2009
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Cover Vol.6, No.1, Februery 2009
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ROBUST $H_{infty}$ CONTROL FOR T–S TIMEVARYING DELAY
SYSTEMS WITH NORM BOUNDED UNCERTAINTY BASED ON
LMI APPROACH
ROBUST $H_{infty}$ CONTROL FOR T–S TIMEVARYING DELAY
SYSTEMS WITH NORM BOUNDED UNCERTAINTY BASED ON
LMI APPROACH
2
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In this paper we consider the problem of delaydependent robustH1 control for uncertain fuzzy systems with timevarying delay. The Takagi–Sugeno (T–S) fuzzy model is used to describe such systems. Timedelay isassumed to have lower and upper bounds. Based on the LyapunovKrasovskiifunctional method, a sufficient condition for the existence of a robust $H_{infty}$controller is obtained. The fuzzy state feedback gains are derived by solvingpertinent LMIs. The proposed method can avoid restrictions on the derivativeof the timevarying delay assumed in previous works. The effectiveness of ourmethod is demonstrated by a numerical example.
1
In this paper we consider the problem of delaydependent robustH1 control for uncertain fuzzy systems with timevarying delay. The Takagi–Sugeno (T–S) fuzzy model is used to describe such systems. Timedelay isassumed to have lower and upper bounds. Based on the LyapunovKrasovskiifunctional method, a sufficient condition for the existence of a robust H1controller is obtained. The fuzzy state feedback gains are derived by solvingpertinent LMIs. The proposed method can avoid restrictions on the derivativeof the timevarying delay assumed in previous works. The effectiveness of ourmethod is demonstrated by a numerical example.
1
14
HanLiang
Huang
HanLiang
Huang
Department of Mathematics, Beijing Institute of Technology,
Beijing 100081, China
Department of Mathematics, Beijing Institute
China
hl_huang1980.student@sina.com
FuGui
Shi
FuGui
Shi
Department of Mathematics, Beijing Institute of Technology, Beijing
100081, China
Department of Mathematics, Beijing Institute
China
f.g.shi@263.net
$H_{infty}$ control
linear matrix inequality (LMI)
Delaydependent
T–S fuzzy systems
Uncertainty
[[1] Y. Y. Cao and P. M. Frank, Stability analysis and synthesis of nonlinear timedelay systems##via linear Takagi–Sugeno fuzzy models, Fuzzy Sets and Systems, 124 (2001), 213219. ##2] P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, LMI control toolbox, MathWorks Inc,##Natick, MA, 1995.##[3] H. Gao, P. Shi and J. Wang, Parameterdependent robust stability of uncertain timedelay##systems, J. Comput. Appl. Math., 206 (2007), 366373.##[4] T. M. Guerra and L. Vermeiren, LMIbased relaxed nonquadratic stabilization conditions for##nonlinear systems in the Takagi–Sugeno’s form, Automatica, 40 (2004), 823–829.##[5] A. T. Han, G. D. Chen, M. Y. Yang and L. Yu, Stabilizing fuzzy controller design for uncertain##timedelay systems, Proceedings of the 3rd World Congress on Intelligent Control and##Automation, 2000, 15401543.##[6] Q. L. Han and K. Gu, Stability of linear systems with timevarying delay: A generalized##discretized Lyapunov functional approach, Asian J. Contr., 3 (2001), 170180.##[7] S. K. Hong and R. Langari, An LMIbased H1 fuzzy control system design with TS framework,##Inform. Sci., 123 (2000), 163179.##[8] X. F. Jiang, Q. L. Han and X. H. Yu, Robust H1 control for uncertain Takagi–Sugeno fuzzy##systems with interval timevarying delay, 2005 American Control Conference June 810, 2005,##Portland, OR, USA, 2005, 11141119.