2012
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Cover Special Issue vol. 9, no. 6, December 2012
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CREDIBILITYBASED FUZZY PROGRAMMING MODELS TO
SOLVE THE BUDGETCONSTRAINED FLEXIBLE
FLOW LINE PROBLEM
CREDIBILITYBASED FUZZY PROGRAMMING MODELS TO
SOLVE THE BUDGETCONSTRAINED FLEXIBLE
FLOW LINE PROBLEM
2
2
This paper addresses a new version of the exible ow line prob lem, i.e., the budget constrained one, in order to determine the required num ber of processors at each station along with the selection of the most eco nomical process routes for products. Since a number of parameters, such as due dates, the amount of available budgets and the cost of opting particular routes, are imprecise (fuzzy) in practice, they are treated as fuzzy variables. Furthermore, to investigate the model behavior and to validate its attribute, we propose three fuzzy programming models based upon credibility measure, namely expected value model, chanceconstrained programming model and dependent chanceconstrained programming model, in order to transform the original mathematical model into a fuzzy environment. To solve these fuzzy models, a hybrid metaheuristic algorithm is proposed in which a genetic al gorithm is designed to compute the number of processors at each stage; and a particle swarm optimization (PSO) algorithm is applied to obtain the op timal value of tardiness variables. Finally, computational results and some concluding remarks are provided.
1
This paper addresses a new version of the exible ow line prob lem, i.e., the budget constrained one, in order to determine the required num ber of processors at each station along with the selection of the most eco nomical process routes for products. Since a number of parameters, such as due dates, the amount of available budgets and the cost of opting particular routes, are imprecise (fuzzy) in practice, they are treated as fuzzy variables. Furthermore, to investigate the model behavior and to validate its attribute, we propose three fuzzy programming models based upon credibility measure, namely expected value model, chanceconstrained programming model and dependent chanceconstrained programming model, in order to transform the original mathematical model into a fuzzy environment. To solve these fuzzy models, a hybrid metaheuristic algorithm is proposed in which a genetic al gorithm is designed to compute the number of processors at each stage; and a particle swarm optimization (PSO) algorithm is applied to obtain the op timal value of tardiness variables. Finally, computational results and some concluding remarks are provided.
1
29
Ali
Ghodratnama
Ali
Ghodratnama
Department of Industrial Engineering, College of Engineering,
University of Tehran, Tehran, Iran
Department of Industrial Engineering, College
Iran
ghodratn@ut.ac.ir
Seyed Ali
Torabi
Seyed Ali
Torabi
Department of Industrial Engineering, College of Engineering,
University of Tehran, Tehran, Iran
Department of Industrial Engineering, College
Iran
satorabi@ut.ac.ir
Raza
TavakkoliMoghaddam
Raza
TavakkoliMoghaddam
Department of Industrial Engineering, College of En
gineering, University of Tehran, Tehran, Iran
Department of Industrial Engineering, College
Iran
tavakoli@ut.ac.ir
Budgetconstrained exible ow lines
Credibilitybased fuzzy pro gramming
Metaheuristic
Genetic Algorithm
particle swarm optimization
