzy Sets and##Systems, 158(1) (2007), 99114.##]
Coincidence point theorem in ordered fuzzy metric spaces and its application in integral inclusions
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2
The purpose of this paper is to present some coincidence point and common fixed point theorems for multivalued contraction maps in complete fuzzy metric spaces endowed with a partial order. As an application, we give an existence theorem of solution for general classes of integral inclusions by the coincidence point theorem.
1

141
154


Z.
Sadeghi
Young Researchers and Elite Club, Roudehen Branch, Islamic Azad
University, Roudehen, Iran.
Young Researchers and Elite Club, Roudehen
Iran


S. M.
Vaezpour
Department of Mathematics and Computer Sciences, Amirkabir Uni
versity of Technology, Tehran, Iran
Department of Mathematics and Computer Sciences,
Iran
Coincidence point
Fixed point
Multivalued mapping
Ordered fuzzy metric space
Volterra integral inclusion
[[1] A. Beitollahi, P. Azhdari, Multivalued ( ; ; ; )contractin in probabilistic metric space,##Fixed Point Theory and Applications, (2012), 2012:10.##[2] S. S. Chang, B. S. Lee, Y. J. Cho, Y. Q. Chen, S. M. Kang and J. S. Jung, Generalized##contraction mapping principle and diferential equations in probabilistic metric spaces, Pro##ceedings of the American Mathematical Society, 124(8) (1996), 2367{2376.##[3] L. Ciric, Some new results for Banach contractions and Edelestein contractive mappings on##fuzzy metric spaces, Chaos, Solutons and Fractals, 42 (2009), 146{154.##[4] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy sets and systems,##64 (1994), 395{399.##[5] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy sets and systems, 27 (1988), 385{389.##[6] V. Gregori and A. Sapena, On xed point theorems in fuzzy metric spaces, Fuzzy Sets and##Systems, 125 (2002), 245{252.##[7] O. Hadzic and E. Pap, A xed point theorem for multivalued mappings in probabilistic metric##spaces and an application in fuzzy metric spaces, Fuzzy Sets and Systems, 127 (2002), 333{##[8] O. Hadzic, E. Pap and M. Budincevic, Countable extension of triangular norms and their ap##plications to the xed point theory in probabilistic metric spaces, Kybernetika, 38(3) (2002),##[9] O. Hadzic and E. Pap, Fixed point theorems for singlevalued and multivalued mappings in##probabilistic metric spaces, Atti Sem. Mat. Fiz. Modena LI, (2003), 377395.##[10] O. Hadzic and E. Pap, Fixed point theory in probabilistic metric space, Kluwer Academic##Publishers, Dordrecht, 2001.##[11] O. Hadzic and E. Pap, New classes of probabilistic contractions and applications to random##operators, in: Y.J. Cho, J.K. Kim, S.M. Kong (Eds.), Fixed Point Theory and Application,##Vol. 4, Nova Science Publishers, Hauppauge, New York, (2003), 97119.##[12] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984),##[13] Y. Liu and Zh. Li, Coincidence point theorems in probabilistic and fuzzy metric spaces, Fuzzy##Sets and Systems, 158 (2007), 5870.##[14] D. Mihet, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems,##144 (2004), 431439. ##[15] D. Mihet, A class of contractions in fuzzy metric spaces, Fuzzy Sets and Systems, 161 (2010),##11311137.##[16] D. Mihet, Multivalued generalizations of probabilistic contractions, J. Mathematic Analysis##Application, 304 (2005), 464{472.##[17] J. RodrguezLpez and S. Romaguera, The Hausdor fuzzy metric on compact sets, Fuzzy##Sets and Systems, 147 (2004), 273{283.##[18] R. Saadati and S. M. Vaezpour, Some results on fuzzy Banach spaces, J. Application Math##ematic and Computing, 17(12) (2005), 475484.##[19] B. Schweizer, A. Sklar, Statistical metric spaces, Pac. J. Math., 10 (1960), 313{334.##[20] S. L. Singh and S. N. Mishra, Coincidence and xed points of nonself hybrid contractions, J.##Math. Anal. Appl., 256 (2001), 486{497.##[21] P. Tirado, Contraction mappings in fuzzy quasimetric spaces and [0; 1]fuzzy posets, Fixed##Point Theory, 13(1) (2012), 273{283.##[22] T. ZikicDosenovic, A multivalued generalization of Hick's Ccontraction, Fuzzy Sets and##Systems, 151(3) (2005), 549562.##[23] T. ZikicDosenovic, Fixed point theorems for contractive mappings in Menger probabilistic##metric spaces, Proceeding of IPMU'08, June 22{27, (2008), 1497{1504.##]
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