2016
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Arithmetic Aggregation Operators for Intervalvalued Intuitionistic Linguistic Variables and Application to Multiattribute Group Decision Making
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The intuitionistic linguistic set (ILS) is an extension of linguisitc variable. To overcome the drawback of using single real number to represent membership degree and nonmembership degree for ILS, the concept of intervalvalued intuitionistic linguistic set (IVILS) is introduced through representing the membership degree and nonmembership degree with intervals for ILS in this paper. The operation law, score function, accuracy function , and certainty function for intervalvalued intuitionistic linguistic varibales (IVILVs) are defined. Hereby a lexicographic method is proposed to rank the IVILVs. Then, three kinds of intervalvalued intuitionistic linguistic arithmetic average operators are defined, including the intervalvalued intuitionistic linguistic weighted arithmetic average (IVILWAA) operator, intervalvalued intuitionistic linguistic ordered weighted arithmetic (IVILOWA) operator, and intervalvalued intuitionistic linguistic hybrid arithmetic (IVILHA) operator, and their desirable properties are also discussed. Based on the IVILWAA and IVILHA operators, two methods are proposed for solving multiattribute group decision making problems with IVILVs. Finally, an investment selection example is illustrated to demonstrate the applicability and validity of the methods proposed in this paper.
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1
23


Jiuying
Dong
School of Statistics, Jiangxi University of Finance and Economics,
Nanchang 330013, China and Research Center of Applied Statistics, Jiangxi University
of Finance and Economics, Nanchang 330013, China
School of Statistics, Jiangxi University
China
jiuyingdong@126.com


ShuPing
Wan
College of Information Technology, Jiangxi University of Finance
and Economics, Nanchang 330013, China
College of Information Technology, Jiangxi
China
Multiattribute group decision making
Intervalvalued intuitionistic linguistic set
Intuitionistic fuzzy set
Arithmetic operators
[[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1)(1986), 8796.##[2] K. Atanassov and G. Gargov, Intervalvalued intuitionistic fuzzy sets, Fuzzy Sets and##Systems, 31 (3) (1989), 343349.##[3] D. P. Filev and R. R. Yager, On the issue of obtaining OWA operator weights, Fuzzy Sets##and Systems, 94(2) (1998), 157169.##[4] F. Herrera and L. Martnez, A 2tuple fuzzy linguistic representation model for computing##with words, IEEE Transactions on Fuzzy Systems, 8 (2000), 746752.##[5] F. Herrera, L. Martnez and P. J. Snchez, Managing nonhomogeneous information in group##decisionmaking, European Journal of Operational Research, 166(1) (2005), 115132.##[6] P. D. Liu and F. Jin, Methods for aggregating intuitionistic uncertain linguistic variables##and their application to group decision making, Information Sciences, 205 (2012), 5871.##[7] J. M. Merigo, M. Casanovas and L. Martnez, Linguistic aggregation operators for linguistic##decision making based on the DempsterShafer theory of evidence, International Journal##of Uncertainty, Fuzziness and KnowledgeBased Systems, 18(3) (2010), 287304.##[8] J. M. Merigo, A unied model between the weighted average and the induced OWA oper##ator, Expert Systems with Applications, 38(9) (2011), 1156011572.##[9] J. M. Merigo and A. M. GilLafuente, Decision making techniques in business and econom##ics based on the OWA operator, SORT C Statistics and Operations Research Transactions,##36(1) (2012), 81102.##[10] J. M. Merigo, A. M. GilLafuente, L. G. Zhou and H. Y. Chen, Induced and linguistic##generalized aggregation operators and their application in linguistic group decision making,##Group Decision and Negotiation, 21 (2012), 531549.##[11] M. OHagan, Aggregating template rule antecedents in realtime expert systems with fuzzy##set logic. In: Proc 22nd Annual IEEE Asilomar Conference on Signals, Systems and Computers.##Pacic Grove, CA: IEEE and Maple Press,(1988), 681689.##[12] Z. Pei, D. Ruan, J. Liu and Y. Xu, A linguistic aggregation operator with three kinds of##weights for nuclear safeguards evaluation, KnowledgeBased Systems, 28 (2012), 1926.##[13] V. Torra, The weighted OWA operator, International Journal of Intelligent Systems, 12##(1997), 153166.##[14] S. P. Wan, 2tuple linguistic hybrid arithmetic aggregation operators and application to##multiattribute group decision making, KnowledgeBased Systems, 45 (2013), 3140.##[15] S. P.Wan, Some hybrid geometric aggregation operators with 2tuple linguistic Information##and their applications to multiattribute group decision making, International Journal of##Computational Intelligence Systems, 6(4) (2013), 750763.##[16] S. P. Wan and J. Y. Dong. Intervalvalued intuitionistic fuzzy mathematical programming##method for hybrid multicriteria group decision making with intervalvalued intuitionistic##fuzzy truth degrees, Information Fusion, 26 (2015), 4965.##[17] S. P. Wan and D. F. Li. Fuzzy mathematical programming approach to heterogeneous##multiattribute decisionmaking with intervalvalued intuitionistic fuzzy truth degrees, Information##Sciences, 325 (2015), 484503.##[18] S. P. Wan, G. L Xu, F. Wang and J. Y. Dong, A new method for Atanassov's interval##valued intuitionistic fuzzy MAGDM with incomplete attribute weight information, Information##Sciences, 316 (2015), 329347.##[19] J. Q. Wang and J. J. Li, The multicriteria group decision making method based on multi##granularity intuitionistic two semantics, Science and Technology Information, 33 (2009),##[20] J. Q.Wang and H. B. Li, Multicriteria decision making based on aggregation operators for##intuitionistic linguistic fuzzy numbers, Control and Decision, 25(10) (2010), 15711574. ##[21] Y. Wang and Z. S. Xu, A new method of giving OWA weights, Mathematics in Practice##and Theory, 38 (2008), 5161.##[22] G. W. Wei, A method for multiple attribute group decision making based on the ET##WG and ETOWG operators with 2tuple linguistic information, Expert Systems with##Application, 37(12) (2010), 78957900.##[23] G. W. Wei, Some generalized aggregating operators with linguistic information and their##application to multiple attribute group decision making, Computers and Industrial Engineering,##61(1) (2011), 3238.##[24] G. W. Wei, Grey relational analysis method for 2tuple linguistic multiple attribute##group decision making with incomplete weight information, Expert Systems with Application,##38(5) (2011), 78957900.##[25] G. W. Wei and X. F. Zhao, Some dependent aggregation operators with 2tuple linguistic##information and their application to multiple attribute group decision making, Expert##Systems with Applications, 39 (2012), 58815886.##[26] Z. S. Xu, Uncertain linguistic aggregation operators based approach to multiple attribute##group decision making under uncertain linguistic environment, Information Sciences, 168##(2004), 171184.##[27] Z. S. Xu, An overview of methods for determining OWA weights, International Journal of##Intelligent Systems,20(8) (2005), 843865.##[28] Z. S. Xu,Induced uncertain linguistic OWA operators applied to group decision making,##Information Fusion, 7 (2006), 231238.##[29] Z. S. Xu,An approach based on the uncertain LOWG and induced uncertain LOWG oper##ators to group decision making with uncertain multiplicative linguistic preference relation,##Decision Support Systems, 41 (2006), 488499.##[30] Z. S. Xu, An interactive approach to multiple attribute group decision making with multi##granular uncertain linguistic information, Group Decision and Negotiation, 18 (2009),##[31] Y. J. Xu and H. M. Wang, Power geometric operators for group decision making under##multiplicative linguistic preference relations, International Journal of Uncertainty, Fuzziness##and KnowledgeBased Systems, 20(1) (2012), 139159.##[32] R. R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision##making, IEEE Trans Syst Man Cybern,18 (1988), 183190.##[33] R. R. Yager, Including importances in OWA aggregation using fuzzy systems modeling,##IEEE Transactions on Fuzzy Systems, (1998), 6286294.##[34] R. R. Yager, K. J. Engemann and D. P. Filev, On the concept of immediate probabilities,##International Journal of Intelligent Systems 10 (1995), 373397.##[35] W. E. Yang, J. Q.Wang and X. F.Wang, An outranking method for multicriteria decision##making with duplex linguistic information, Fuzzy Sets and Systems, 198 (2012), 2033.##[36] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338356.##[37] L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning##Part I, Information Sciences, 8(3) (1975), 199249.##[38] S. Z. Zeng and Balezentis, C. H. Zhang, A method based on OWA operator and distance##measures for multiple attribute decision making with 2Tuple linguistic information, Informatica,##23(4) (2012), 665681.##]
Decision making in medical investigations using new divergence measures for intuitionistic fuzzy sets
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In recent times, intuitionistic fuzzy sets introduced by Atanassov has been one of the most powerful and flexible approaches for dealing with complex and uncertain situations of real world. In particular, the concept of divergence between intuitionistic fuzzy sets is important since it has applications in various areas such as image segmentation, decision making, medical diagnosis, pattern recognition and many more. The aim of this paper is to introduce a new divergence measure for Atanassov's intuitionistic fuzzy sets (textit{AIFS)}. The properties of the proposed divergence measure have been studied and the findings are applied in medical diagnosis of some diseases with a common set of symptoms.
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25
44