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Zaeimazad, A note on zimmermann method for solving fuzzy##linear programming problems, Iranian Journal of Fuzzy Systems, 4(2) (2007), 3145.##[41] M. Sakawa and R. Kubota, Fuzzy programming for multi objective job shop scheduling with##fuzzy processing time and fuzzy due date through genetic algorithms, European Journal of##Operational Research, 120(2) (2000), 393407.##[42] J. Salerno, Using the particle swarm optimization technique to train a recurrent neural model,##IEEE Transactions, International Conference on Tools with Articial Intelligence, (1997), 45##[43] T. Sawik, Mixed integer programming for scheduling flexible flow lines with limited intermediate##buffers, Mathematical and Computer Modelling, 31 (2000), 3952.##[44] Y. Shi and R. Eberhart, Particle swarm optimization: development, applications and resources##, IEEE Transaction, 3(1) (2001), 8186. ##[45] E. Shivanian, E. Khorram and A. Ghodousian, Optimization of linear objective function subject##to fuzzy relation inequalities constraints with maxaverage composition, Iranian Journal##of Fuzzy Systems, 4(2) (2007), 1529.##[46] H. Tanaka, H. Ichihashi and K. Asai, A formulation of fuzzy linear programming problem##bases on comparison of fuzzy numbers, Control and Cybernetics, 13 (1984), 185194.##[47] S. A. Torabi and E. Hassini, An interactive possibilistic programming approach for multiple##objective supply chain master planning, Fuzzy Sets and Systems, 159 (2008), 193214.##[48] S. A. Torabi and E. Hassini, Multisite production planning integrating procurement and##distribution plans in multiechelon supply chains: an interactive fuzzy goal programming##approach, International Journal of Production Research, 47(19) (2009), 54755499.##[49] S. A. Torabi, M. Ebadian and R. Tanha, Fuzzy hierarchical production planning (with a case##study), Fuzzy Sets and Systems, 161 (2010), 15111529.##[50] S. Vob and A. Witt, Hybrid flow shop scheduling as a multimode multiproject scheduling##problem with batching requirements: a realworld application, International Journal of##Production Economics, 105(2) (2007), 445458.##[51] L. Zadeh,Fuzzy sets, Information and Control, 8 (1965), 338353.##[52] H. Zimmermann, Fuzzy programming and linear programming with several objective functions##, Fuzzy Sets and Systems, 1(1) (1978), 4556.##]
AN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC
FUZZY LOGIC
AN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC
FUZZY LOGIC
2
2
In this paper we extend the notion of degrees of membership and nonmembership of intuitionistic fuzzy sets to lattices and introduce a residuated lattice with appropriate operations to serve as semantics of intuitionistic fuzzy logic. It would be a step forward to find an algebraic counterpart for intuitionistic fuzzy logic. We give the main properties of the operations defined and prove some theorems to demonstrate our goal.
1
In this paper we extend the notion of degrees of membership and nonmembership of intuitionistic fuzzy sets to lattices and introduce a residuated lattice with appropriate operations to serve as semantics of intuitionistic fuzzy logic. It would be a step forward to find an algebraic counterpart for intuitionistic fuzzy logic. We give the main properties of the operations defined and prove some theorems to demonstrate our goal.
31
41
Esfandiar
Eslami
Esfandiar
Eslami
Department of Mathematics, Faculty of Mathematics and Com
puter, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Mathematics, Faculty of Mathematics
Iran
esfandiar.eslami@uk.ac.ir
Intuitionistic fuzzy logic
residuated lattice
Intuitionistic fuzzy implication
[bibitem{1 } K. T. Atanassov, {it Intuitionistic fuzzy sets}, Fuzzy Sets and Systems, {bf20} (1986), 8796. ##bibitem{2 } K. T. Atanassov and S. Stoeva, {it Intuitionistic L  fuzzy sets}, In: R. Trappl, ed., Elsevier Science Publishers B.V., Noth Holland, 1984. ##bibitem{3 } K. T. Atanassov and G. Gargov, {it Elements of intuitionistic fuzzy logic. part I}, Fuzzy Sets and Systems, {bf95} (1998), 3952. ##bibitem{4 } M. Baczynski, {it Residual implications revisited}, Fuzzy Sets and Systems, {bf 145} (2004), 267 277. ##bibitem{5} P. Burillo and H. Bustince, {it Intuitionistic fuzzy relations. effects of Atanassov's operators on the properties of intuitionistic fuzzy relations}, Mathware & Soft Computing, {bf2} (1995), 117 148. ## bibitem{6} R. Cignoli and F. Esteva, {it Commutative integral bounded residuated lattices with an added involution}, Annals of Pure and Applied Logic, {bf161} (2009), 150160. ## bibitem{7} P. Cintula, {it From fuzzy logic to fuzzy mathematics}, Ph.D. Thesis, Technical University, Prague, 2005. ## bibitem{8} C. Cornelis, G. Deschrijver and E. E. Kerre, {it Classification on intuitionistic fuzzy implicators: an algebraic approach}, In Proceedings of the FT & T' 02, Durham, North Carolina, 105108.