##[9] M. Li and H. G. Zhang, Fuzzy H1 robust control for nonlinear timedelay system via fuzzy##performance evaluator, IEEE International Conference on Fuzzy Systerms, 2003, 555560.##[10] C. H. Lien, Stabilization for uncertain Takagi–Sugeno fuzzy systems with timevarying delays##and bounded uncertainties, Chaos Solitons Fractals, 32 (2007), 645652.##[11] C. Lin, Q. G. Wang and T. H. Lee, Delaydependent LMI conditions for stability and stabilization##of T–S fuzzy systems with bounded timedelay, Fuzzy Sets and Systems, 157 (2006),##12291247.##[12] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling##and control, IEEE Trans. Systems Man Cybernet., 15 (1985), 116163.##[13] R. J. Wang, W. W. Lin and W. J. Wang, Stabilizability of linear quardratic state feedback##for uncertain fuzzy timedelay systems, IEEE Trans. Systems Man Cybernet., (Part B), 34##(2004), 12881292.##[14] W. J. Wang, K. Tanaka and M. F. Griffin, Stabilization, estimation and robustness for largescale##timedelay systems, ControlTheory Adv. Technol., 7 (1991), 569585.##[15] H. O. Wang, K. Tanaka and M. F. Griffin, An approach to fuzzy control of nonlinear systems:##Stability and design issues, IEEE Trans. Fuzzy Syst., 4 (1996), 1423.##[16] J. Yoneyama, Robust control analysis and synthesis for uncertain fuzzy systems with timedelay,##The IEEE International Conference on Fuzzy Systerms, 2003, 396401.##[17] K. W. Yu and C. H. Lien, Robust H1 control for uncertain T–S fuzzy systems with state##and input delays, Chaos Solitons Fractals, 37 (2008), 150156.##[18] L. Yu, Robust control An LMI method, Tsinghua University Press, Beijing, China, 2002.##[19] D. Yue and J. Lam, Reliable memory feedback design for a class of nonlinear time delay##systems, Internat. J. Robust and Nonlinear Control, 14 (2004), 3960.##[20] Y. Zhang and A. H. Pheng, Stability of fuzzy control systems with bounded uncertain delays,##IEEE Trans. Fuzzy Syst., 10 (2002), 9297.##]
COMBINING FUZZY QUANTIFIERS AND NEAT OPERATORS
FOR SOFT COMPUTING
COMBINING FUZZY QUANTIFIERS AND NEAT OPERATORS
FOR SOFT COMPUTING
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This paper will introduce a new method to obtain the order weightsof the Ordered Weighted Averaging (OWA) operator. We will first show therelation between fuzzy quantifiers and neat OWA operators and then offer anew combination of them. Fuzzy quantifiers are applied for soft computingin modeling the optimism degree of the decision maker. In using neat operators,the ordering of the inputs is not needed resulting in better computationefficiency. The theoretical results will be illustrated in a water resources managementproblem. This case study shows that more sensitive decisions areobtained by using the new method.
1
This paper will introduce a new method to obtain the order weightsof the Ordered Weighted Averaging (OWA) operator. We will first show therelation between fuzzy quantifiers and neat OWA operators and then offer anew combination of them. Fuzzy quantifiers are applied for soft computingin modeling the optimism degree of the decision maker. In using neat operators,the ordering of the inputs is not needed resulting in better computationefficiency. The theoretical results will be illustrated in a water resources managementproblem. This case study shows that more sensitive decisions areobtained by using the new method.