A.
Srivastava
Department of Mathematics, Jaypee Institute of Information Tech
nology, Noida, Uttar Pradesh
Department of Mathematics, Jaypee Institute
India


S.
Maheshwari
Department of Mathematics, Jaypee Institute of Information Tech
nology, Noida, Uttar Pradesh
Department of Mathematics, Jaypee Institute
India
Fuzzy sets
Intuitionistic fuzzy sets
Divergence measure
Medical diagnosis
[[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 8796.##[2] D. Bhandari and N. R. Pal, Some new information measure for fuzzy sets, Information##Science, 67 (1993), 209228.##[3] F. E. Boran and D. Akay, A biparametric similarity measure on intuitionistic fuzzy sets with##applications to pattern recognition, Information Sciences, 255(10) (2014) 4557.##[4] T. Chaira and A. K. Ray, Segmentation using fuzzy divergence, Pattern Recognition Letters,##24(12) (2003), 18371844.##[5] 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), 209213.##[6] J. Fan and W. Xie, Distance measure and induced fuzzy entropy, Fuzzy Sets and Systems,##104(2) (1999), 305314.##[7] A. G. Hatzimichailidis, G. A. Papakostas and V. G. Kaburlasos, A novel distance measure##of intuitionistic fuzzy sets and its application to pattern recognition problems, International##Journal of Intelligent Systems 27(4) (2012), 396409.##[8] W. L. Hung and M. S. Yang, On the J Divergence of intuitionistic fuzzy sets with its##applications to pattern recognition, Information sciences, 178(6) (2008), 16411650.##[9] K. C. Hung, Medical pattern recognition: applying an improved intuitionistics fuzzy cross##entropy approach, Advances in fuzzy Systems Article ID, 863549 (2012).##[10] M. Junjun, Y. Dengbao and W. Cuicui, A novel crossentropy and entropy measures of IFSs##and their applications, KnowledgeBased Systems, 48 (2013), 3745.##[11] S. Kullback, Information theory and statistics, Dover publications, New York, USA, 1968.##[12] J. Lin, Divergence measures based on the Shannon entropy, IEEE Transactions on Informa##tion Theory, 37(1) (1991), 145151.##[13] S. Montes, I. Couso, P. Gil and C. Bertoluzza, Divergence measure between fuzzy sets, Inter##national Journal of Approximate Reasoning, 30(2) (2002), 91105.##[14] I. Montes, V. Janis and S. Montes, An axiomatic denition of divergence for intuitionistic##fuzzy sets, Advances in Intelligent Systems Research, EUSFLAT 2011, Atlantis Press, AixLes##Bains, ISBN 9789078677000, (2011), 547553.##[15] I. Montes, N. R. Pal, V. Janis and S. Montes, Divergence measures for intuitionistic fuzzy##sets, IEEE Transactions on Fuzzy Systems, 23 (2015) 444456.##[16] 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), 16091622.##[17] X. G. Shang, W. S. Jiang, A note on fuzzy information measures, Pattern Recognition##Letters, 18(5) (1997), 425432##[18] E. Szmidt and J. Kacprzyk, Intuitionistic fuzzy sets in intelligent data analysis for medical##diagnosis, Proceedings of the Computational Science ICCS. Springer, Berlin, Germany, 2074,##(2001), 263271.##[19] E. Szmidt and J. Kacprzyk, Intuitionistic fuzzy sets in some medical applications, Pro##ceedings of the 7th Fuzzy Days, 2206 Computational intelligence: theory and applications.##Springer, Berlin, Germany (2001), 148151.##[20] E. Szmidt and J. Kacprzyk, A Similarity Measure for Intuitionistic Fuzzy Sets and its Ap##plication in Supporting Medical Diagnostic Reasoning, Articial Intelligence and Soft Com##puting { ICAISC, 3070 (2004), 388393.##[21] K. Vlachos and G. D. Sergiadis, Intuitionistic fuzzy informationapplications to pattern##recognition, Pattern Recognition Letters, 28(2) (2007), 197206.##[22] P. Wei and J. Ye, Improved intuitionistic fuzzy crossentropy and its application to pattern##recognition, International Conference on Intelligent Systems and Knowledge Engineering,##(2010), 114116.##[23] M. Xia and Z. Xu,Entropy/cross entropybased group decision making under intuitionistic##fuzzy environment, Information Fusion, 13(1) (2012), 3147.##[24] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338353. ##[25] Q. S. Zhang and S. Y. Jiang, A note on information entropy measures for vague sets and its##applications, Information Sciences, 178(21) (2008), 41844191.##]
Designing a model of intuitionistic fuzzy VIKOR in multiattribute group decisionmaking problems
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Multiple attributes group decision making (MAGDM) is regarded as the process of determining the best feasible solution by a group of experts or decision makers according to the attributes that represent different effects. In assessing the performance of each alternative with respect to each attribute and the relative importance of the selected attributes, quantitative/qualitative evaluations are often required to handle uncertainty, imprecise and inadequate information, which are well suited to represent with fuzzy values. This paper develops a VIKOR method based on intuitionistic fuzzy sets with multijudges and multiattributes in reallife situations. Intuitionistic fuzzy weighted averaging (IFWA) operator is used to aggregate individual judgments of experts to rate the importance of attributes and alternatives. Then, an intuitionistic ranking index is introduced to obtain a compromise solution to solve MAGDM problems. For application and validation, this paper presents two application examples and solves the practical portfolio selection and material handling selection problems to verify the proposed method. Finally, the intuitionistic fuzzy VIKOR method is compared with the existing intuitionistic fuzzy MAGDM method for two application examples, and their computational results are discussed.
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45
65