##bibitem{9} G. Deschrijver, C. Cornelis and E. E. Kerre, {it Intuitionistic fuzzy connectives revisited}, In Proceedings of IPMU'02, July 15, 2002.## bibitem{10} J. A. Goguen, {it L  Fuzzy sets}, Journal of Math. Anal. And Applications, {bf18} (1967), 145173. ## bibitem{11} P. Hajek, {it Metamathematics of fuzzy logic}, Trends in Logic, Kluwer Acad.Publ., Drdrecht, {bf4} (1998). ## bibitem{12} P. Hajek, {it What is mathematical fuzzy logic?}, Fuzzy Sets and Systems, {bf157} ( 2006), 597603. ## bibitem{13} Y. Hong, X. Ruiping and F. Xianwen, {it Characterizing ordered semigroups by means of intuitionistic fuzzy bi ideals}, Mathware & Soft Computing, {bf14} (2007), 5766. ## bibitem{14} H. Ono, {it Subsructural logics and ResiduatedLatticesan introduction}, Trends in Logic, {bf20} (2003), 177212. ## bibitem{15} P. Smets and P. Magrez, {it Implications in fuzzy logic}, Int. J. of Approximate Reasoning, {bf1} (1987), 327347. ## bibitem{16} E. Szmidt and J. Kacprzyk, {it Intuitiinistic fuzzy sets in some medical applications, computational intelligence}, Theory and Applications, Lecture Notes in Computer Science, (2001), V. 2206/2001, 148151. ## bibitem{17} E. Szmidt and K. Marta, {it Atanassov's intuitionistic fuzzy sets in classification of imbalanced and overlapping classes}, Studies in Computational Intelligence (SCI), {bf109} (2008), 455 471. ## bibitem{18}A. Tepavcevic and M. G. Ranitovic, {it General form of lattice valued intuitionistic fuzzy sets}, Computational Intelligence, Theory and Applications, {bf14} (2006), 375381.## bibitem{19}A. Tepavcevic and T. Gerstenkorn, {it Lattice valued intuitionistic fuzzy sets}, Central European Journal of Mathematics, {bf2(3)} (2004), 388398.##bibitem{20} E. Turunen, {it Mathematics behind fuzzy logic}, Advances in Soft Computing, PhysicaVerlag, Heidelberg, 1999.##bibitem{21} I. K. Vlachos and G. D. Sergiadis, {it Towards Intuitionistic fuzzy image processing}, Proceedings of the 2005 International Conference on Computational Intelligence for Modelling, Control and Automation.## ## bibitem{22}M. Ward and R. P. Dilworth, {it"Residuated lattices", Trans. Amer. Math. Soc.}, {bf45} (1939), 33554, Reprinted in Bogart, K, Freese, R., and Kung, J., eds., 1990.## ## bibitem{23} L. A. Zadeh, {it Fuzzy sets}, Information and Control, {bf8(3)} (1965), 338353.## ## bibitem{24} L. A. Zadeh, {it Fuzzy sets, fuzzy logic and fuzzy systems}, Selected papers by Lotfi A. Zadeh, Editors George J. Klir and Bo Yuan, World Scientific, 1996.##]
FUZZY GRADE OF THE COMPLETE HYPERGROUPS
FUZZY GRADE OF THE COMPLETE HYPERGROUPS
2
2
This paper continues the study of the connection between hyper groups and fuzzy sets, investigating the length of the sequence of join spaces associated with a hypergroup. The classes of complete hypergroups and of 1hypergroups are considered and analyzed in this context. Finally, we give a method to construct a nite hypergroup with the strong fuzzy grade equal to a given natural number
1
This paper continues the study of the connection between hyper groups and fuzzy sets, investigating the length of the sequence of join spaces associated with a hypergroup. The classes of complete hypergroups and of 1hypergroups are considered and analyzed in this context. Finally, we give a method to construct a nite hypergroup with the strong fuzzy grade equal to a given natural number.