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25
Ferenc
szidarovszky
Ferenc
szidarovszky
Systems and Industrial Engineering Department, University of
Arizona, Tucson, Az 857210020, USA
Systems and Industrial Engineering Department,
United States
szidar@sie.arizona.edu
Mahdi
Zarghami
Mahdi
Zarghami
Faculty of Civil Engineering, University of Tabriz, Tabriz 51664,
Iran
Faculty of Civil Engineering, University
United States
mzarghami@tabrizu.ac.ir
OWA operator
Fuzzy quantifiers
Neat operator
Multi criteria decision making
Watershed management
[[1] D. Filev and R. R. Yager, On the issue of obtaining OWA operator weights, Fuzzy Sets and##Systems, 94 (1988), 157169.##[2] R. Fullr and P. Majlender, An analytic approach for obtaining maximal entropy OWA operator##weights, Fuzzy Sets and Systems, 124 (2001), 5357.##[3] X. W. Liu, On the methods of decision making under uncertainty with probability information,##Int. J. Intell. Syst., 19 (2004), 12171238.##[4] X. W. Liu and H. Lou, Parameterized additive neat OWA operators with different orness##levels, Int. J. Intell. Syst., 21 (2006), 10451072.##[5] M. Marimin, M. Umano, I. Hatono and H. Tamura, Linguistic labels for expressing fuzzy##preference relations in fuzzy group decision making, IEEE Trans. Syst. Man. Cybern. B, 28##(1998), 205218.##[6] M. O’Hagan, Aggregating template or rule antecedents in realtime expert systems with fuzzy##set, In: Grove P, editor. Proc 22nd Annual IEEE Asilomar Conf on Signals, Systems, Computers.##California, 1988, 681689.##[7] J. I. Pel´aez and J. M. Do˜na, Majority additiveordered weighting averaging: A new neat##ordered weighting averaging operators based on the majority process, Int. J. Intell. Syst., 18##(2003), 469481.##[8] J. I. Pel´aez and J. M. Do˜na, A majority model in group decision making using QMAOWA##operators, Int. J. Intell. Syst., 21 (2006), 193208.##[9] C. E. Shannon, A mathematical theory of communication, Bell System Tech., 27 (1948),##379423 and 623656.##[10] J. Wu, C. Y. Liang and Y. Q. Huang, An argumentdependent approach to determining OWA##operator weights based on the rule of maximum entropy, Int. J. of Intell. Syst., 22 (2007),##[11] Z. Xu, An overview of methods for determining OWA weights, Int. J. Intell. Syst., 20 (2005),##[12] R. R. Yager, On ordered weighted averaging aggregation operators in multicriteria decisionmaking,##IEEE Trans. Syst. Man. Cybern., 18 (1988), 183190. ##[13] R. R. Yager, Families of OWA operators, Fuzzy Sets and Systems, 59 (1993), 125143.##[14] R. R. Yager and D. P. Filev, Parameterized andlike and orlike OWA operators, Int. J. Gen.##Sys., 22 (1994), 297316.##[15] R. R. Yager, Quantifier guided aggregation using OWA operators, Int. J. Intell. Syst., 11##(1996), 4973.##[16] R. R. Yager, On the cardinality and attituditional charactersitics of fuzzy measures, Int. J.##Gen. Sys., 31 (2002), 303329.##[17] R. R. Yager, Centered OWA operators, Soft Comp., 11 (2007), 631639.##[18] L. A. Zadeh, A computational approach to fuzzy quantifiers in natural languages, Comput.##and Math. with App., 9 (1983), 149184.##[19] M. Zarghami, F. Szidarovszky and R. Ardakanian, Sensitivity analysis of the OWA operator,##IEEE Trans. Syst. Man. Cybern. B, 38(2) (2008), 547552.##[20] M. Zarghami and F. Szidarovszky, Revising the OWA operator for multi criteria decision##making problems under uncertainty, Euro. J. Oper. Res., 2008, (Article in press: doi:##10.1016/j.ejor.2008.09.014).##]
THE PERCENTILES OF FUZZY NUMBERS AND THEIR
APPLICATIONS
THE PERCENTILES OF FUZZY NUMBERS AND THEIR
APPLICATIONS
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The purpose of this study is to find the percentiles of fuzzy numbersand to demonstrate their applications, which include finding weightedmeans, dispersion indices, and the percentile intervals of fuzzy numbers. Thecrisp approximations of fuzzy numbers introduced in this paper are new andinteresting for the comparison of fuzzy environments, such as a variety of economic,financial, and engineering systems control problems.
