2016
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Cover for Volume.13, No.4
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Hesitant Fuzzy Linguistic Arithmetic Aggregation Operators in Multiple Attribute Decision Making
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In this paper, we investigate the multiple attribute decision making (MADM) problem based on the arithmetic and geometric aggregation operators with hesitant fuzzy linguistic information. Then, motivated by the idea of traditional arithmetic operation, we have developed some aggregation operators for aggregating hesitant fuzzy linguistic information: hesitant fuzzy linguistic weighted average (HFLWA) operator, hesitant fuzzy linguistic ordered weighted average (HFLOWA) operator and hesitant fuzzy linguistic hybrid average (HFLHA) operator. Furthermore, we propose the concept of the dual hesitant fuzzy linguistic set and develop some aggregation operators with dual hesitant fuzzy linguistic information. Then, we have utilized these operators to develop some approaches to solve the hesitant fuzzy linguistic multiple attribute decision making problems. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.
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1
16


Guiwu
Wei
School of Business, Sichuan Normal University, Chengdu, 610101, P.R.
China; Communications Systems and Networks (CSN) Research Group, Department of
Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah
School of Business, Sichuan Normal University,
Saudi Arabia
weiguiwu@163.com


Fuad E.
Alsaadi
Communications Systems and Networks (CSN) Research Group, Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Communications Systems and Networks (CSN)
Saudi Arabia


Tasawar
Hayat
Department of Mathematics, QuaidIAzam University 45320, Islam
abad 44000, Pakistan; Nonlinear Analysis and Applied Mathematics (NAAM) Research
Group, Department of Mathematics, Faculty of Science, King Abdulaziz University,
Jeddah 21589, Saudi
Department of Mathematics, QuaidIAzam University
Saudi Arabia