43
56
Carmen
Angheluta
Carmen
Angheluta
Faculty of Mathematics and Computer Science, University of
Bucharest, Str. Academiei 14, 010014 Bucharest, Romania
Faculty of Mathematics and Computer Science,
Romania
floryangheluta@yahoo.com
Irina
Cristea
Irina
Cristea
Center for Systems and Information Technologies, University of
Nova Gorica, Vipavska 13, SI5000, Nova Gorica, Slovenia
Center for Systems and Information Technologies,
Slovenia
irinacri@yahoo.co.uk
Complete hypergroup
Join space
Fuzzy set
Fuzzy grade
[[1] R. Ameri and M. M. Zahedi,typergroup and join space induced by a fuzzy subset, PU.M.A.,8(1997). ##[2] R. Ameri and O. R. Dehghan,Dimension of fuzzy hypervector spaces, Iranian Journal of Fuzzy Systems,8(5) (2011), 149166. ##[3] C. Angheluta and I. Cristea,On Atanassov's intuitionistic fuzzy grade of complete hypergroups, J. Mult.Valued Logic Soft Comput., 20 (2013), 5574. ##[4] P. Corsini,Prolegomena of Hypergroups Theory, Aviani Editore, Tricesimo, 1993. ##[5] P. Corsini,Join spaces, power sets, fuzzy sets , Proc. Fifth International Congress on A.H.A.,1993, Iasi, Romania, Hadronic Press, (1994), 4552. ##[6] P. Corsini,A new connection between hypergroups and fuzzy sets, Southeast Asian Bull.Math.,27(2003), 221229. ##[7] P. Corsini and I. Cristea,Fuzzy grade of i.p.s. hypergroups of order 7, Iranian Journal of Fuzzy Systems,1(2) (2004), 1532. ##[8] P. Corsini and I. Cristea,Fuzzy sets and non complete 1hypergroups, An. St. Univ. vidiusConstanta,13(1)(2005), 2754. ##[9] P. Corsini and V. Leoreanu,Join spaces associated with fuzzy sets, J. Combin. Inform. Syst.Sci.,20(14) 1995), 293303. ##[10] P. Corsini and V. Leoreanu,Applications of hyperstructure theory, Kluwer Academic Publishers, Advances in Mathematics, 2003. ##[11] P. Corsini and V. LeoreanuFotea,On the grade of a sequence of fuzzy sets and join spacesdetermined by a hypergraph, Southeast Asian Bull. Math., 34(2) (2010), 231242. ##[12] P. Corsini, V. LeoreanuFotea and A. Iranmanesh,On the sequence of hypergroups and membership functions determined by a hypergraph, J. Mult.Valued Logic Soft Comput., 14(6)(2008), 565577. ##[13] P. Corsini and R. Mahjoob,Multivalued functions, fuzzy subsets and join spaces, Ratio Math.,20(2010), 141. ##[14] I. Cristea,Complete hypergroups, 1Hypergroups and fuzzy sets, An. St. Univ. Ovidius Constanta,10(2) (2002), 2538. ##[15] I. Cristea,A property of the connection between fuzzy sets and hypergroupoids, Ital. J. Pure Appl. Math.,21 (2007), 7382. ##[16] I. Cristea,About the fuzzy grade of the direct product of two hypergroupoids, Iranian Journal of Fuzzy Systems,7(2) (2010), 95108. ##[17] I. Cristea and B. Davvaz,Atanassov's intuitionistic fuzzy grade of hypergroups, Information Sciences,180 (2010), 15061517. ##[18] I. Cristea, M. Jafarpour and S. S. Mousavi,On fuzzy preordered structures and (fuzzy) hyperstructures, Acta Math. Sin. (Engl. Ser.), 28(9) (2012), 17871798. ##[19] B. Davvaz and M. Karimian,On the ncomplete hypergroups, European J. Combin., 28(1)(2007), 8693.