Seyed Meysam
Mousavi
Department of Industrial Engineering, Faculty of Engi
neering, Shahed University, Tehran, Iran
Department of Industrial Engineering, Faculty
Iran


Behnam
Vahdani
Faculty of Industrial and Mechanical Engineering, Qazvin Branch,
Islamic Azad University, Qazvin, Iran
Faculty of Industrial and Mechanical Engineering,
Iran
b.vahdani@ut.ac.ir


Shadan Sadigh
Behzadi
Department of Mathematics, Qazvin Branch, Islamic Azad
University, Qazvin, Iran
Department of Mathematics, Qazvin Branch,
Iran
Intuitionistic fuzzy sets
VIKOR method
Group decision making
Portfolio selection
Material handling selection
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Atanassov, On Intuitionistic Fuzzy Sets Theory, Studies in Fuzziness and Soft Computing.##SpringerVerlag., 2012.##[8] KT. Atanassov and C. Georgiev, Intuitionistic fuzzy prolog, Fuzzy Sets and Systems., 53##(1993), 121128.##[9] KT. Atanassov, NG. Nikolov and HT. Aladjov, Remark on two operations over intuitionistic##fuzzy sets, International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems.,##9(1) (2003), 7175.##[10] KT. Atanassov, G. Pasi and RR. Yager, Intuitionistic fuzzy interpretations of multicriteria##multiperson and multimeasurement tool decision making, International Journal of Systems##Science., 36 (2005), 859868.##[11] FE. 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##(2009), 1136311368.##[12] P. Burillo and H. Bustince H, Entropy on intuitionistic fuzzy sets and on intervalvalued##fuzzy sets, Fuzzy Sets and Systems., 78 (1996), 305316.##[13] G. Buyukozkan and D. Ruan, Evaluation of software development projects using a fuzzy##multicriteria decision approach, Mathematics and Computers in Simulation., 77 (2008),##[14] CL. Chang and CH. Hsu, Multicriteria analysis via the VIKOR method for prioritizing##landuse restraint strategies in the TsengWen reservoir watershed, Journal of Environmental##Management., 90 (2009), 32263230.##[15] LY. Chen and TC. Wang, Optimizing partners choice in IS/IT outsourcing projects: The##strategic decision of fuzzy VIKOR, International Journal of Production Economics., 120##(2009), 233242.##[16] SJ. Chen and CL. Hwang, Fuzzy multiple attribute decision making: methods and applica##tions, SpringerVerlag, Berlin., 1992.##[17] T. Chen, Remarks on the Subtraction and Division Operations over Intuitionistic Fuzzy Sets##and IntervalValued Fuzzy Sets, International Journal of Fuzzy Systems., 9 (2007), 169172.##[18] R. De SK Biswas and AR. Roy, Some operations on intuitionistic fuzzy sets, Fuzzy Sets and##Systems., 114 (2000), 477484.##[19] R. Degani and G. Bortolan, The problem of linguistic approximation in clinical decision##making, International Journal of Approximate Reasoning., 2 (1988), 143162.##[20] M. Delgado, J. Verdegay and M. Vila, On aggregation operations of linguistic labels, International##Journal of Intelligent Systems., 8 (1993), 351370.##[21] G. Deschrijver and EE. Kerre, On the representation of intuitionistic fuzzy tnorms and##tconorms, IEEE Transactions on Fuzzy Systems., 12 (2004), 4561.##[22] Y. Dong, Y. Xu and S. Yu, Computing the numerical scale of the linguistic term set for##the 2tuple fuzzy linguistic representation model, IEEE Transactions on Fuzzy Systems., 17##(2009), 13661378.##[23] E. Gurkan, I. Erkmen and AM. Erkmen, Twoway fuzzy adaptive identication and control##exiblejoint robot arm, Information Science., 145 (2002), 1343.##[24] F. Herrera and E. HerreraViedma, Linguistic decision analysis: steps for solving decision##problems under linguistic information, Fuzzy Sets and Systems., 115 (2000), 6782.##[25] F. Herrera, E. HerreraViedma and L. Martinez, A fuzzy linguistic methodology to deal with##unbalanced linguistic term sets, IEEE Transactions on Fuzzy Systems., 16 (2008), 354370.##[26] F. Herrera and L. Martnez, A 2tuple fuzzy linguistic representation model for computing##with words, IEEE Transactions on Fuzzy Systems., 8 (2000), 746 752.##[27] WL. Hung and MS. Yang, Similarity measures of intuitionistic fuzzy sets based on Lp metric,##International Journal of Approximate Reasoning., 46 (2007), 120136. ##[28] MS. Kuo, GH. Tzeng and WC. Huang Group decisionmaking based on concepts of ideal##and antiideal points in a fuzzy environment, Mathematical and Computer Modelling., 45##(2007), 324339.##[29] DF. Li, Compromise ratio method for fuzzy multiattribute group decision making, Applied##Soft Computing., 7 (2006), 807817.##[30] DF. Li, Multiattribute group decision making method using extended linguistic variables##International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems., 17 (2007),##[31] L. Lin, XH. Yuan and ZQ. Xia, M ulticriteria fuzzy decisionmaking methods based on intuitionistic##fuzzy sets, Journal of Computer and System Sciences., 73 (2007), 8488.##[32] H. Malekly, SM. Mousavi and H. Hashemi, A fuzzy integrated methodology for evaluating##conceptual bridge design, Expert Systems with Applications., 37 (2010), 49104920.##[33] J. Mendel, LA. Zadeh, E. Trillas, R. Yager, J. Lawry, H. Hagras and S. Guadarrama What##computing with words means to me, IEEE Computational Intelligence Magazine., 5 (2010),##[34] L. Mikhailov and P. Tsvetinov, Evaluation of services using a fuzzy analytic hierarchy process,##Applied Soft Computing., 5 (2004), 2333.##[35] SM. Mousavi SM, F. Jolai and R. TavakkoliMoghaddam, A fuzzy stochastic multiattribute##group decisionmaking approach for selection problems, Group Decision and Negotiation., 22##(2004), 207233.##[36] SM. Mousavi, SA. Torabi and R. TavakkoliMoghaddam, A hierarchical group decision##making approach for new product selection in a fuzzy environment, Arabian Journal for##Science and Engineering., 38 (2013), 32333248.##[37] SM. Mousavi, B. Vahdani, R. TavakkoliMoghaddam, S. Ebrahimnejad and M. Amiri, A##multistage decision making process for multiple attributes analysis under an intervalvalued##fuzzy environment, International Journal of Advanced Manufacturing Technology., 64 (2013),##12631273.##[38] S. Opricovic S, Multicriteria optimization of civil engineering systems, Faculty of Civil##Engineering, Belgrade (1998).##[39] S. Opricovic S and GH. Tzeng, Compromise solution by MCDM methods: A comparative##analysis of VIKOR and TOPSIS, European Journal of Operational Research., 156 (2004),##[40] S. Opricovic and GH. Tzeng, Extended VIKOR method in comparison with outranking meth##ods, European Journal of Operational Research., 178 (2007), 514529.##[41] A. Sanayei, SF. Mousavi and A. Yazdankhah, Group decision making process for supplier se##lection with VIKOR under fuzzy environment, Expert Systems with Applications., 37 (2010),##[42] MS. Shu, CH. Cheng and JR. Chang, Using intuitionistic fuzzy sets for faulttree analysis on##printed circuit board assembly, Microelectronics Reliability., 46 (2006), 21392148.##[43] E. Szmidt and J. Kacprzyk, Using intuitionistic fuzzy sets in group decision making, Control##Cybernetics., 31 (2002), 10371053.##[44] LT. Tong, CC. Chen and CH. Wang, Optimization of multiresponse processes using the##VIKOR method, International Journal of Advanced Manufacturing Technology., 31 (2007),##10491057.##[45] GH. Tzeng, MH. Teng, JJ. Chen and S. Opricovic, Multicriteria selection for a restaurant##location in Taipei, International Journal of Hospitality Management., 21 (2002), 171187.##[46] B. Vahdani, SM. Mousavi, R. TavakkoliMoghaddam and R. Hashemi, A new design of the##elimination and choice translating reality method for multiple criteria group decisionmaking##in an intuitionistic fuzzy environment, Applied Mathematical Modelling., 37 (2013), 1781##[47] B. Vahdani, R. TavakkoliMoghaddam, SM. Mousavi and A. Ghodratnama, Soft computing##based on new fuzzy modied multicriteria decision making method, Applied Soft Computing.,##13 (2013), 165172. ##[48] B. Vahdani and H. Hadipour, Extension of the ELECTRE method based on intervalvalued##fuzzy sets, Soft Computing., 15 (2011), 569579.##[49] J. Wang and J. Hao J, A new version of 2tuple fuzzy linguistic representation model for##computing with words, IEEE Transactions on Fuzzy Systems., 14 (2010), 435445.##[50] X. Wang, Z. Gao and G. Wei, An approach to archives websites performance evaluation in##our country with interval intuitionistic fuzzy information, Advances in information sciences##and service sciences., 3 (2011), 112117.##[51] Z. Xu Z, A method based on linguistic aggregation operators for group decision making with##linguistic preference relations, Information Sciences., 166 (2004), 1930.##[52] ZS. Xu ZS, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems.,##15 (2007), 11791187.##[53] ZS. Xu and RR. Yager, Some geometric aggregation operators based on intuitionistic fuzzy##sets, International Journal of General Systems., 35 (2006), 417433.##[54] R. Yager, A new methodology for ordinal multiobjective decisions based on fuzzy sets, Decision##Sciences., 12 (1981), 589600.##[55] L. Zadeh, Fuzzy logic = computing with words, IEEE Transactions on Fuzzy Systems., bf 94##(1996), 103111.##[56] L. Zadeh and J. Kacprzyk, Computing with words in information / Intelligent systems 1##(Foundations), Studies in Fuzziness and Soft Computing., 33 (1999), SpringerVerlag.##[57] L. Zadeh and J. Kacprzyk, Computing with words in information / Intelligent systems 2##(Applications), Studies in Fuzziness and Soft Computing., 34 (1999), SpringerVerlag.##[58] LA. Zadeh, Fuzzy sets, Information and Control., 8 (1965), 338353.##[59] CY. Zhang and HY. Fu, Similarity measures on three kinds of fuzzy sets, Pattern Recognition##Letters., 27 (2006), 13071317.##[60] SF. Zhang and SY Liu, A GRAbased intuitionistic fuzzy multicriteria group decision##making method for personnel selection, Expert Systems with Applications., 38 (2011), 11401##]
Piecewise cubic interpolation of fuzzy data based on Bspline basis functions
2
2
In this paper fuzzy piecewise cubic interpolation is constructed for fuzzy data based on Bspline basis functions. We add two new additional conditions which guarantee uniqueness of fuzzy Bspline interpolation.Other conditions are imposed on the interpolation data to guarantee that the interpolation function to be a welldefined fuzzy function. Finally some examples are given to illustrate the proposed method.
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67
76