Ahmed
Alsaedi
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group,
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Nonlinear Analysis and Applied Mathematics
Saudi Arabia
Multiple attribute decision making (MADM)
Hesitant fuzzy linguistic values
Hesitant fuzzy linguistic hybrid average (HFLHA) operator
Dual hesitant fuzzy linguistic set
[[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87{96.##[2] K. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33 (1989), 37{46.##[3] K. Atanassov, Two theorems for intuitionistic fuzzy sets, Fuzzy Sets and Systems, 110##(2000), 267{269.##[4] S. M. Chen, S. H. Cheng and C. H. Chiou, Fuzzy multiattribute group decision making based##on intuitionistic fuzzy sets and evidential reasoning methodology, Information Fusion, 27##(2016), 215{227.##[5] T. Y. Chen, Bivariate models of optimism and pessimism in multicriteria decisionmaking##based on intuitionistic fuzzy sets, Information Sciences, 181 (2011), 2139{2165.##[6] T. Y. Chen, The inclusionbased TOPSIS method with intervalvalued intuitionistic fuzzy##sets for multiple criteria group decision making, Appl. Soft Comput., 26 (2015), 57{73.##[7] T. Y. Chen, An intervalvalued intuitionistic fuzzy permutation method with likelihood##based preference functions and its application to multiple criteria decision analysis, Appl.##Soft Comput., 42 (2016), 390409.##[8] F. Chiclana, F. Herrera and E. HerreraViedma, The ordered weighted geometric opera##tor: Properties and application, In: Proc of 8th Int Conf on Information Processing and##Management of Uncertainty in Knowledgebased Systems, Madrid, (2000), 985991.##[9] M. Gabroveanu, I. Iancu and M. Cosulschi, An Atanassov's intuitionistic fuzzy reasoning##model, Journal of Intelligent and Fuzzy Systems, 30(1) (2016), 117{128.##[10] F. Herrera and E. HerreraViedma, Linguistic decision analysis: Steps for solving decision##problems under linguistic information, Fuzzy Sets and Systems, 115 (2000), 67{82.##[11] F. Herrera and L. Martnez, A 2tuple fuzzy linguistic representation model for computing##with words, IEEE Transactions on Fuzzy Systems, 8 (2000), 746{752.##[12] J. Y. Huang, Intuitionistic fuzzy hamacher aggregation operators and their application##to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems, 27(1)##(2014), 505{513.##[13] X. P. Jiang and G. W. Wei, Some Bonferroni mean operators with 2tuple linguistic infor##mation and their application to multiple attribute decision making, Journal of Intelligent##and Fuzzy Systems, 27 (2014), 2153{2162.##[14] D. F. Li, TOPSISbased nonlinearprogramming methodology for multiattribute decision##making with intervalvalued intuitionistic fuzzy sets, IEEE Transactions on Fuzzy Systems,##18 (2010), 299{311. ##[15] D. F. Li and H. P. Ren, Multiattribute decision making method considering the amount##and reliability of intuitionistic fuzzy information, Journal of Intelligent and Fuzzy Systems,##28(4) (2015), 1877{1883.##[16] R. Lin, G. W. Wei, H. J. Wang and X. F. Zhao, Choquet integrals of weighted triangular##fuzzy linguistic information and their applications to multiple attribute decision making,##Journal of Business Economics and Management, 15(5) (2014),795{809.##[17] L. Martinez and F. Herrera, An overview on the 2tuple linguistic model for computing with##words in decision making: Extensions, applications and challenges, Information Sciences,##207(10) (2012), 1{18.##[18] J. M. Merigo, A unied model between the weighted average and the induced OWA operator,##Expert Systems with Applications, 38(9) (2011), 11560{11572.##[19] J. M. Merigo and M. Casanovas, Fuzzy generalized hybrid aggregation operators and its##application in decision making, International Journal of Fuzzy Systems, 12(1) (2010),##[20] J. M. Merigo and M. Casanovas, The fuzzy generalized OWA operator and its application##in strategic decision making, Cybernetics & Systems, 41(5) (2010), 359{370.##[21] J. M. Merigo, M. Casanovas and L. Martinez, 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), 287{304.##[22] J. M. Merigo and A. M. GilLafuente, Decision making techniques in business and econom##ics based on the OWA operator, SORTStatistics and Operations Research Transactions,##36(1) (2012), 81{102.##[23] J. M. Merigo and A. M. GilLafuente, Induced 2tuple linguistic generalized aggregation##operators and their application in decisionmaking, Information Sciences, 236(1) (2013),##[24] 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(4) (2012), 531{549.##[25] L. D. Miguel, H. Bustince, J. Fernandez, E. Indurain, A. Kolesarova and R. Mesiar,##Construction of admissible linear orders for intervalvalued Atanassov intuitionistic fuzzy##sets with an application to decision making, Information Fusion, 27 (2016), 189{197.##[26] V. Torra, The weighted OWA operator, International Journal of Intelligent Systems, 12##(1997), 153{166.##[27] V. Torra, Hesitant fuzzy sets, International Journal of Intelligent Systems, 25 (2010),##[28] V. Torra and Y. Narukawa, On hesitant fuzzy sets and decision, In: The 18th IEEE##International Conference on Fuzzy Systems, Jeju Island, Korea, (2009), 1378{1382.##[29] H. J. Wang, X. F Zhao and G. W. Wei, Dual hesitant fuzzy aggregation operators in##multiple attribute decision making, Journal of Intelligent and Fuzzy Systems, 26(5) (2014),##2281{2290.##[30] X. R. Wang, Z. H. Gao, X. F. Zhao and G. W. Wei, Model for Evaluating the Government##Archives Websites Construction Based on the GHFHWD Measure with Hesitant Fuzzy##Information, International Journal of Digital Content Technology and its Applications,##5(12) (2011), 418{425.##[31] G. W. Wei, Some geometric aggregation functions and their application to dynamic mul##tiple attribute decision making in intuitionistic fuzzy setting, International Journal of Un##certainty, Fuzziness and KnowledgeBased Systems, 17(2) (2009), 179{196.##[32] G. W. Wei, Some induced geometric aggregation operators with intuitionistic fuzzy infor##mation and their application to group decision making, Applied Soft Computing, 10(2)##(2010), 423{431.##[33] G. W. Wei, Gray relational analysis method for intuitionistic fuzzy multiple attribute##decision making, Expert Systems with Applications, 38 (2011), 11671{11677. ##[34] G. W. Wei, Some harmonic averaging operators with 2tuple linguistic assessment infor##mation and their application to multiple attribute group decision making, International##Journal of Uncertainty, Fuzziness and Knowledge Based Systems, 19(6) (2011), 977{998.##[35] G. W. Wei, Hesitant fuzzy prioritized operators and their application to multiple attribute##group decision making, KnowledgeBased Systems, 31 (2012), 176{182.##[36] G. W. Wei, Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute deci##sion making with incomplete weight information, International Journal of Fuzzy Systems,##17(3) (2015), 484{489.##[37] G. W. Wei, R. Lin, X. F. Zhao and H. J. Wang, An approach to multiple attribute deci##sion making based on the induced Choquet integral with fuzzy number intuitionistic fuzzy##information, Journal of Business Economics and Management, 15(2) (2014), 277{298.##[38] G. W. Wei and N. Zhang, A multiple criteria hesitant fuzzy decision making with Shapley##valuebased VIKOR method, Journal of Intelligent and Fuzzy Systems, 26(2) (2014), 1065{##[39] G. W. Wei and X. F. Zhao, Some induced correlated aggregating operators with intuition##istic fuzzy information and their application to multiple attribute group decision making,##Expert Systems with Applications, 39(2) (2012), 2026{2034.##[40] 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), 5881{5886.##[41] G. W. Wei and X. F. Zhao, Induced hesitant intervalvalued fuzzy einstein aggregation##operators and their application to multiple attribute decision making, Journal of Intelligent##and Fuzzy Systems, 24 (2013), 789{803.##[42] G. W. Wei, X. F. Zhao, H. J. Wang and R. Lin, Hesitant fuzzy choquet integral aggregation##operators and their applications to multiple attribute decision making, Information: An##International Interdisciplinary Journal, 15(2) (2012), 441{448.##[43] G. W. Wei, H. J. Wang, X. F. Zhao and R. Lin, Hesitant triangular fuzzy information ag##gregation in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems,##26(3) (2014), 1201{1209.##[44] G. W. Wei, H. J. Wang, X. F. Zhao and R. Lin, Approaches to hesitant fuzzy multiple##attribute decision making with incomplete weight information, Journal of Intelligent and##Fuzzy Systems, 26(1) (2014), 259{266.##[45] M. Xia and Z. S. Xu, Hesitant fuzzy information aggregation in decision making, Interna##tional Journal of Approximate Reasoning, 52(3) (2011), 395{407.##[46] Z. S. Xu, A method based on linguistic aggregation operators for group decision making##with linguistic preference relations, Information Sciences, 166(1) (2004), 19{30.##[47] Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transations on Fuzzy Systems,##15(6) (2007), 1179{1187.##[48] Z. S. Xu and Q. L. Da, An overview of operators for aggregating information, International##Journal of Intelligent System, 18 (2003), 953{969.##[49] Z. S. Xu and M. Xia, Distance and similarity measures for hesitant fuzzy sets, Information##Sciences, 181(11) (2011), 2128{2138.##[50] Z. S. Xu and M. Xia, On distance and correlation measures of hesitant fuzzy information,##International Journal of Intelligence Systems, 26(5) (2011), 410{425.##[51] Z. S. Xu, M. Xia and N. Chen, Some hesitant fuzzy aggregation operators with their##application in group decision making, Group Decision and Negotiation, 22(2) (2013),##[52] Z. S. Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic##fuzzy sets, International Journal of General System, 35 (2006), 417{433.##[53] R. R. Yager, Including importances in OWA aggregation using fuzzy systems modelling,##IEEE Transactions on Fuzzy Systems, 6 (1998), 286{294.##[54] R. R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision##making, IEEE Transactions on Systems Man and Cybernetics, 18 (1988), 183{190. ##[55] R. R. Yager, K. J. Engemann and D. P. Filev, On the concept of immediate probabilities,##International Journal of Intelligent Systems, 10 (1995), 373{397.##[56] R. R. Yager and D. P. Filev, Induced ordered weighted averaging operators, IEEE Trans##actions on Systems, Man, and Cybernetics Part B, 29 (1999), 141{150.##[57] R. R. Yager, Prioritized aggregation operators, International Journal of Approximate Rea##soning, 48 (2008), 263{274.##[58] O. Y. Yao and W. Pedrycz, A new model for intuitionistic fuzzy multiattributes decision##making, European Journal of Operational Research, 249(2) (2016), 677{682.##[59] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{356.##[60] X. F. Zhao, Q. X. Li, G. W. Wei, Some prioritized aggregating operators with linguistic##information and their application to multiple attribute group decision making, Journal of##Intelligent and Fuzzy Systems, 26(4) (2014), 1619{1630.##[61] L. Y. Zhou, R. Lin, X. F. Zhao and G. W. Wei, Uncertain linguistic prioritized aggregation##operators and their application to multiple attribute group decision making, International##Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, 21(4) (2013), 603{627.##[62] B. Zhu, Z. S. Xu and M. M. Xia, Hesitant fuzzy geometric Bonferroni means, Information##Sciences, Information Sciences, 205(1) (2012), 72{85.##]
A satisfactory strategy of multiobjective two person matrix games with fuzzy payoffs
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The multiobjective two person matrix game problem with fuzzy payoffs is considered in this paper. It is assumed that fuzzy payoffs are triangular fuzzy numbers. The problem is converted to several multiobjective matrix game problems with interval payoffs by using the $alpha$cuts of fuzzy payoffs. By solving these problems some $alpha$Pareto optimal strategies with some interval outcomes are obtained. An interactive algorithm is presented to obtain a satisfactory strategy of players. Validity and applicability of the method is illustrated by a practical example.
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33


Hamid
Bigdeli
Department of Mathematics, University of Birjand, Birjand, I.R. Iran
Department of Mathematics, University of
Iran