[20] B. Davvaz and M. Karimian,On the ncomplete hypergroups and K(H) hypergroups, ActaMath. Sin. (Engl. Ser.),24(11) (2008), 19011908. ##[21] B. Davvaz, E. Hassani Sadrabadi and I. Cristea,Atanassov's intuitionistic fuzzy grade of the complete hypergroups of order less than or equal to6, submitted. ##[22] B. Davvaz, E. Hassani Sadrabadi and I. Cristea,Atanassov's intuitionistic fuzzy grade of i.p.s. pergroups of order less than or equal to6, Iranian Journal of Fuzzy Systems,9(4)(2012), 7197. ##[23] B. Davvaz, E. Hassani Sadrabadi and I. Cristea,Atanassov's intuitionistic fuzzy grade of i.p.s.hypergroups of order7, J. Mult.Valued Logic Soft Comput., accepted. ##[24] H. Hedayati,Generalized fuzzy kideals of semirings with intervalvalued membership functions, Bull.Malays.Math.Sci.Soc., 32(3) (2009), 409424. ##[25] J. Jantosciak,Homomorphism, equivalence and reductions in hypergroups, Riv. Mat. PuraAppl.,9(1991), 2347. ##[26] V. Leoreanu Fotea,t hypermodules, Comput. Math. Appl., 57(3) (2009), 46647. ##[27] V. Leoreanu Fotea and B. Davvaz,Fuzzy hyperrings, Fuzzy Sets and Systems, 160(16)(2009), 23662378.[28] F. Marty,Sur une generalization de la notion de groupe, Eight Congress Math. Scandenaves,Stockholm, (1934), 4549. ##[29] R. Migliorato,On the complete hypergroups, Riv. Mat. Pura Appl., 14 (1994), 2131. ##[30] W. Prenowits and J. Jantosciak,Geometries and join spaces, J. Reind und Angew Math.,257(1972), 100128. ##[31] A. Rosenfeld,Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517. ##[32] M. Shabir, Y. B. Yun and M. Bano,On prime fuzzy biideals of semigroups, Iranian Journal of Fuzzy Systems,7(3) (2010), 115128. ##[33] M. Stefanescu and I. Cristeon the fuzzy grade of the hypergroup, Fuzzy Sets and System159((2008), 10971106. ##[34] Y. Yin, J. Zhan and X. Huang,A new way to fuzzy hideals of hemirings, Iranian Journal of Fuzzy Systems,8(5) (2011), 81101. ##[35] L. A. Zadeh,Fuzzy sets, Information and Control, 8 (1965), 338353. ##[36] J. Zhan and B. Davvaz,Study of fuzzy algebraic hypersystems from a general viewpoint, Int.J. Fuzzy Syst.,12(1) (2010), 7379. ##]
DEFUZZIFICATION METHOD FOR RANKING FUZZY
NUMBERS BASED ON CENTER OF GRAVITY
DEFUZZIFICATION METHOD FOR RANKING FUZZY
NUMBERS BASED ON CENTER OF GRAVITY
2
2
Ranking fuzzy numbers plays a very important role in decision making and some other fuzzy application systems. Many different methods have been proposed to deal with ranking fuzzy numbers. Constructing ranking indexes based on the centroid of fuzzy numbers is an important case. But some weaknesses are found in these indexes. The purpose of this paper is to give a new ranking index to rank various fuzzy numbers effectively. Finally, several numerical examples following the procedure indicate the ranking results to be valid.
1
Ranking fuzzy numbers plays a very important role in decision making and some other fuzzy application systems. Many different methods have been proposed to deal with ranking fuzzy numbers. Constructing ranking indexes based on the centroid of fuzzy numbers is an important case. But some weaknesses are found in these indexes. The purpose of this paper is to give a new ranking index to rank various fuzzy numbers effectively. Finally, several numerical examples following the procedure indicate the ranking results to be valid.