Masoumeh
Zeinali
Faculty of mathematical sciences, University of Tabriz, Tabriz,
Iran
Faculty of mathematical sciences, University
Iran
zeinali@tabrizu.ac.ir


Sedaghat
Shahmorad
Faculty of mathematical sciences, University of Tabriz, Tabriz,
Iran
Faculty of mathematical sciences, University
Iran
shahmorad@tabrizu.ac.ir


Kamal
Mirnia
Faculty of mathematical sciences, University of Tabriz, Tabriz, Iran
Faculty of mathematical sciences, University
Iran
mirniakam@tabrizu.ac.ir
Fuzzy number
Fuzzy interpolation
Bspline basis functions
[[1] A. M. Anile, B. Falcidieno, G. Gallo, M. Spagnuolo and S. Spinello, Modeling uncertain data##with fuzzy Bspline, Fuzzy Sets and Systems, 113 (2000), 397410.##[2] B. Bede and S. G. Gal, Generalizations of the dierentiability of fuzzynumbervalued func##tions with applications to fuzzy dierential equations, Fuzzy Sets and Systems, 151 (2005),##[3] P. Blaga and B. Bede, Approximation by fuzzy Bspline series, J. Appl. Math. & Computing,##20 (2006), 157169.##[4] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986),##[5] O. Kaleva, Interpolation of fuzzy data, Fuzzy Sets and Systems, 61 (1994), 6370.##[6] W. A. Lodwick and J. Santos, Constructing consistenct fuzzy surfaces from fuzzy data, Fuzzy##Sets and Systems, 135 (2003), 259277.##[7] R. Lowen, A fuzzy Lagrange interpolation theorem, Fuzzy Sets and Systems, 34 (1990), 3338.##[8] P. M. Prenter, Spline and variational methods, A WileyInterscience publication, 1975.##[9] C. Wu and Z. Gong, On Henstock integral of fuzzynumbervalued functions I, Fuzzy Sets##and Systems, 120 (2001), 523532.##[10] M. Zeinali, S. Shahmorad and K. Mirnia, Hermite and piecewise cubic Hermite interpolation##of fuzzy data, Journal of Intelligent & Fuzzy Systems, 26 (2014), 28892898.##]
Further results on $L$ordered fuzzifying convergence spaces
2
2
In this paper, it is shown that the category of $L$ordered fuzzifying convergence spaces contains the category of pretopological $L$ordered fuzzifying convergence spaces as a bireflective subcategory and the latter contains the category of topological $L$ordered fuzzifying convergence spaces as a bireflective subcategory. Also, it is proved that the category of $L$ordered fuzzifying convergence spaces can be embedded in the category of stratified $L$ordered convergence spaces as a coreflective subcategory.
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77
92