Hassan
Hassanpour
Department of Mathematics, University of Birjand, Birjand,
I.R. Iran
Department of Mathematics, University of
Iran
hhassanpour@birjand.ac.ir
Fuzzy multiobjective game
Interval multiobjective programming
Satisfactory strategy
Security level
[[1] C. R. Bector and S. Chandra, Fuzzy mathematical programming and fuzzy matrix games,##Springer Verlag, Berlin, 2005.##[2] S. M. Belenson and K. C. Kapur, An algorithm for solving multicriterion linear programming##problems with examples, Operational Research Quarterly, 24(1) (1973), 65{77.##[3] A. Billot, Economic theory of fuzzy equilibria, SpringerVerlag, 1992.##[4] J. J. Buckley, Multiple goal noncooperative con##icts under uncertainty: a fuzzy set approach,##Fuzzy Sets and Systems, 13(2) (1984), 107{124.##[5] D. Butnariu, Fuzzy games; a description of the concept, Fuzzy Sets and Systems, 1(3) (1978),##[6] D. Butnariu, Stability and Shapley value for an npersons fuzzy game, Fuzzy Sets and Systems,##4(1) (1980), 63{72.##[7] L. Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and##Systems, 32(3) (1989), 275{289.##[8] L. Chandra and A. Aggarwal, On solving matrix games with payos of triangular fuzzy num##bers: Certain observations and generalizations, European Journal of Operational Research,##246 (2015), 575{581.##[9] M. Clemente and F. R. Fernandez, Paretooptimal security strategies in matrix games with##fuzzy payos, Fuzzy Sets and Systems, 176(1) (2011), 36{45.##[10] W. D. Collins and C. Y. Hu, Studying interval valued matrix games with fuzzy logic, Soft##Computing, 12(2) (2008), 147{155.##[11] W. D. Cook, Zerosum games with multiple goals, Naval Research Logistics Quarterly, 23(4)##(1976), 615{622.##[12] L. Cunlin and Z. Qiang, Nash equilibrium strategy for fuzzy noncooperative games, Fuzzy##Sets and Systems, 176(1) (2011), 46{55.##[13] B. Dutta and S. K. Gupta, On Nash equilibrium strategy of two person zero sum games with##trapezoidal fuzzy payos, Fuzzy Information and Engineering, 6(3) (2014), 299{314.##[14] K. Fahem and M. S. Radjef, Properly ecient nash equilibrium in multicriteria noncooper##ative games, Mathematical Methods of Operations Research, 82(2) (2015), 175{193.##[15] F. R. Fernandez and J. Puerto, Vector linear programming in zerosum multicriteria matrix##games, Journal of Optimization Theory and Application, 89(1) (1996), 115{127.##[16] D. Fudenberg and J. Tirole, Game theory, The MIT Press, 1991.##[17] C. L. Hwang and K. Yoon, Multiple attribute decision making: methods and applications, a##state of the art survey, SpringerVerlag, Berlin, 1981.##[18] S. Kumar, Maxmin solution approach for multiobjective matrix game with fuzzy goals, Yugoslav##Journal of Operations Research, 2015.##[19] M. Larbani, Non cooperative fuzzy games in normal form: a survey, Fuzzy Sets and Systems,##160(22) (2009), 3184{3210.##[20] D. F. Li, A fuzzy multiobjective programming approach to solve fuzzy matrix games, The##Journal of Fuzzy Mathematics, 7(4) (1999), 907{912.##[21] D. F. Li, Lexicographic method for matrix games with payos of triangular fuzzy numbers,##International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, 16(3) (2008),##[22] D. F. Li, A fast approach to compute fuzzy values of matrix games with payos of triangular##fuzzy numbers, European Journal of Operational Research, 223(2) (2012), 421{429.##[23] D. F. Li and J. X. Nan, An intervalvalued programming approach to matrix games with##payos of triangular intuitionistic fuzzy numbers, Iranian Journal of Fuzzy Systems, 11(2)##(2014), 45{57.##[24] T. Maeda, On characterization of equilibrium strategy of twoperson zero sum games with##fuzzy payos, Fuzzy Sets and Systems, 139(2) (2003), 283{296.##[25] P. K. Nayak and M. Pal, Linear programming technique to solve twoperson matrix games##with interval payos, AsiaPacic Journal of Oprational Research, 26(2) (2009), 285{305. ##[26] J. V. Neumann and O. Morgenstern, Theory of games and economic behavior, Wiley, New##York, 1944.##[27] I. Nishizaki and M. Sakawa, Fuzzy and multiobjective games for con##ict resolution, Springer##Verlag, Berlin, 2001.##[28] G . Owen, Game theory, Academic Press, San Diego, Third Edition, 1995.##[29] M. Sakawa, Fuzzy sets and interactive multiobjective optimization, Plenum press, New York##and london, 1993.##[30] M. R. Seikh, P. K. Nayak and M. Pal, An alternative approach for solving fuzzy matrix##games, International Journal of Mathematics and Soft Computing, 5(1) (2015), 79{92.##[31] M. R. Seikh, P. K. Nayak and M. Pal, Matrix games with intuitionistic fuzzy payos, Journal##of Information & Optimization Sciences, 36(12) (2015), 159{181.##[32] M. R. Seikh, P. K. Nayak and M. Pal, Application of intuitionistic fuzzy mathematical pro##gramming with exponential membership and quadratic nonmembership functions in matrix##games, Annals of Fuzzy Mathematics and Informatics, 9(2) (2015), 183{195.##[33] M. Zeleny, Games with multiple payos, International Journal of Game Theory, 4(4) (1975),##[34] X. Zhou, Y. Song, Q. Zhang and X. Gao, Multiobjective matrix game with vague payos,##Fuzzy Information and Engineering, 40 (2007), 543{550.##]
Bisimulation for BLgeneral fuzzy automata
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In this note, we define bisimulation for BLgeneral fuzzy automata and show that if there is a bisimulation between two BLgeneral fuzzy automata, then they have the same behavior.For a given BLgeneral fuzzy automata, we obtain the greatest bisimulation for the BLgeneral fuzzy automata. Thereafter, if we use the greatest bisimulation, then we obtain a quotient BLgeneral fuzzy automata and this quotient is minimal, furthermore there is a morphism from the first one to its quotient.Also, for two given BLgeneral fuzzy automata we present an algorithm, which determines bisimulation between them.Finally, we present some examples to clarify these new notions.
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M.
Shamsizadeh
Department of Mathematics, Graduate University of Advanced
Technology, Kerman, Iran
Department of Mathematics, Graduate University
Iran


M. M.
Zahedi
Department of Mathematics, Graduate University of Advanced Technology, Kerman, Iran
Department of Mathematics, Graduate University
Iran
zahedi_mm@ mail.uk.ac.ir