1
The purpose of this study is to find the percentiles of fuzzy numbersand to demonstrate their applications, which include finding weightedmeans, dispersion indices, and the percentile intervals of fuzzy numbers. Thecrisp approximations of fuzzy numbers introduced in this paper are new andinteresting for the comparison of fuzzy environments, such as a variety of economic,financial, and engineering systems control problems.
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44
Eynollah
Pasha
Eynollah
Pasha
Department of Mathematics, The teacher Training University,
Tehran, Iran
Department of Mathematics, The teacher Training
Iran
pasha@saba.tmu.ac.ir
Abolfazl
Saiedifar
Abolfazl
Saiedifar
Department of Statistics, Science and Research branch, Islamic
Azad University, Tehran 14515775, Iran
Department of Statistics, Science and Research
Iran
asaiedi@iauarak.ac.ir or saiedifar1349@yahoo.com
Babak
Asady
Babak
Asady
Department of Mathematics, Islamic Azad University, Arak, Iran
Department of Mathematics, Islamic Azad University
Iran
babakmz2002@yahoo.com
Trimmed mean
Winsorized mean
Interquartile range
Skewness
Kurtosis
Percentile interval
[[1] S. Bodjanova, Median value and median interval of a fuzzy number, Information Sciencees,##172 (2005), 7389.##[2] G. K. Bhattacharyya and R. A. Johnson, Statistical concepts and methods, John Wiley and##Sons, 1977.##[3] C. Carlsson and R. Full´er, On possibilistic mean value and variance of fuzzy numbers, Fuzzy##Sets and Systems, 122 (2001), 315326.##[4] D. Dubois and H. Prade, Fuzzy sets and systems, theory and applications, Academikc press,##New York, 1980.##[5] D. Dubois and H. Prade, The mean value of a fuzzy number, Fuzzy Sets and Systems, 24##(1987), 279300.##[6] D. Dubois and H. Prade, Fundamentals of fuzzy sets, The Handbooks of Fuzzy Sets Series,##Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.##[7] A. Galvan, Univariate equtions, Internet, 2005.##[8] M. Ma, A. Kandel and M. Friedman, A new approach for defuzzification, Fuzzy Sets and##Systems, 111 (2000), 351356.##[9] C. R. Marques, P. D. Neves and L. M. Sarmento, Evaluating core inflation indicator, Economic##Modelling, 20 (2003), 765775.##[10] P. McAdam and P. McNelis, Forecasting inflation with thick model and neural networks,##Economic Modelling, 22 (2005), 848867.##[11] A. V. Patel and B. M. Mohan, Some numerical aspects of center of area defuzzification##method, Fuzzy Sets and Systems, 132 (2002), 401409.##[12] A. K. Rose, A stable internatinal monetary systems emerges: Inflation targeting is Bretton##Woods, reversed, Journal of International Money and Finance, 26 (2007), 663681.##[13] A. Saiedifar and E. Pasha, The percentiles of trapezoidal fuzzy numbers and their applications,##ICREM3, Kuala Lumpur, Malaysia, Proceedings of Pure Mathemathics Statistics, 2007, 61##[14] A. Stuart, J. K. Ord and Kendall’s, Advanced theory of statistics, Distribution Theory 6th##ed. New York, Oxford University Press, 1 (1998).##[15] L. A. Zadeh, A fuzzy settheoritic interpretation of linguistic hedges, Journal of Cybernetics,##2 (1972), 434.##]
ABSORBENT ORDERED FILTERS AND THEIR
FUZZIFICATIONS IN IMPLICATIVE SEMIGROUPS
ABSORBENT ORDERED FILTERS AND THEIR FUZZIFICATIONS IN IMPLICATIVE SEMIGROUPS
2
2
The notion of absorbent ordered filters in implicative semigroupsis introduced, and its fuzzification is considered. Relations among (fuzzy) orderedfilters, (fuzzy) absorbent ordered filters, and (fuzzy) positive implicativeordered filters are stated. The extensionproperty for (fuzzy) absorbent orderedfilters is established. Conditions for (fuzzy) ordered filters to be (fuzzy)absorbent ordered filters are provided. The notions of normal/maximal fuzzyabsorbent ordered filters and complete absorbent ordered filters are introducedand their properties are investigated.