Bin
Pang
Shenzhen Graduate School, Harbin Institute of Technology, 518055 Shen
zhen, P.R. China
Shenzhen Graduate School, Harbin Institute
China
pangbin1205@163.com


Yi
Zhao
Shenzhen Graduate School, Harbin Institute of Technology, 518055 Shen
zhen, P.R. China
Shenzhen Graduate School, Harbin Institute
China
zhaoyisz420@sohu.com
$L$ordered fuzzifying convergence space
Stratified $L$ordered convergence space
$L$filter
Category theory
[[1] J. Adamek, H. Herrlich and G. E. Strecker, Abstract and concrete categories, Wiley, New##York, 1990.##[2] J. M. Fang, Stratied Lordered convergence structures, Fuzzy Sets Syst., 161 (2010), 2130{##[3] J. M. Fang, Relationships between Lordered convergence structures and strong Ltopologies,##Fuzzy Sets Syst., 161(22) (2010), 2923{2944.##[4] J. Gutierrez Garca, I. Mardones Perez and M. H. Burton, The relationship between various##lter notions on a GLMonoid, J. Math. Anal. Appl., 230(1999), 291{302.##[5] U. Hohle and A. P. Sostak, Axiomatic foudations of xedbasis fuzzy topology, in: U. Hohle,##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{##[6] U. Hohle, Characterization of Ltopologies by Lvalued neighborhoods, Chapter 5, In [5],##[7] U. Hohle, Many valued topology and its applications, Kluwer Academic Publishers, Boston,##Dordrecht, London, 2001.##[8] G. Jager, A category of Lfuzzy convergence spaces, Quaest. Math., 24 (2001), 501{517.##[9] G. Jager, Subcategories of latticevalued convergence spaces, Fuzzy Sets Syst., 156 (2005),##[10] G. Jager, Pretopological and topological latticevalued convergence spaces, Fuzzy Sets Syst.,##158 (2007), 424{435.##[11] B. Y. Lee, J. H. Park and B. H. Park, Fuzzy convergence structures, Fuzzy Sets Syst., 56##(1993), 309{315.##[12] L. Q. Li and Q. Jin, On stratied Lconvergence spaces: Pretopological axioms and diagonal##axioms, Fuzzy Sets Syst., 204 (2012), 40{52.##[13] E. Lowen, R. Lowen and P. Wuyts, The categorical topological approach to fuzzy topology##and fuzzy convergence, Fuzzy Sets Syst., 40 (1991), 347{373.##[14] K. C. Min, Fuzzy limit spaces, Fuzzy Sets Syst., 32 (1989), 343{357. ##[15] B. Pang and J. M. Fang, Lfuzzy Qconvergence structures, Fuzzy Sets Syst., 182 (2011),##[16] B. Pang, On (L;M)fuzzy convergence spaces, Fuzzy Sets Syst., 238 (2014), 46{70.##[17] B. Pang, Enriched (L;M)fuzzy convergence spaces, J. Intell. Fuzzy Syst., 27(1) (2014),##[18] B. Pang and F. G. Shi Degrees of compactness of (L;M)fuzzy convergence spaces and its##applications, J. Intell. Fuzzy Syst., 251 (2014), 1{22.##[19] W. C. Wu and J. M. Fang, Lordered fuzzifying convergence spaces, Iranian Journal of Fuzzy##Systems, 9(2) (2012), 147{161.##[20] L. S. Xu, Characterizations of fuzzifying topologies by some limit structures, Fuzzy Sets Syst.,##123 (2001), 169{176.##[21] W. Yao, On manyvalued stratied Lfuzzy convergence spaces, Fuzzy Sets Syst., 159 (2008),##2503{2519.##[22] W. Yao, On Lfuzzifying convergence spaces, Iranian Journal of Fuzzy Systems, 6(1) (2009),##]
On The Relationships Between Types of $L$convergence Spaces
2
2
This paper focuses on the relationships between stratified $L$convergence spaces, stratified strong $L$convergence spaces and stratifiedlevelwise $L$convergence spaces. It has been known that: (1) astratified $L$convergence space is precisely a leftcontinuousstratified levelwise $L$convergence space; and (2) a stratifiedstrong $L$convergence space is naturally a stratified $L$convergence space, but the converse is not true generally.In this paper, a strong leftcontinuity condition for stratified levelwise $L$convergence space is given. It is proved that a stratified strong $L$convergence space is precisely a strongly leftcontinuous stratifiedlevelwise $L$convergence space. Then a sufficient and necessary condition for a stratified $L$convergence space to be a stratified strong $L$convergence space is presented.
1

93
103


Qiu
Jin
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
Department of Mathematics, Liaocheng University,
China
jinqiu79@126.com


Lingqiang
Li
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
Department of Mathematics, Liaocheng University,
China
lilingqiang@126.com