K.
Abolpour
Department of Mathematics, Kazerun Branch, Islamic Azad University,
Kazerun, Iran
Department of Mathematics, Kazerun Branch,
Iran
BLgeneral fuzzy automata
Bisimulation
Reduction
General fuzzy automata
Quotient automata
[[1] K. Abolpour and M. M. Zahedi, BLgeneral fuzzy automata and accept behavior, Journal##Applied Mathematics and Computing, 38 (2012), 103118.##[2] K. Abolpour and M. M. Zahedi, Isomorphism between two BLgeneral fuzzy automata, Soft##Computing, 16 (2012), 729736.##[3] C. Baier, B. Engelen and M. Majster Cederbaum, Deciding bisimilarity and similarity for##probabilistic processes, Journal of Computer and System Sciences, 60 (2000), 187231.##[4] P. Buchholz, Bisimulation relations for weighted automata, Theoretical Computer Science,##393 (2008), 109123.##[5] Y. Cao, G. Chen and E. Kerre, Bisimulations for fuzzy transition systems, IEEE Transactions##on Fuzzy Systems, 19 (2011), 540552.##[6] Y. Cao, H. Wang, S. X. Sun and G. Chen, A behavioral distance for fuzzytransition systems,##IEEE Transactions on Fuzzy Systems, 21 (2012), 735747.##[7] M. Ciric, J. Ignjatovic, M. Basic and I. Jancic, Nondeterministic automata: equivalence,##bisimulations, and uniform relations, Information Sciences, 261 (2013), 185218.##[8] M. Ciric, J. Ignjatovic, N. Damljanovic and M. Basic, Bisimulations for fuzzy automata,##Fuzzy Sets and Systems, 186 (2012) 100139.##[9] M. Ciric, J. Ignjatovic, I. Jancic and N. Damljanovic, Computation of the greatest simulations##and bisimulations between fuzzy automata, Fuzzy Sets and Systems, 208 (2012), 2242.##[10] N. Damljanovic, M. Ciric and J. Ignjatovic, Bisimulations for weighted automata over an##additively idempotent semiring, Theoretical Computer Science, 534 (2014), 86100.##[11] W. Deng and D. W. Qiu, Supervisory control of fuzzy discrete event systems for simulation##equivalence, IEEE Transactions on Fuzzy Systems, 23 (2015), 178192.##[12] M. Doostfatemeh and S. C. Kremer, New directions in fuzzy automata, International Journal##of Approximate Reasoning, 38 (2005), 175214.##[13] C. L. Giles, C. W. Omlin and K. K. Thornber, Equivalence in knowledge representation:##automata, recurrent neural networks, and dynamical fuzzy systems, Proceedings of IEEE, 87##(1999), 16231640.##[14] M. M. Gupta, G. N. Saridis and B. R. Gaines, Fuzzy Automata and Decision Processes,##North Holland, New York, (1977), 111175.##[15] P. Hajek, Metamathematics of fuzzy logic, Trends in Logic, Kluwer, Dordercht, 4 (1998).##[16] J. Hgberg, A. Maletti and J. May, Backward and forward bisimulation minimisation of tree##automata, In: J. Holub, J. drek (Eds.), IAA07, in: Lecture Notes in Computer Science, 4783##(2007), 109121. ##[17] J. Hgberg, A. Maletti and J. May, Backward and forward bisimulation minimisation of tree##automata, Theoretical Computer Science, 410 (2009) 35393552.##[18] D. C. Kozen, Automata and computability, Springer, USA, 1997.##[19] E. T. Lee and L. A. Zadeh, Note on fuzzy languages, Information Sciences, 1 (1969), 421434.##[20] L. Li and D. Qiu, On the state minimization of fuzzy automata, IEEE Transactions on Fuzzy##Systems, 23 (2015), 434443.##[21] N. Lynch and F. Vaandrager, Forward and backward simulations, Information and Computation,##121 (1995), 214233.##[22] D. S. Malik and J. N. Mordeson, Fuzzy Automata and Languages: Theory and Applications,##Chapman Hall, CRC Boca Raton, London, New York, Washington DC, 2002.##[23] D. S. Malik and J. N. Mordeson, Fuzzy discrete structures, PhysicaVerlag, New York, (2000),##London, 2002.##[24] R. Milner, Acalculus of communicating systems, In: G. Goos, J. Hartmanis (Eds.), Lecture##Notes in Computer Science, Springer, 92 (1980).##[25] C. W. Omlin, K. K. Thornber and C. L. Giles, Fuzzy nitestate automata can be deter##ministically encoded in recurrent neural networks, IEEE Transactions on Fuzzy Systems, 5##(1998), 7689.##[26] D. Park, Concurrency and automata on innite sequences, In: P.Deussen(Ed.), Proceedings##of the 5th GI Conference, Karlsruhe, Germany, Lecture Notesin Computer Science, Springer##Verlag, 104, (1981), 167183.##[27] W. Pedrycz and A. Gacek, Learning of fuzzy automata, International Journal of Computational##Intelligence and Applications, 1 (2001), 1933.##[28] K. Peeva, Behavior, reduction and minimization of nite Lautomata, Fuzzy Sets and Systems,##28 (1988), 171181.##[29] K. Peeva, Equivalence, reduction and minimization of nite automata over semirings, Theoretical##Computer Science, 88 (1991), 269285.##[30] D. Qiu, Automata theory based on complete residuated latticevalued logic, Science in China##Series: Information Sciences, 44 (2001), 419429.##[31] D. Qiu, Automata theory based on complete residuated latticevalued logic (II), Science in##China Series F: Information Sciences, 45 (2002), 442452.##[32] D. Qiu, Characterizations of fuzzy nite automata, Fuzzy Sets and Systems, 141 (2004),##[33] D. Qiu, Pumping lemma in automata theory based on complete residuated latticevalued logic:##A not, Fuzzy Sets and Systems, 157 (2006), 21282138.##[34] D. Qiu, Supervisory control of fuzzy discrete event systems: a formal approach, IEEE Transactions##on Systems, Man and CyberneticsPart B, 35 (2005), 7288.##[35] E. S. Santos, Maxmin automata, Information Control, 13 (1968), 363377.##[36] V. Topencharov and K. Peeva, Equivalence, reduction and minimization of nite fuzzy au##tomata, Journal of Mathematical Analysis and Applications, 84 (1981), 270281.##[37] E. Turunen, Boolean deductive systems of BLalgebras, Archive for Mathematical Logic, 40##(2001), 467473.##[38] W. G. Wee, On generalization of adaptive algorithm and application of the fuzzy sets concept##to pattern classication, Ph.D. Thesis, Purdue University, Lafayette, IN, 1967.##[39] W. G. Wee and K. S. Fu, A formulation of fuzzy automata and its application as a model of##learning systems, IEEE Transactions on Systems, Man and Cybernetics, 5 (1969), 215223.##[40] L. Wu and D. Qiu, Automata theory based on complete residuated latticevalued logic: Re##duction and minimization, Fuzzy Sets and Systems, 161 (2010), 16351656.##[41] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965,) 338353.##]
Characterizations of $L$convex spaces
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In this paper, the concepts of $L$concave structures, concave $L$interior operators and concave $L$neighborhood systems are introduced. It is shown that the category of $L$concave spaces and the category of concave $L$interior spaces are isomorphic, and they are both isomorphic to the category of concave $L$neighborhood systems whenever $L$ is a completely distributive lattice. Also, it is proved that these categories are all isomorphic to the category of $L$convex spaces whenever $L$ is a completely distributive lattice with an orderreversing involution operator.
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51
61