1
The notion of absorbent ordered filters in implicative semigroupsis introduced, and its fuzzification is considered. Relations among (fuzzy) orderedfilters, (fuzzy) absorbent ordered filters, and (fuzzy) positive implicativeordered filters are stated. The extensionproperty for (fuzzy) absorbent orderedfilters is established. Conditions for (fuzzy) ordered filters to be (fuzzy)absorbent ordered filters are provided. The notions of normal/maximal fuzzyabsorbent ordered filters and complete absorbent ordered filters are introducedand their properties are investigated.
45
61
Young Bae
Jun
Young Bae
Jun
Department of Mathematics Education and (RINS), Gyeongsang National
University, Chinju 660701, Korea
Department of Mathematics Education and (RINS),
Korea
skywine@gmail.com
Chul Hwan
Park
Chul Hwan
Park
Department of Mathematics, University of Ulsan, Ulsan 680749,
Korea
Department of Mathematics, University of
Korea
skyrosemary@gmail.com
D. R.
Prince Williams
D. R.
Prince Williams
Department of Information Technology, Salalah College of
Technology, Post Box: 608, Salalah211, Sultanate of Oman
Department of Information Technology, Salalah
Oman
princeshree1@gmail.com
Implicative semigroup
(fuzzy) positive implicative ordered filter
(fuzzy) absorbent ordered filter
Normal fuzzy absorbent ordered filter
Maximal fuzzy absorbent ordered filter
Complete fuzzy absorbent ordered filter
[[1] G. Birkhoff, Lattice theory, Amer. Math. Soc. Coll. Publ. Vol. XXV, Providence, 1967.##[2] T. S. Blyth, Pseudoresiduals in semigroups, J. London Math. Soc., 40 (1965), 441454.##[3] M. W. Chan and K. P. Shum, Homomorphisms of implicative semigroups, Semigroup Forum,##46 (1993), 715.##[4] H. B. Curry, Foundations of mathematics logic, McGrowHill, New York, 1963.##[5] Y. B. Jun, Implicative ordered filters of implicative semigroups, Comm. Korean Math. Soc.,##14(1) (1999), 4755.##[6] Y. B. Jun, Fuzzy implicative ordered filters in implicative semigroups, Southeast Asian. Bull.##Math., 26 (2003), 935943.##[7] Y. B. Jun, Folding theory applied to implicative ordered filters of implicative semigroups,##Southeast Asian. Bull. Math., 31 (2007), 893901.##[8] Y. B. Jun and K. H. Kim, Positive implicative ordered filters of implicative semigroups,##Internat. J. Math. Math. Sci., 23(12) (2000), 801806.##[9] Y. B. Jun, Y. H. Kim and H. S. Kim, Fuzzy positive implicative ordered filters of implicative##semigroups, Internat. J. Math. Math. Sci., 32(5) (2002), 263270.##[10] Y. B. Jun, J. Meng and X. L. Xin, On ordered filters of implicative semigroups, Semigroup##Forum, 54 (1997), 7582.##[11] S. W. Kuresh, Y. B. Jun and W. P. Huang, Fuzzy ordered filters in implicative semigroups,##Chinese Quartely J. Math., 13(2) (1998), 5357.##[12] J. Meng, Implicative commutative semigroups are equivalent to a class of BCKalgebras,##Semigroup Forum, 50 (1995), 8996.##[13] W. C. Nemitz, Implicative semilattices, Trans. Amer. Math. Soc., 117 (1965), 128142.##]
ON ($epsilon, epsilon vee q$)FUZZY IDEALS OF BCIALGEBRAS
ON ($epsilon, epsilon vee q$)FUZZY IDEALS OF BCIALGEBRAS
2
2
The aim of this paper is to introduce the notions of ($epsilon, epsilon vee q$)fuzzy pideals, ($epsilon, epsilon vee q$)fuzzy qideals and ($epsilon, epsilon vee q$)fuzzy aideals in BCIalgebras and to investigate some of their properties. Several characterizationtheorems for these generalized fuzzy ideals are proved and the relationshipamong these generalized fuzzy ideals of BCIalgebras is discussed. It is shownthat a fuzzy set of a BCIalgebra is an ($epsilon, epsilon vee q$)fuzzy aideal if and only if itis both an ($epsilon, epsilon vee q$)fuzzy pideal and an ($epsilon, epsilon vee q$)fuzzy qideal. Finally, the concept of implicationbased fuzzy aideals in BCIalgebras is introduced and,in particular, the implication operators in Lukasiewicz system of continuousvaluedlogic are discussed.