Guangwu
Meng
Department of Mathematics, Liaocheng University, Liaocheng, P.R.China
Department of Mathematics, Liaocheng University,
China
$L$topology
Stratified $L$filter
Stratified $L$convergence space
[[1] R. Belohlavek, Fuzzy relational systems: Foundations and Principles, New York: Kluwer##Academic Publishers, (2002), 75212.##[2] J. M. Fang, Stratied Lordered convergence structures, Fuzzy Sets and Systems, 161 (2010),##2130{2149.##[3] J. M. Fang, Relationships between Lordered convergence structures and strong Ltopologies,##Fuzzy Sets and Systems, 161 (2010), 2923{2944.##[4] J. M. Fang, Latticevalued semiuniform convergence spaces, Fuzzy Sets and Systems, 195##(2012), 33{57.##[5] J. M. Fang, Stratied Lordered quasiuniform limit spaces, Fuzzy Sets and Systems, 227##(2013), 51{73.##[6] P. V. Flores, R. N. Mohapatra and G. Richardson, Latticevalued spaces: Fuzzy convergence,##Fuzzy Sets and Systems, 157 (2006), 2706{2714.##[7] U. Hohle, Commutative, residuated lmonoids, In: U. Hohle, E.P. Klement (Eds.), Nonclassical##Logics and Their Applications to Fuzzy Subsets: A Handbook of the Mathematical##Foundations of Fuzzy Set Theory, Dordrecht: Kluwer Academic Publishers, (1995), 53{105.##[8] U. Hohle and A. Sostak, Axiomatic foundations of xedbasis fuzzy topology, In: U. Hohle,##S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory,##The Handbooks of Fuzzy Sets Series, Vol.3, Boston, Dordrecht, London: Kluwer Academic##Publishers, (1999), 123{273.##[9] G. Jager, A category of Lfuzzy convergence spaces, Quaestiones Mathematicae, 24 (2001),##[10] G. Jager, Subcategories of latticevalued convergence spaces, Fuzzy Sets and Systems, 156##(2005), 1{24.##[11] G. Jager, Fischer's diagonal condition for latticevalued convergence spaces, Quaestiones##Mathematicae, 31 (2008), 11{25.##[12] G. Jager, Gahler's neighbourhood condition for latticevalued convergence spaces, Fuzzy Sets##and Systems, 204 (2012), 27{39.##[13] G. Jager, Diagonal conditions for latticevalued uniform convergence spaces, Fuzzy Sets and##Systems, 210 (2013), 39{53.##[14] G. Jager, Stratied LMNconvergence tower spaces, Fuzzy Sets and Systems, 282 (2016),##[15] L. Li and Q. Li, A new regularity (pregularity) of stratied Lgeneralized convergence spaces,##Journal of Computational Analysis and Applications, 20(2) (2016), 307318.##[16] L. Li and Q. Jin, On adjunctions between Lim, SLTop, and SLLim, Fuzzy Sets and Systems,##182 (2011), 66{78.##[17] L. Li and Q. Jin, On stratied Lconvergence spaces: Pretopological axioms and diagonal##axioms, Fuzzy Sets and Systems, 204 (2012), 40{52.##[18] L. Li and Q. Jin, ptopologicalness and pregularity for latticevalued convergence spaces,##Fuzzy Sets and Systems, 238 (2014), 26{45.##[19] L. Li and Q. Jin, latticevalued convergence spaces: weaker regularity and pregularity, Abstract##and Applied Analysis, Volume 2014, Article ID 328153, 11 pages. ##[20] L. Li, Q. Jin and K. Hu, On stratied Lconvergence spaces: Fischer's diagonal axiom, Fuzzy##Sets and Systems, 267 (2015), 31{40.##[21] L. Li, Q. Jin, G. Meng and K. Hu, The lower and upper ptopological (pregular) modications##for latticevalued convergence spaces, Fuzzy Sets and Systems, 282 (2016), 47{61.##[22] D. Orpen and G. Jager, Latticevalued convergence spaces: extending the lattice context,##Fuzzy Sets and Systems, 190 (2012), 1{20.##[23] B. Pang and F. Shi, Degrees of compactness in (L;M)fuzzy convergence spaces, Fuzzy Sets##and Systems, 251 (2014), 1{22.##[24] G. D. Richardson and D. C. Kent, Probabilistic convergence spaces, Journal of the Australian##Mathematical Society, 61 (1996), 400{420.##[25] W. Yao, On manyvalued stratied Lfuzzy convergence spaces, Fuzzy Sets and Systems, 159##(2008), 2503{2519.##[26] W. Yao, Quantitative domains via fuzzy sets: Part I: continuity of fuzzy directed complete##posets, Fuzzy Sets and Systems, 161 (2010), 973{987.##[27] W. Yao and F. Shi, Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topology on##fuzzy directedcomplete posets, Fuzzy Sets and Systems, 173 (2011), 60{80.##[28] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems,##158 (2007), 349{366.##[29] Q. Zhang, W. Xie and L. Fan, Fuzzy complete lattices, Fuzzy Sets and Systems, 160 (2009),##2275{2291.##]
Correspondence between probabilistic norms and fuzzy norms
2
2
In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and WuFang fuzzy (semi) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a WuFang fuzzy seminorm and conversely, a WuFang fuzzy norm can generate a probabilistic norm.
1

105
114


HuaPeng
Zhang
School of Science, Nanjing University of Posts and Telecommuni
cations, Nanjing 210023, China
School of Science, Nanjing University of
China
huapengzhang@163.com
Probabilistic norm
Fuzzy norm
[[1] C. Alegre and S. Romaguera, Characterizations of metrizable topological vector spaces and##their asymmetric generalizations in terms of fuzzy (quasi)norms, Fuzzy Sets and Systems,##161 (2010), 2181{2192.##[2] C. Alsina, M. J. Frank and B. Schweizer, Associative Functions: Triangular Norms and##Copulas, World Scientic Publishing, Singapore, 2006.##[3] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math.,##11(3) (2003), 687{705.##[4] T. Bag and S. K. Samanta, A comparative study of fuzzy norms on a linear space, Fuzzy##Sets and Systems, 159 (2008), 670{684.##[5] S. C. Cheng and J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces,##Bull. Calcutta Math. Soc., 86 (1994), 429{436.##[6] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48 (1992),##[7] O. Hadzic and E. Pap, Fixed point theory in probabilistic metric spaces, Kluwer Academic##Publishers, Dordrecht, 2001.##[8] U. Hohle, Minkowski functionals of Lfuzzy sets, in: P.P. Wang, S.K. Chang (Eds.), Fuzzy##sets: theory and applications to policy analysis and information systems, Plenum Press, New##York, (1980), 1324.##[9] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984),##[10] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12 (1984), 143{##[11] A. K. Katsaras, Linear fuzzy neighborhood spaces, Fuzzy Sets and Systems, 16 (1985), 25{40.##[12] A. K. Katsaras, Locally convex topologies induced by fuzzy norms, Global Journal of Mathematical##Analysis, 1(3) (2013), 83{96.##[13] E. P. Klement, R. Mesiar and E. Pap, Triangular norms, Kluwer Academic Publishers,##Dordrecht, 2000.##[14] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11##(1975), 336{344.##[15] B. LafuerzaGuillen and P. K. Harikrishnan, Probabilistic normed spaces, World Scientic##Publishing, Singapore, 2014.##[16] M. Ma, A comparison between two denitions of fuzzy normed spaces, J. Harbin Inst. Technology##Suppl. Math., (in Chinese), (1985), 47{49. ##[17] S. Nadaban and I. Dzitac, Atomic decompositions of fuzzy normed linear spaces for wavelet##applications, Informatica, 25(4) (2014), 643{662.##[18] R. Saadati and S. M. Vaezpour, Some results on fuzzy Banach spaces, J. Appl. Math. &##Computing, 17(12) (2005), 475{484.##[19] B. Schweizer and A. Sklar, Probabilistic metric spaces, NorthHolland series in Probability##and Applied Mathematics, NorthHolland, New York, 1983.##[20] C. Sempi, A short and partial history of probabilistic normed spaces, Mediterr. J. Math., 3##(2006), 283{300.##[21] C. X. Wu and J. X. Fang, Fuzzy generalization of Kolmogoro's theorem, J. Harbin Inst.##Technology, (in Chinese), 1 (1984), 1{7.##[22] C. X. Wu and M. Ma, Fuzzy norms, probabilistic norms and fuzzy metrics, Fuzzy Sets and##Systems, 36 (1990), 137{144.##[23] C. H. Yan and J. X. Fang, Generalization of Kolmogoro's theorem to Ltopological vector##spaces, Fuzzy Sets and Systems, 125 (2002), 177{183.##]
$L$fuzzy approximation spaces and $L$fuzzy topological spaces
2
2
The $L$fuzzy approximation operator associated with an $L$fuzzy approximation space $(X,R)$ turns out to be a saturated $L$fuzzy closure (interior) operator on a set $X$ precisely when the relation $R$ is reflexive and transitive. We investigate the relations between $L$fuzzy approximation spaces and $L$(fuzzy) topological spaces.
1

115
129


A. A.
Ramadan
Department of Mathematics, Faculty of Science, BeniSuef Univer
sity, BeniSuef, Egypt
Department of Mathematics, Faculty of Science,
Egypt