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
$L$convex structure
$L$concave structure
Convex $L$closure operator
Concave $L$interior operator
Concave $L$neighborhood system
[[1] S. Abramsky and A. Jung, Domain theory, S. Abramsky, D. Gabbay, T.S.E. Mailbaum (Eds.),##Handbook of Logic in Computer Science, Oxford University Press, Oxford (1994), 1{168.##[2] V. Chepoi, Separation of two convex sets in convexity structures, J. Geom., 50 (1994), 30{51.##[3] E. Ellis, A general setseparation theorem, Duke Math. J., 19 (1952), 417{421.##[4] J. Eckho, Radon's theorem in convex product structures I, Monatsh. Math., 72 (1968),##[5] J. Eckho, Radon's theorem in convex product structures II, Monatsh. Math., 73 (1969),##[6] R. E. Jamison, A general theory of convexity, Dissertation, University ofWashington, Seattle,##Washington, 1974.##[7] D. C. Kay and E. W. Womble, Axiomatic convexity theory and the relationship between the##Caratheodory, Helly and Radon numbers, Pacic J. Math., 38 (1971), 471{485.##[8] M. Lassak, On metric Bconvexity for which diameters of any set and its hull are equal, Bull.##Acad. Polon. Sci., 25 (1977), 969{975. ##[9] F. W. Levi, On Helly's theorem and the axioms of convexity, J. Indian Math. Soc., 15 Part##A (1951), 65{76.##[10] Y. Maruyama, Latticevalued fuzzy convex geometry, RIMS Kokyuroku, 164 (2009), 22{37.##[11] K. Menger, Untersuchungen uber allgemeine Metrik, Math. Ann., 100(1928), 75{163.##[12] B. Pang and F. G. Shi, Subcategories of the category of Lconvex spaces, Fuzzy Sets Syst.,##(2016), http://dx.doi.org/10.1016/j.fss.2016.02.014.##[13] B. Pang and F. G. Shi, Lhull operators and Linterval operators in Lconvex spaces, Sub##[14] M. V. Rosa, On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets##Syst., 62 (1994), 97{100.##[15] F. G. Shi and Z. Y. Xiu, A new approach to the fuzzication of convex structures, J. Appl.##Math., 2014 (2014), 12 pages.##[16] G. Sierkama, Caratheodory and Hellynumbers of convexproductstructures, Pacic J. Math.,##61 (1975), 272{282.##[17] G. Sierkama, Relationships between Caratheodory, Helly, Radon and Exchange numbers of##convex spaces, Nieuw Archief Wisk, 25 (1977), 115{132.##[18] V. P. Soltan, Some questions in the abstract theory of convexity, Soviet Math. Dokl., 17##(1976), 730{733.##[19] V. P. Soltan, Dconvexity in graphs, Soviet Math. Dokl., 28 (1983), 419{421.##[20] V. P. Soltan, Introduction to the axiomatic theory of convexity, (Russian) Shtiinca, Kishinev##[21] M. Van De Vel, Finite dimensional convex structures II: the invariants, Topology Appl., 16##(1983), 81{105.##[22] M. Van De Vel, Binary convexities and distributive lattices, Proc. London Math. Soc., 48##(1984), 1{33.##[23] M. Van De Vel, Theory of convex structures, NorthHolland, Amsterdam 1993.##[24] J. C. Varlet, Remarks on distributive lattices, Bull. Acad. Polon. Sci., 23 (1975), 1143{1147.##]
Multiple Fuzzy Regression Model for Fuzzy InputOutput Data
2
2
A novel approach to the problem of regression modeling for fuzzy inputoutput data is introduced.In order to estimate the parameters of the model, a distance on the space of intervalvalued quantities is employed.By minimizing the sum of squared errors, a class of regression models is derived based on the intervalvalued data obtained from the $alpha$level sets of fuzzy inputoutput data.Then, by integrating the obtained parameters of the intervalvalued regression models, the optimal values of parameters for the main fuzzy regression model are estimated.Numerical examples and comparison studies are given to clarify the proposed procedure, and to show the performance of the proposed procedure with respect to some common methods.
1

63
78


Jalal
Chachi
Department of Mathematics, Statistics and Computer Sciences, Sem
nan University, Semnan, Semnan 35195363, Iran
Department of Mathematics, Statistics and
Iran
taheri.chachi@gmail.com