1
The aim of this paper is to introduce the notions of ($epsilon, epsilon vee q$)fuzzy pideals, ($epsilon, epsilon vee q$)fuzzy qideals and ($epsilon, epsilon vee q$)fuzzy aideals in BCIalgebras and to investigate some of their properties. Several characterizationtheorems for these generalized fuzzy ideals are proved and the relationshipamong these generalized fuzzy ideals of BCIalgebras is discussed. It is shownthat a fuzzy set of a BCIalgebra is an ($epsilon, epsilon vee q$)fuzzy aideal if and only if itis both an ($epsilon, epsilon vee q$)fuzzy pideal and an ($epsilon, epsilon vee q$)fuzzy qideal. Finally, the concept of implicationbased fuzzy aideals in BCIalgebras is introduced and,in particular, the implication operators in Lukasiewicz system of continuousvaluedlogic are discussed.
81
94
Jianming
Zhan
Jianming
Zhan
Department of Mathematics, Hubei Institute for Nationalities, Enshi,
Hubei Province,445000, P. R. China
Department of Mathematics, Hubei Institute
China
zhanjianming@hotmail.com
Young Bae
Jun
Young Bae
Jun
Department of Mathematics Education, Gyeongsang National University,
Chinju 660701, Korea
Department of Mathematics Education, Gyeongsang
Korea
skywine@gmail.com
Bijan
Davvaz
Bijan
Davvaz
Department of Mathematics, Yazd University, Yazd, Iran
Department of Mathematics, Yazd University,
Iran
davvaz@yazduni.ac.ir
BCIalgebra
($epsilon
epsilon vee q$)fuzzy (p
q and a) ideal
Fuzzy Logic
Implication operator
[[1] S. K. Bhakat, (2, 2_ q)fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and##Systems, 112 (2000), 299312.##[2] S. K. Bhakat and P. Das, (2, 2 _ q)fuzzy subgroups, Fuzzy Sets and Systems, 80 (1996),##[3] C. C. Chang, Algebraic analysis of many valued logic, Trans. Amer. Math. Soc., 88 (1958),##[4] B. Davvaz, (2, 2 _q)fuzzy subnearrings and ideals, Soft Computing, 10 (2006), 206211.##[5] B. Davvaz and P. Corsini, Redefined fuzzy Hvsubmodules and many valued implications,##Inform. Sci., 177 (2007), 865875.##[6] F. Esteva and L. Godo, Monoidal tnorm based logic: towards a logic for leftcontinuous##tnorms, Fuzzy Sets and Systems, 124 (2001), 271288.##[7] P. H´ajek, Metamathematics of fuzzy logic, Kluwer Academic Press, Dordrecht, 1998.##[8] Y. Imai and K. Iseki, On axiom system of propositional calculus, Proc. Japan Acad., 42##(1966), 1922.##[9] A. Iorgulescu, Some direct ascendents of wajsberg and MV algebras, Sci. Math. Japon., 57##(2003), 583647. ##[10] A. Iorgulescu, PseudoIseki algebras. connection with pseudoBL algebras, MultipleValued##Logic and Soft Computing, 11 (2005), 263308.##[11] K. Iseki, An algebra related with a propositional calculus, Proc. Japan Acad., 42 (1966),##[12] K. Iseki and S. Tanaka, Ideal theory of BCKalgebras, Math. Japon., 21 (1966), 351366.