57
67
Tofigh
Allahviranloo
Tofigh
Allahviranloo
Department of Mathematics, Science and Research Branch,
Islamic Azad University, Tehran, Iran
Department of Mathematics, Science and Research
Iran
allahviranloo@yahoo.com
Rahim
Saneifard
Rahim
Saneifard
Department of Mathematics, Science and Research Branch, Islamic
Azad University, Tehran, Iran
Department of Mathematics, Science and Research
Iran
srsaneeifard@yahoo.com
Ranking
Fuzzy numbers
Centroid point
Defuzzification
[[1] S. Abbasbandy and B. Asady,Ranking of fuzzy numbers by sign distance, Information Sciences,176(2006), 24052416. ##[2] T. Allahviranloo and M. Afshar,Numerical methods for fuzzy linear partial differential equationsunder new ition for derivative, Iranian Journal of Fuzzy Systems, 7 (2010), 3350. ##[3] T. Allahviranloo, S. Abbasbandy and R. Saneifard,A method for ranking fuzzy numbers using new weighted distance , Mathematical and Computational Applications, 2 (2011), 359369. ##[4] T. Allahviranloo, S. Abbasbandy and R. Saneifard,An approximation approach for ranking fuzzy numbers based on weighted intervalvalue, Mathematical and Computational Applications,3(2011), 588597. ##[5] I. Altun,Some fixed point theorems for single and multi valued mappings on ordered nonarchimedean fuzzy metric spaces , Iranian Journal of Fuzzy Systems, 1 (2010), 9196. ##[6] J. F. Baldwin and N. C. Guild,Comparison of fuzzy sets on the same decision space, Fuzzy Sets and Systems,2 (1979), 213231. ##[7] S. M. Bass and H. Kwakernaak,Rating and ranking of multiple aspect alternatives using fuzzy sets Automatica, 13 (1977), 4758. ##[8] W. Chang,Ranking of fuzzy utilities with triangular membership function , Proceeding of the International conference on policy analysis information system,105 (1981), 263272. ##[9] S. H. Chen,Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets and Systems,17 985), 113129. ##[10] L. H. Chen and H. W. Lu,An approximate approach for ranking fuzzy numbers based on left and right dominance, Comput Math Appl., 41(2001), 15891602. ##[11] C. H. Cheng,A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems,95 998), 307317. ##[12] C. H. Cheng,Ranking alternatives with fuzzy weights using maximizing set and minimizing set, Fuzzy Sets and Systems, 105 (1999), 365375. ##[13] T. Chu and C. Tsao,Ranking fuzzy numbers with an area between the centroid point andoriginal point, Comput. Math. Appl., 43 (2002), 111117. ##[14] D. Dubois and H. Prade,Operation on fuzzy numbers, Internat. J. Syst. Sci., 9 (1978),613626. ##[15] R. Ezatti and R. Saneifard,A new approach for ranking of fuzzy numbers with continuousweighted quasiarithmetic means, Mathematical Sciences, 4 (2010), 143158. ##[16] R. Ezzati and R. Saneifard,Defuzzification through a novel approach, Proc.10th Iranian Conference on Fuzzy Systems, (2010), 343348. ##[17] E. C. Lee and R. L. Li,Comparison of fuzzy numbers based on the probability measure of fuzzy events , Comput Math Appl., 105 (1988), 887896. ##[18] F. Merghadi and A. Alioughe,A related fixed point theorem in n fuzzy metric spaces, Iranian Journal of Fuzzy Systems,3 (2010), 7386. ##[19] E. Pasha, A. Saiedfar and B. Asady,The percentiles of fuzzy numbers and their applications,Iranian Journal of Fuzzy Systems,6 (2009), 2744. ##[20] R. Saneiafard,Some properties of neural networks in designing fuzzy systems, Neural Computing and Applications, doi:10.1007/s0052101107771, 2011. ## [21] R. Saneifard,A method for defuzzification by weighted distance, Int. J. Industrial Mathematics,3(2010), 209217. ##[22] R. Saneiafrd,Designing an algorithm for evaluating decisionmaking units based on neuralweighted function, Neural Computing and Applications, doi:10.1007/s0052101208785, 2012. ##[23] R. Saneifard,Ranking LR fuzzy numbers with weighted averaging based on levels, Int. J.Industrial Mathematics,2(2009), 163173. ##[24] Rahim Saneiafrd and Rasoul Saneifard,A new effect of radius of gyration with neural networks,Neural Computing and Applications, doi:10.1007/s0052101210672, 2012. ##[25] R. Saneifard and R. Ezzati,Defuzzification through a biSymmetrical weighted function, Aust.J. Basic