S. Mahmoud
Taheri
Faculty of Engineering Science, College of Engineering, University of Tehran, Tehran, P.O. Box 113654563, Iran
Faculty of Engineering Science, College of
Iran
taher@cc.iut.ac.ir;sm_taheri@ut.ac.ir
Fuzzy regression
Intervalvalued regression
Least squares method
$LR$Fuzzy number
Multiple regression
Predictive ability
[[1] A. R. Arabpour and M. Tata, Estimating the parameters of a fuzzy linear regression model,##Iranian Journal of Fuzzy Systms, 5(2) (2008), 119.##[2] M. Are and S. M. Taheri, Leastsquares regression based on Atanassov's intuitionistic fuzzy##inputsoutputs and Atanassov's intuitionistic fuzzy parameters, IEEE Trans. on Fuzzy Syst.,##23 (2015), 11421154.##[3] A. Bargiela, W. Pedrycz and T. Nakashima, Multiple regression with fuzzy data, Fuzzy Sets##Syst., 158 (2007), 21692188.##[4] A. Bisserier, R. Boukezzoula and S. Galichet, A revisited approach to linear fuzzy regression##using trapezoidal fuzzy intervals, Inf. Sci., 180 (2010), 36533673.##[5] J. Chachi and M. Roozbeh, A fuzzy robust regression approach applied to bed##load transport data, Communications in StatisticsSimulation and Computation, DOI:##10.1080/03610918.2015.1010002, 2015.##[6] J. Chachi and S. M. Taheri, A leastabsolutes approach to multiple fuzzy regression, in: Proc.##58th ISI Congress, Dublin, Ireland, CPS07701, 2011.##[7] J. Chachi and S. M. Taheri, A leastabsolutes regression model for imprecise response based##on the generalized Hausdormetric, J. Uncertain Syst., 7 (2013), 265276.##[8] J. Chachi, S. M. Taheri and N. R. Arghami, A hybrid fuzzy regression model and its appli##cation in hydrology engineering, Applied Soft Comput., 25 (2014), 149{158.##[9] J. Chachi, S. M. Taheri and H. Rezaei Pazhand, Suspended load estimation using L1Fuzzy##regression, L2Fuzzy regression and MARSFuzzy regression models, Hydrological Sciences##J., 61(8) (2016), 14891502.##[10] J. Chachi, S. M. Taheri and R. H. Rezaei Pazhand, An intervalbased approach to fuzzy##regression for fuzzy inputoutput data, in: Proc. IEEE Int. Conf. Fuzzy Syst., Taipei, Taiwan,##(2011), 28592863.##[11] S. P. Chen and J. F. Dang, A variable spread fuzzy linear regression model with higher##explanatory power and forecasting accuracy, Inf. Sci., 178 (2008), 39733988.##[12] R. Coppi, P. D'Urso, P. Giordani and A. Santoro, Least squares estimation of a linear re##gression model with LR fuzzy response, Comp. Stat. Data Anal., 51 (2006), 267286.##[13] P. D'Urso, Linear regression analysis for fuzzy/crisp input and fuzzy/crisp output data,##Comp. Stat. Data Anal., 42 (2003), 4772.##[14] P. D'Urso and Gastaldi T., An orderwise polynomial regression procedure for fuzzy data,##Fuzzy Set Syst., 130 (2002), 119.##[15] P. D'Urso, R. Massari and A. Santoro, A class of fuzzy clusterwise regression models, Inf.##Sci., 180 (2010), 47374762.##[16] P. D'Urso, R. Massari and A. Santoro, Robust fuzzy regression analysis, Inf. Sci., 181 (2011),##41544174.##[17] P. D'Urso and A. Santoro, Fuzzy clusterwise regression analysis with symmetrical fuzzy output##variable, Comp. Stat. Data Anal., 51 (2006), 287313. ##[18] M. B. Ferraro, R. Coppi, G. Gonzalez Rodrguez and A. Colubi, A linear regression model##for imprecise response, Int. J. Approx. Reason., 51 (2010), 759770.##[19] H. Hassanpour, H. R. Maleki and M. A. Yaghoobi, Fuzzy linear regression model with crisp##coecients: A programming approach, Iranian J. Fuzzy Syst., 7 (2010), 1939.##[20] H. Hassanpour, H. R. Maleki and M. A. Yaghoobi, A goal programming approach to fuzzy##linear regression with fuzzy inputoutput data, Soft Comput., 15 (2011), 15691580.##[21] Y. C. Hu, Functionallink nets with geneticalgorithmbased learning for robust nonlinear##interval regression analysis, Neurocomputin, 72 (2009), 18081816.##[22] C. Kao and C. L. Chyu, A fuzzy linear regression model with better explanatory power, Fuzzy##Sets Syst., 126 (2002), 401409.##[23] C. Kao and C. L. Chyu, Leastsquares estimates in fuzzy regression analysis, European J.##Oper. Res., 148 (2003), 426435.##[24] M. Kelkinnama and S. M. Taheri, Fuzzy leastabsolutes regression using shape preserving##operations, Inf. Sci., 214 (2012), 105120.##[25] B. Kim and R. R. Bishu, Evaluation of fuzzy linear regression models by comparison mem##bership function, Fuzzy Sets Syst., 100 (1998), 343352.##[26] K. S. Kula and A. Apaydin, Fuzzy robust regression analysis based on the ranking of fuzzy##sets, Int. J. Uncertain., Fuzziness KnowledgeBased Syst., 16 (2008), 663681.##[27] J. Lu and R. Wang, An enhanced fuzzy linear regression model with more ##exible spreads,##Fuzzy Sets Syst., 160 (2009), 25052523.##[28] M. H. Mashinchi, M. A. Orgun, M. Mashinchi and W. Pedrycz, A tabuharmony searchbased##approach to fuzzy linear regression, IEEE Trans. Fuzzy Syst., 19 (2011), 432448.##[29] MATLAB, The Language of Technical Computing, The MathWorks Inc., MA, 2009.##[30] M. Modarres, E. Nasrabadi and M. M. Nasrabadi, Fuzzy linear regression analysis from the##point of view risk, Int. J. Uncertain., Fuzziness KnowledgeBased Syst., 12 (2004), 635649.##[31] M. Modarres, E. Nasrabadi and M. M. Nasrabadi, Fuzzy linear regression with least squares##errors, Appl. Math. Comput., 163 (2005), 977989.##[32] R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for##Industrial and Applied Mathematics, Philadelphia, PA, 2009.##[33] M. Namdari, J. H. Yoon, A. Abadi, S. M. Taheri and S. H. Choi, Fuzzy logistic regression##with least absolute deviations estimators, Soft Comput., 19 (2015), 909917.##[34] E. Nasrabadi and S. M. Hashemi, Robust fuzzy regression analysis using neural networks,##Int. J. Uncertain., Fuzziness KnowledgeBased Syst., 16 (2008), 579598.##[35] E. Nasrabadi, S. M. Hashemi and M. Ghatee, An LPbased approach to outliers detection##in fuzzy regression analysis, Int. J. Uncertain., Fuzziness KnowledgeBased Syst., 15 (2007),##[36] M. M. Nasrabadi and E. Nasrabadi, A mathematicalprogramming approach to fuzzy linear##regression analysis, Appl. Math. Comput., 155 (2004), 873881.##[37] M. M. Nasrabadi, E. Nasrabadi and A. R. Nasrabadi, Fuzzy linear regression analysis: a##multiobjective programming approach, Appl. Math. Comput., 163 (2005), 245251.##[38] S. Pourahmad, S. M. T. Ayatollahi and S. M. Taheri, Fuzzy logistic regression: A new##possibilistic model and its application in clinical vague status, Iranian J. Fuzzy Syst., 8##(2011), 117.##[39] S. Pourahmad, S. M. T. Ayatollahi, S. M. Taheri and Z. Habib Agahi, Fuzzy logistic regression##based on the least squares approach with application in clinical studies, Comput. Math. Appl.,##62 (2011), 33533365.##[40] M. R. Rabiei, N. R. Arghami, S. M. Taheri and B. Sadeghpour Gildeh, Leastsquares approach##to regression modeling in full intervalvalued fuzzy environment, Soft Comput., 18 (2014),##20432059.##[41] M. Sakawa and H. Yano, Multiobjective fuzzy linear regression analysis for fuzzy inputoutput##data, Fuzzy Sets Syst., 157 (1992), 173181.##[42] H. Shakouri and R. Nadimi, A novel fuzzy linear regression model based on a nonequality##possibility index and optimum uncertainty, Appl. Soft Comput., 9 (2009), 590598. ##[43] S. M. Taheri and M. Kelkinnama, Fuzzy linear regression based on least absolute deviations,##Irannian Journal of Fuzzy Systems, 9(1) (2012), 121140.##[44] H. Tanaka, I. Hayashi and J. Watada, Possibilistic linear regression analysis for fuzzy data,##European J. Oper. Res., 40 (1989), 389396.##[45] H. Tanaka, S. Vejima and K. Asai, Linear regression analysis with fuzzy model, IEEE Trans.##Syst., Man, Cybernetics, 12 (1982), 903907.##[46] H. J. Zimmermann, Fuzzy set theory and its applications, 4th ed., Kluwer Niho, Boston,##]
On impulsive fuzzy functional differential equations
2
2
In this paper, we prove the existence and uniqueness of solution to the impulsive fuzzy functional differential equations under generalized Hukuhara differentiability via the principle of contraction mappings. Some examples are provided to illustrate the result.
1

79
94


Ho
Vu
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam; Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Division of Computational Mathematics and
Viet Nam
hovumath@gmail.com