##[13] Y. B. Jun, Closed fuzzy ideals in BCIalgebras, Math. Japon., 38 (1993), 199202.##[14] Y. B. Jun, On (, )fuzzy ideals of BCK/BCIalgebras, Sci. Math. Japon., 60 (2004), 613##[15] Y. B. Jun, On (, )fuzzy subalgebras of BCK/BCIalgebras, Bull. Korean Math. Soc., 42##(2005), 703711.##[16] Y. B. Jun and J. Meng, Fuzzy pideals in BCIalgebras, Math. Japon, 40 (1994), 271282.##[17] Y. B. Jun and J. Meng, Fuzzy commutative ideals in BCIalgebras, Comm. Korean Math.##Soc., 9 (1994), 1925.##[18] Y. B. Jun and W. H. Shim, Fuzzy strong implicative hyper BCKideals of hyper BCKalgebras,##Inform. Sci., 170 (2005), 351361.##[19] Y. B. Jun, Y. Xu and J. Ma, Redefined fuzzy implicative filters, Inform. Sci., 177 (2007),##14221429.##[20] T. D. Lei and C. C. Xi, pradical in BCIalgebras, Math. Japon., 30 (1995), 511517.##[21] Y. L. Liu, Some results on psemisimple BCIalgebras, Math. Japon, 30 (1985), 511517.##[22] Y. L. Liu, S. Y. Liu and J. Meng, FSIideals and FSCideals of BCIalgebras, Bull. Korean##Math. Soc., 41 (2004), 167179.##[23] Y. L. Liu and J. Meng, Fuzzy qideals of BCIalgebras, J. Fuzzy Math., 8 (2000), 873881.##[24] Y. L. Liu and J. Meng, Fuzzy ideals in BCIalgebras, Fuzzy Sets and Systems, 123 (2001),##[25] Y. L. Liu, J. Meng, X. H. Zhang and Z. C. Yue, qideals and aideals in BCIalgebras, SEA##Bull. Math., 24 (2000), 243253.##[26] Y. L. Liu, Y. Xu and J. Meng, BCIimplicative ideals of BCIalgebras, Inform. Sci., 177##(2007), 49874996.##[27] Y. L. Liu and X. H. Zhang, Fuzzy aideals in BCIalgebras, Adv. in Math.(China), 31 (2002),##[28] J. Meng and X. Guo, On fuzzy ideals in BCKalgebras, Fuzzy Sets and Systems, 149 (2005),##[29] J. Meng and Y. B. Jun, BCKalgebras, Kyung Moon Sa Co., Seoul, Korean, 1994.##[30] D. Mundici, MV algebras are categorically equivalent to bounded commutative BCKalgebras,##Math. Japon., 31 (1986), 889894.##[31] P. M. Pu and Y. M. Liu, Fuzzy topology I: Neighourhood structure of a fuzzy point and##MooreSmith convergence, J. Math. Anal. Appl., 76 (1980), 571599.##[32] X. H. Yuan, C. Zhang and Y. H. Ren, Generalized fuzzy groups and many valued applications,##Fuzzy Sets and Systems, 138 (2003), 205211.##[33] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338353.##[34] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU)an outine, Inform. Sci., 172##(2005), 140.##[35] J. Zhan and Y. L. Liu, On fderivation of BCIalgebras, Int. J. Math. Math. Sci., 2005,##16751684.##[36] J. Zhan and Z. Tan, Fuzzy aideals of ISalgebras, Sci. Math. Japon, 58 (2003), 8587.##[37] J. Zhan and Z. Tan, Intuitionistic fuzzy aideals in BCIalgebras, Soochow Math. J., 30##(2004), 207216.##[38] X. H. Zhang, H. Jiang and S. A. Bhatti, On pideals of BCIalgebras, Punjab Univ. J. Math.,##27 (1994), 121128.##]
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