E. H.
Elkordy
Department of Mathematics, Faculty of Science, BeniSuef Univer
sity, BeniSuef, Egypt
Department of Mathematics, Faculty of Science,
Egypt


M.
ElDardery
Department of Mathematics, Faculty of Science, Fayoum University,
Fayoum, Egypt
Department of Mathematics, Faculty of Science,
Egypt
Complete residuated lattice
$L$fuzzy approximation spaces
$L$fuzzy topology
continuity
[[1] R. Belohlavek, Fuzzy relational systems: foundations and principles, Kluwer Academic/##Plenum Press, New York (2002).##[2] K. Blount and T. Tsinakis, The structure of residuated lattices, Int. J. Algebra and Computation,##13(4) (2004), 473{461.##[3] D. Boixader, J. Jacas and J. Recasens, Upper and lower approximations of fuzzy sets, Int.##Jour. of Gen. Sys., 29 (2000), 555{568.##[4] X. Chen and Q. Li, Construction of rough approximations in fuzzy setting, Fuzzy Sets and##Systems, 158 (2007), 2641{2653.##[5] M. Chuchro, On rough sets in topological Boolean algebra. In: Ziarko, W.(ed.): Rough Sets,##Fuzzy Sets and Knowledge Discovery, SpringerVerlage, New York, (1994), 157{160.##[6] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst., 17(23)##(1990), 191{208.##[7] J. Fang, Ifuzzy Alexadrov topologies and specialization orders, Fuzzy Sets and Systems, 158##(2007), 2359{2374.##[8] P. Hajek, Metamathematics of fuzzy logic, Kluwer, Dordrecht (1998).##[9] U. Hohle and A. P. Sostak, Axiomatic foundations of xedbasis fuzzy topology, In: Hohle,##S. E. Rodabaugh (Eds), Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory,##The Handbooks of Fuzzy Sets Series, Chapter 3, Kluwer Academic Publisher, Dordrechet##(1999), 123{272.##[10] Y. C. Kim and Y. S. Kim, (L,)approximation spaces and (L,)fuzzy quasiuniform spaces,##Information Sciences, 179 (2009), 2028{2048.##[11] H. Lai and D. Zhang, Fuzzy pre order and fuzzy topology, Fuzzy Sets and Systems, 157##(2006), 1865{1885.##[12] Z. M. Ma and B. Q. Hu, Topological and lattice structures of Lfuzzy rough sets determined##by lower and upper sets, Information Sciences, 218 (2013), 194{204. ##[13] N. N. Morsi and M. M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Sets Systems, 100##(1998), 327342.##[14] Z. Pawlak, Rough sets, Inter. J. Comp. Info. Sci., 161 (2010), 29232944.##[15] K. Qin and Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems,##151 (2005), 601{613.##[16] K. Qin, J. Yang and Z. Pei, Generalized rough sets based on re##exive and transitive, Info.##Sci., 178 (2008), 4138{4141.##[17] A. M. Radzikowoska and E. E. Kerre, Fuzzy rough sets based on residuated lattices, In:##Transaction on Rough sets II, in Lincs, 3135 (2004), 278{296.##[18] A. A. Ramadan, Smooth topological Spaces, Fuzzy Sets and Systems, 48(3) (1992), 371{357.##[19] A. A. Ramadan, Lfuzzy interior systems, Comp. and Math. with Appl., 62 (2011), 4301{##[20] S. E. Rodabaugh and E. P. Kelment, Topological and algebraic structures in fuzzy sets, The##Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Trends in Logic 20,##Kluwer Academic Publisher, Boston (2003).##[21] A. Sostak, On a fuzzy topological structure, Rend. Circ. Mat. Palermo (Supp. Ser.II), 11##(1985), 89{103.##[22] E. Turunen, Mathematics behind fuzzy logic, A Springer Verlag Co., Hiedelberg (1999).##[23] C. Y. Wang and B. Q. Hu, Fuzzy rough sets based on generalized residuated lattices, Information##Sciences, 248 (2013), 31{49.##[24] W. Z. Wu, A study on relationship between fuzzy rough approximation operators and fuzzy##topological spaces, ##c SpringerVerlag, Berlin, Heidelberg (2005).##[25] W. Z. Wu, J. S. Mi and W. X. Zhang, Generalized fuzzy rough sets, Information Sciences,##151 (2003), 263{282.##[26] Y. Y. Yao, Constructive and Algebraic methods of the theory of rough sets, Information##Sciences, 109 (1998), 21{27.##[27] W. X. Zhang, Y. Leung and W. Z.Wu, Information systems and knowledge discovery, Science##Press,Beijing (2003).##[28] P. Zhi, P. Daowu and Z. Li, Topology vs generalized rough sets, Fuzzy Sets and Systems,##52(2) (2011), 231{239.##]
Commutative pseudo BEalgebras
2
2
The aim of this paper is to introduce the notion of commutative pseudo BEalgebras and investigate their properties.We generalize some results proved by A. Walendziak for the case of commutative BEalgebras.We prove that the class of commutative pseudo BEalgebras is equivalent to the class of commutative pseudo BCKalgebras. Based on this result, all results holding for commutative pseudo BCKalgebras also hold for commutative pseudo BEalgebras. For example, any finite commutative pseudo BEalgebra is a BEalgebra, and any commutative pseudo BEalgebra is a joinsemilattice. Moreover, if a commutative pseudo BEalgebra is a meetsemilattice, then it is a distributive lattice. We define the pointed pseudoBE algebras, and introduce and study the relative negations on pointed pseudo BEalgebras. Based on the relative negations we construct two closure operators on a pseudo BEalgebra.We also define relative involutive pseudo BEalgebras, we investigate their properties and prove equivalent conditions for a relative involutive pseudo BEalgebra.We define the relative Glivenko property for a relative good pseudo BEalgebra and show that any relativeinvolutive pseudo BEalgebra has the relative Glivenko property.
1