Ngo
VanHoa
Division of Computational Mathematics and Engineering, Institute
for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam; Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Division of Computational Mathematics and
Viet Nam
Impulsive fuzzy functional differential equations
impulsive functional differential equations
impulsive differential equations
[[1] T. Allahviranloo, S. Abbasbandy, S. Salahshour and A. Hakimzadeh, A new method for##solving fuzzy linear dierential equations, Computing, 92 (2010), 181{197.##[2] T. Allahviranloo, S. Salahshour and S. Abbasbandy, Explicit solutions of fractional dieren##tial equations with uncertainty, Soft Computing, 16 (2011), 297{302.##[3] T. Allahviranloo, S. Abbasbandy, O. Sedaghgatfar and P. Darabi, A new method for solving##fuzzy integrodierential equation under generalized dierentiability, Neural Computing and##Applications, 21 (2011), 191{196.##[4] L. C. Barros, R. C. Bassanezi and P. A. Tonelli, Fuzzy modelling in population dynamics,##Ecological Modelling, 128 (2000), 27{33.##[5] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive dierential equations and inclusions,##Hindawi Publishing Corporation, USA, 2006.##[6] M. Benchohra, J. J. Nieto and A. Ouahab, Fuzzy solutions for impulsive dierential equations,##Communications in Applied Analysis, 11 (2007), 379{394.##[7] J. J. Buckley and T. Feuring, Fuzzy dierential equations, Fuzzy Sets and Systems, 110##(2000), 43 { 54.##[8] V. J. Devi and A. S. Vatsala, Method of vector lyapunov functions for impulsive fuzzy systems,##Dynamic Systems and Applications, 13 (2004), 521{531.##[9] L. S. Dong, H. Vu and N. V. Hoa, The formulas of the solution for linearorder random fuzzy##dierential equations, Journal of Intelligent & Fuzzy Systems, 28 (2015), 795{807.##[10] M. Guo, X. Xue and R. Li, Impulsive functional dierential inclusions and fuzzy population##models, Fuzzy Sets and Systems, 138 (2003), 601{615.##[11] N. V. Hoa, Fuzzy fractional functional dierential equations under Caputo gH##dierentiability, Communications in Nonlinear Science and Numerical Simulation, 22 (2015),##11341157.##[12] N. V. Hoa, Fuzzy fractional functional integral and dierential equations, Fuzzy Sets and##Systems, 280 (2015), 5890.##[13] N. V. Hoa and N. D. Phu, Fuzzy functional integrodierential equations under generalized##Hdierentiability, Journal of Intelligent & Fuzzy Systems, 26 (2014), 2073{2085.##[14] N. V. Hoa, N. D. Phu, T. T. Tung and L. T. Quang, Intervalvalued functional integro##dierential equations, Advance in Dierence Equations, (2014), 2014:177.##[15] N. V. Hoa, P. V. Tri, T. T. Dao and I. Zelinka, Some global existence results and stability##theorem for fuzzy functional dierential equations, Journal of Intelligent & Fuzzy Systems,##28 (2015), 393{409.##[16] O. Kaleva, Fuzzy dierential equations, Fuzzy Sets and Systems, 24 (1987), 301{317.##[17] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of impulsive dierential##equations, World Scientic, 1989. ##[18] V. Lakshmikantham, T. Gnana Bhaskar and Devi J. Vasundhara, Theory of set dierential##equations in a metric space, Cambridge Scientic Publishing, UK, 2006.##[19] V. Lakshmikantham and F. A. McRae, Basic results for fuzzy impulsive dierential equations,##Mathematical Inequalities & Applications, 4 (2001), 239{246.##[20] V. Lupulescu, On a class of fuzzy functional dierential equations, Fuzzy Sets and Systems,##160 (2009), 1547{1562.##[21] R. N. Mohapatra and V. Lakshmikantham, Theory of fuzzy dierential equations and inclu##sions, CRC Press, Singapore, 2003.##[22] J. J. Nieto, A. Khastan and K. Ivaz, Numerical solution of fuzzy dierential equations under##generalized dierentiability, Nonlinear Analysis: Hybrid Systems, 3 (2009), 700{707.##[23] J. J. Nieto and R. RodrguezLopez, Periodic boundary value problem for nonLipschitzian##impulsive functional dierential equations, Journal of Mathematical Analysis and Applica##tions, 318 (2006), 593610.##[24] R. RodrguezLopez, Periodic boundary value problems for impulsive fuzzy dierential equa##tions, Fuzzy Sets and Systems, 159 (2008), 1384{1409.##[25] A. M. Samoilenko and N. A. Perestyuk, Impulsive dierential equations, World Scientic,##Singapore, 1995.##[26] P. V. Tri, N. V. Hoa and N. D. Phu, Sheaf fuzzy problems for functional dierential equations,##Advance in Dierence Equation, 2014 2014:156##[27] A. S. Vatsala, Impulsive hybrid fuzzy dierential equations, Facta Univ. Ser Mech, Automatic##Control Robot, 3 (2003), 851{859.##[28] H. Vu, L. S. Dong and N. V. Hoa, Random fuzzy functional integrodierential equations##under generalized Hukuhara dierentiability , Journal of Intelligent & Fuzzy Systems, 27##(2014), 14911506.##[29] H. Vu and L. S. Dong, Random setvalued functional dierential equations with the second##type hukuhara derivative, Dierential Equations & Applications, 5 (2013), 501{518.##[30] H. Vu and L. S. Dong, Initial value problem for secondorder random fuzzy dierential equa##tions, Advances in Dierence Equations, 2015 2015:373##]
Stratified $(L,M)$fuzzy Qconvergence spaces
2
2
This paper presents the concepts of $(L,M)$fuzzy Qconvergence spaces and stratified $(L,M)$fuzzy Qconvergence spaces. It is shown that the category of stratified $(L,M)$fuzzy Qconvergence spaces is a bireflective subcategory of the category of $(L,M)$fuzzy Qconvergence spaces, and the former is a Cartesianclosed topological category. Also, it is proved that the category of stratified $(L,M)$fuzzy topological spaces can be embedded in the category of stratified $(L,M)$fuzzy Qconvergence spaces as a reflective subcategory, and the former is isomorphic to the category of topological stratified $(L,M)$fuzzy Qconvergence spaces.
1

95
111


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
(Stratified) $(L
M)$fuzzy topology
M)$fuzzy Qconvergence structure
Topological category
Cartesianclosedness
[[1] J. Adamek, H. Herrlich and G. E. Strecker, Abstract and concrete categories, Wiley, New##York, 1990.##[2] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182{190.##[3] J. M. Fang, Categories isomorphic to LFTOP, Fuzzy Sets Syst., 157 (2006), 820{831.##[4] J. M. Fang, Stratied Lordered convergence structures, Fuzzy Sets Syst., 161 (2010), 2130{##[5] J. M. Fang, Relationships between Lordered convergence structures and strong Ltopologies,##Fuzzy Sets Syst., 161 (2010), 2923{2944.##[6] M. Guloglu and D. Coker, Convergence in Ifuzzy topological spaces, Fuzzy Sets Syst., 151##(2005), 615{623.##[7] 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),##[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] G. Jager, Latticevalued convergence spaces and regularity, Fuzzy Sets Syst., 159 (2008),##2488{2502.##[12] G. Jager, Fischer's diagonal condition for latticevalued convergence spaces, Quaest. Math.,##31 (2008), 11{25.##[13] G. Jager, Stratied LMNconvergence tower spaces, Fuzzy Sets Syst., 282 (2016), 62{73.##[14] T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985.##[15] L. Q. Li and Q. Jin, On adjunctions between Lim, SLTop, and SLLim, Fuzzy Sets Syst.,##182 (2011), 66{78.##[16] L. Q. Li and Q. Jin, On stratied Lconvergence spaces: Pretopological axioms and diagonal##axioms, Fuzzy Sets Syst., 204 (2012), 40{52.##[17] R. Lowen, Convergence in fuzzy topological spaces, Gen. Topl. Appl., 10 (1979),147{160.##[18] K. C. Min, Fuzzy limit spaces, Fuzzy Sets Syst., 32 (1989), 343{357.##[19] B. Pang and J.M. Fang, Lfuzzy Qconvergence structures, Fuzzy Sets Syst., 182 (2011),##[20] B. Pang, Futher study on Lfuzzy Qconvergence structures, Iranian Journal of Fuzzy Systems,##10(5) (2013), 147{164.##[21] B. Pang, On (L;M)fuzzy convergence spaces, Fuzzy Sets Syst., 238 (2014), 46{70.##[22] B. Pang and F. G. Shi, Degrees of compactness of (L;M)fuzzy convergence spaces and its##applications, Fuzzy Sets Syst., 251 (2014), 1{22.##[23] B. Pang, Enriched (L;M)fuzzy convergence spaces, J. Intell. Fuzzy Syst., 27 (2014), 93{103.##[24] G. Preuss, Foundations of topology{an approach to convenient topology, Kluwer Academic##Publisher, Dordrecht, Boston, London, 2002.##[25] A. P. Sostak, On a fuzzy topological structure, Suppl. Rend. Circ. Mat. Palermo Ser. II, 11##(1985), 89{103.##[26] L. S. Xu, Characterizations of fuzzifying topologies by some limit structures, Fuzzy Sets Syst.,##123 (2001), 169{176.##[27] W. Yao, On manyvalued stratied Lfuzzy convergence spaces, Fuzzy Sets Syst., 159 (2008),##2503{2519.##[28] W. Yao, On Lfuzzifying convergence spaces, Iranian Journal of Fuzzy Systems, 6(1) (2009),##[29] W. Yao, MooreSmith convergence in (L;M)fuzzy topology, Fuzzy Sets Syst., 190 (2012),##]
Fuzzy Topology Generated by Fuzzy Norm
2
2
In the current paper, consider the fuzzy normed linear space $(X,N)$ which is defined by Bag and Samanta. First, we construct a new fuzzy topology on this space and show that these spaces are Hausdorff locally convex fuzzy topological vector space. Some necessary and sufficient conditions are established to illustrate that the presented fuzzy topology is equivalent to two previously studied fuzzy topologies.
1

113
123


M.
Saheli
Department of Mathematics, ValieAsr University of Rafsanjan, Rafsanjan, Iran
Department of Mathematics, ValieAsr University
Iran
Fuzzy norm
Fuzzy topology
locally convex topological vector 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] N. F. Das and P. Das, Fuzzy topology generated by fuzzy norm, Fuzzy Sets and Systems, 107##(1999), 349354.##[5] J. X. Fang, On Itopology generated by fuzzy norm, Fuzzy Sets and Systems, 157 (2006),##27392750.##[6] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48 (1992),##[7] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984),##[8] I. Karmosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11##(1975), 326334.##[9] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12 (1984), 143##[10] M. Saheli, On fuzzy topology and fuzzy norm, Annals of Fuzzy Mathematics and Informatics,##10(4) (2015), 639647.##[11] J. Xiao and X. Zhu, Fuzzy normed space of operators and its completeness, Fuzzy Sets and##Systems, 133 (2003) 389399.##[12] G. H. Xu and J. X. Fang, A new Ivector topology generated by a fuzzy norm, Fuzzy Sets and##Systems, 158 (2007), 23752385.##]
Extended Fuzzy $BCK$subalgebras
2
2
This paper extends the notion of fuzzy $BCK$subalgebras to fuzzy hyper $BCK$subalgebras and defines an extended fuzzy $BCK$subalgebras. This study considers a type of fuzzy hyper $BCK$ideals in this hyperstructure and describes the relationship between hyper $BCK$ideals and fuzzy hyper $BCK$ideals. In fact, it tries to introduce a strongly regular relation on hyper $BCK$algebras. Moreover, by using the fuzzy hyper $BCK$ideals, it defines a congruence relation on (weak commutative) hyper $BCK$algebras that under some conditions is strongly regular and the quotient of any hyper $BCK$algebra via this relation is a $($hyper $BCK$algebra$)$ $BCK$algebra.
1

125
144


Jianming
Zhan
Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China
Department of Mathematics, Hubei University
China
zhanjianming@ hotmail.com


Mohammad
Hamidi
Department of Mathematics, Payame Noor University, Tehran,
Iran
Department of Mathematics, Payame Noor University,
Iran
m.hamidi20@gmail.com


Arsham
Borumand Saeid
Department of Pure Mathematics, Faculty of Mathematics
and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Faculty of
Iran
arsham@iauk.ac.ir
Extended fuzzy $BCK$subalgebra
(Strongly) Fuzzy hyper $BCK$ideal
Fundamental relation $beta^*$
[[1] M. Bakhshi, M. M. Zahdi and R. A. Borzooei, Fuzzy (positive, weak) implicative hyper BCK##ideals, Iranian Journal of Fuzzy Systems, 1(2) (2004), 6379.##[2] R. A. Borzooei, R. Ameri and M. Hamidi, Fundamental relation on hyper BCKalgebras, An.##Univ. oradea, fasc. Mat., 21(1) (2014), 123{136.##[3] G. R. Biyogmam, O. A. HeuboKwegna and J. B. Nganou, Super implicative hyper BCK##algebras, Int. J. Pure Appl. Math., 76(2) (2012), 267275.##[4] J. Chvalina, S. HoskovaMayerova and A. D. Nezhad, General actions of hyperstructures and##some applications, An. St. Univ. Ovidius Constanta, 21(1) (2013), 59{82.##[5] P. Corsini, Prolegomena of hypergroup theory, Second Edition, Aviani Editor, 1993.##[6] P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Klwer Academic Pub##lishers, 2002.##[7] J. Dongho, Category of fuzzy hyper BCKalgebras, arXiv:1101.2471.##[8] P. F. He and X. L. Xin, Fuzzy hyperlattices, Comput. Math. Appl., 62 (2011), 4682{4690.##[9] S. Hoskova, Topological hypergroupoids, Comput. Math. Appl., 64 (2012), 2845{2849.##[10] Y. Imai and K. Iseki, On axiom systems of propositional calculi, XIV, Proc. Japan Acad.##Ser. A Math. Sci., 42 (1966), 19{22.##[11] Y. B. Jun, M. M. Zahedi, X. L. Xin and R. A. Borzooei, On hyper BCKalgebras, Ital. J.##Pure Appl. Math., 10 (2000), 127{136.##[12] Y. B. Jun and X. L. Xin, Fuzzy hyper BCKideals of hyper BCKalgebras, Sci. Math. Jpn.,##53(2) (2001), 353{360.##[13] Y. B. Jun, M. S. Kang and S. Z. Song, Several types of bipolar fuzzy hyper BCKideals in##hyper BCKalgebras, Honam Math. J., 34(2) (2012), 145{159.##[14] V. LeoreanuFotea, Fuzzy hypermodules, Comput. Math. Appl., 57 (2009), 466{475. ##[15] F. Marty, Sur une Generalization de la notion de groupe, 8th Congres Math. Scandinaves,##Stockholm., (1934), 45{49.##[16] J. Meng and Y. B. Jun, BCKalgebra, Kyung Moonsa, Seoul, 1994.##[17] H. T. Nguyen and E. A. Walker, A First Course in Fuzzy Logic, (Third Edition), Chapman##and Hall/CRC. Boca Raton, 2005.##[18] M. K. Sen, R. Ameri and G. Chowdhury, Fuzzy hypersemigroups, Soft Comput., 12 (2008),##[19] X. Xie, Fuzzy ideals extensions in semigroups, Kyungpook Math. J., 42(1) (2002), 39{49.##[20] X. L. Xin and P. Wang, States and measures on hyper BCKalgebras, J. Appl. Math., (2014),##[21] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338{353.##[22] J. Zhan, Fuzzy regular relations on hyperquasigroups, J. Math. Res. Exposition, Nov., 30(6)##(2010), 1083{1090.##]
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