131
144


L. C.
Ciungu
Department of Mathematics, University of Iowa, 14 MacLean Hall,
Iowa City, Iowa 522421419, Usa
Department of Mathematics, University of
United States
lcciungu@yahoo.com
Pseudo BEalgebra
Pseudo BCKalgebra
Commutative pseudo BCKalgebra
Commutative pseudo BEalgebra
Pointed pseudo BEalgebra
Relative involutive pseudo BEalgebra
Relative Glivenko property
[[1] S. S. Ahn and K. S. So, On ideals and upper sets in BEalgebras, Scientiae Mathematicae##Japonicae, 68(2) (2008), 279285. ##[2] S. S. Ahn, Y. H. Kim and J. M. Ko, Filters in commutative BEalgebras, Communications##of the Korean Mathematical Society, 27(2) (2012), 233242.##[3] A. Borumand Saeid, Smarandache BEalgebras, Education Publisher, Columbus, Ohio, USA,##[4] R. A. Borzooei, A. Borumand Saeid, A. Rezaei, A. Radfar and R. Ameri, On pseudo BE##algebras, Discussiones Mathematicae General Algebra and Applicationes, 33(1) (2013), 95##[5] R. A. Borzooei, A. Borumand Saeid and R. Ameri, States on BEalgebras, Kochi Journal of##Mathematics, 9(1) (2014), 2742.##[6] R. A. Borzooei, A. Borumand Saeid, A. Rezaei, A. Radfar and R. Ameri, Distributive pseudo##BE{algebras, Fasciculi Mathematici, 54(1) (2015), 2139.##[7] R. Cignoli and A. Torrens, Glivenko like theorems in natural expansions of BCKlogic, Mathematical##Logic Quaterly, 50(2) (2004), 111125.##[8] R. Cignoli and A. Torrens, Free Algebras in Varieties of Glivenko MTLalgebras Satisfying##the Equation 2(x2) = (2x)2, Studia Logica, 83(13) (2006), 157181.##[9] Z. Ciloglu and Y. Ceven, Commutative and bounded BEalgebras, Hindawi Publishing Corporation,##2013(1) (2013), Article ID 473714.##[10] L. C. Ciungu and A. Dvurecenskij, Measures, states and de Finetti maps on pseudoBCK##algebras, Fuzzy Sets and Systems, 161(22) (2010), 28702896.##[11] L. C. Ciungu, G. Georgescu and C. Muresan, Generalized Bosbach states: part I, Archive for##Mathematical Logic, 52(34) (2013), 335376.##[12] L. C. Ciungu and J. Kuhr, New probabilistic model for pseudoBCK algebras and pseudo##hoops, Journal of MultipleValued Logic and Soft Computing, 20(34) (2013), 373400.##[13] L. C. Ciungu, Noncommutative multiplevalued logic algebras, Springer, Cham, Heidelberg,##New York, Dordrecht, London, 2014.##[14] L. C. Ciungu, Relative negations in noncommutative fuzzy structures, Soft Computing,##18(1) (2014), 1533.##[15] A. Dvurecenskij and O. Zahiri, Pseudo equality algebras: revision, Soft Computing, doi:##10.1007/s005000151888x, (2015).##[16] G. Georgescu and A. Iorgulescu, Pseudo MValgebras, MultipleValued Logic, 6(12) (2001),##[17] G. Georgescu and A. Iorgulescu, PseudoBCK algebras: An extension of BCKalgebras, Proceedings##of DMTCS'01: Combinatorics, Computability and Logic, Springer, London, (2001),##[18] Y. Imai and K. Iseki, On axiom systems of propositional calculi XIV, Proceedings of the##Japan Academy, 42(1) (1966), 1922.##[19] A. Iorgulescu, Classes of pseudoBCK algebras  Part I, Journal of MultipleValued Logic##and Soft Computing, 12(12) (2006), 71130.##[20] A. Iorgulescu, Algebras of logic as BCKalgebras, ASE Ed., Bucharest, 2008.##[21] H. S. Kim and Y. H. Kim, On BEalgebras, Scientiae Mathematicae Japonicae, 66(1) (2007),##[22] K. H. Kim and Y. H. Yon, Dual BCKalgebra and MValgebra, Scientiae Mathematicae##Japonicae, 66(2) (2007), 247254.##[23] J. Kuhr, PseudoBCK semilattices, Demonstratio Mathematica, 40(3) (2007), 495516.##[24] J. Kuhr, PseudoBCK algebras and related structures, Habilitation thesis, Palacky University##in Olomouc, 2007.##[25] J. Kuhr, Commutative pseudo BCKalgebras, Southeast Asian Bulletin of Mathematics,##33(3) (2009), 451475.##[26] K. J. Lee, Pseudovaluations on BEalgebras, Applied Mathematical Sciences, 7(125) (2013),##61996207.##[27] B. L. Meng, CIalgebras, Scientiae Mathematicae Japonicae, 71(1) (2010), 1117.##[28] B. L. Meng, On lters in BEalgebras, Scientiae Mathematicae Japonicae, 71(2) (2010),##[29] J. Rachunek, A noncommutative generalization of MValgebras, Czechoslovak Mathematical##Journal, 52(127) (2002), 255273.##[30] A. Rezaei and A. Borumand Saeid, On fuzzy subalgebras of BEalgebras, Afrika Matematika,##22(2) (2011), 115127.##[31] A. Rezaei and A. Borumand Saeid, Some results in BEalgebras, Annals of Oradea University## Mathematics Fascicola, XIX(1) (2012), 3344.##[32] A. Rezaei and A. Borumand Saeid, Commutative ideals in BEalgebras, Kyungpook Mathematical##Journal, 52(4) (2012), 483494.##[33] A. Rezaei, A. Borumand Saeid and R. A. Borzooei, Relation between Hilbert algebras and##BEalgebras, Applications and Applied Mathematics, 8(2) (2013), 573584.##[34] A. Rezaei, A. Borumand Saeid, A. Radfar and R. A. Borzooei, Congruence relations on##pseudo BEalgebras, Annnals of the University of Craiova, Mathematics and Computer Sciences##Series, 41(2) (2014), 166176.##[35] A. Rezaei, L. C. Ciungu and A. Borumand Saeid, States on pseudo BEalgebras, submitted.##[36] A. Walendziak, On commutative BEalgebras, Scientiae Mathematicae Japonicae, 69(2)##(2009), 281284.##[37] A. Walendziak, On normal lters and congruence relations in BEalgebras, Commentationes##Mathematicae, 52(2) (2012), 199205.##[38] H. Zhou and B. Zhao, Generalized Bosbach and Riecan states based on relative negations in##residuated lattices, Fuzzy Sets and Systems, 187(1) (2012), 3357.##]
SemiGfilters, Stonean filters, MTLfilters, divisible filters, BLfilters and regular filters in residuated lattices
2
2
At present, the filter theory of $BL$textit{}algebras has been widelystudied, and some important results have been published (see for examplecite{4}, cite{5}, cite{xi}, cite{6}, cite{7}). In other works such ascite{BP}, cite{vii}, cite{xiii}, cite{xvi} a study of a filter theory inthe more general setting of residuated lattices is done, generalizing thatfor $BL$textit{}algebras. Note that filters are also characterized byvarious types of fuzzy sets. Most of such characterizations is trivial butsome are nontrivial, for example characterizations obtained in cite{xm}.Both situation have revealed a rich range of classes of filters: Boolean,implicative, Heyting, positive implicative, fantastic (or MVfilter), etc.In this paper we work in the general cases of residuated lattices and put inevidence new types of filters in a residuated lattice (in the spirit of cite{mvl}): semiGfilterstextit{, }Stonean filters, divisible filters,BLfilters and regular filters.
1

145
160


D.
Busneag
Department of Mathematics, Faculty of Mathematics and Natural Sci
ences, University of Craiova, Craiova, Romania
Department of Mathematics, Faculty of Mathematics
Romania


D.
Piciu
Department of Mathematics, Faculty of Mathematics and Natural Sciences,
University of Craiova, Craiova, Romania
Department of Mathematics, Faculty of Mathematics
Romania
Residuated lattice
Boolean algebra
BLalgebra
MValgebra
MTLalgebra
Divisible residuated lattice
Regular residuated lattice
Deductive system
Filter
Boolean filter
MTLfilter
Divisible filter
BLfilter
MVfilter
SemiGfilter
Stonean filter
Regular filter
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