appl. sci.,10 (2009), 49764984. ##[26] R. Saneifard and T. Allahviranloo,A comparative study of ranking fuzzy numbers based on regular weighted function, Fuzzy Information and Engineering, 3 (2012), 235248. ##[27] R. Saneifard, T. Allahviranloo, F. Hosseinzadeh and N. Mikaeilvand,Euclidean ranking DMUs with fuzzy data in DEA, Applied Mathematical Sciences, 60 (2007), 29892998. ##[28] T. Y. Tseng and C. M. Klein,New algorithm for the ranking procedure in fuzzy decision making, IEEE Trans Syst Man Cybernet SMC., 19 (1989), 12891296. ##[29] X. Wang and E. E. Kerre,Reasonable properties for the ordering of fuzzy quantities (I),Fuzzy Sets and Systems,118 (2001), 378405. ##[30] Y. M. Wang, J. B. Yang, D. L. Xu and K. S. Chin,On the centroids of fuzzy numbers, FuzzySets and Systems,157 (2006), 919926. ##[31] R. R. Yager and D. P. Filev,On the issue of defuzzification and selection based on a fuzzy set, Fuzzy Sets and Systems, 55 (2006), 255272. ##[32] J. Zhao and Q. S. Liu,Ranking fuzzy numbers based on the centroid of fuzzy numbers, Fuzzy Systems and Mathematics,22 (2008), 142146. ##]
A FUZZY DIFFERENCE BASED EDGE DETECTOR
A FUZZY DIFFERENCE BASED EDGE DETECTOR
2
2
In this paper, a new algorithm for edge detection based on fuzzyconcept is suggested. The proposed approach defines dynamic membershipfunctions for different groups of pixels in a 3 by 3 neighborhood of the centralpixel. Then, fuzzy distance and cut theory are applied to detect the edgemap by following a simple heuristic thresholding rule to produce a thin edgeimage. A large number of experiments are employed to confirm the robustnessof the proposed algorithm. In the experiments different cases such as normalimages, images corrupted by Gaussian noise, and uneven lightening imagesare involved. The results obtained are compared with some famous algorithmssuch as Canny and Sobel operators, a competitive fuzzy edge detector, and astatistical based edge detector. The visual and quantitative comparisons showthe effectiveness of the proposed algorithm even for those images that werecorrupted by strong noise.
1
In this paper, a new algorithm for edge detection based on fuzzy concept is suggested. The proposed approach de nes dynamic membership functions for dierent groups of pixels in a 3 by 3 neighborhood of the central pixel. Then, fuzzy distance and cut theory are applied to detect the edge map by following a simple heuristic thresholding rule to produce a thin edge image. A large number of experiments are employed to con rm the robustness of the proposed algorithm. In the experiments dierent cases such as normal images, images corrupted by Gaussian noise, and uneven lightening images are involved. The results obtained are compared with some famous algorithms such as Canny and Sobel operators, a competitive fuzzy edge detector, and a statistical based edge detector. The visual and quantitative comparisons show the eectiveness of the proposed algorithm even for those images that were corrupted by strong noise.
69
85
M. A.
Nikouei Mahani
M. A.
Nikouei Mahani
Electrical Engineering Department, Shahid Bahonar Univer
sity of Kerman, Kerman, Iran
Electrical Engineering Department, Shahid
Iran
nikouei@mahani.info
Mohamad
Koohi Moghadam
Mohamad
Koohi Moghadam
School of Computer Engineering, Iran University of
Science and Technology, Tehran, Iran
School of Computer Engineering, Iran University
Iran
m koohi m@comp.iust.ac.ir
Hosein
Nezamabadipour
Hosein
Nezamabadipour
Electrical Engineering Department, Shahid Bahonar Uni
versity of Kerman, Kerman, Iran
Electrical Engineering Department, Shahid
Iran
nezam@mail.uk.ac.ir
Edge detection
Fuzzy edge detection
Dynamic membership function
Fuzzy dierence
Noisy images
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Hildreth, Theory of edge detection, Proceedings Royal Soc.London, 207##(1980), 187217.##[20] A. Maturo, On some structures of fuzzy numbers, Iranian Journal of Fuzzy Systems, 6 (2009),##[21] R. MedinaCarnicer, A. CarmonaPoyato, R. MuozSalinas and F. J. MadridCuevas, Deter##mining hysteresis thresholds for edge detection by combiningthe advantages and disadvantages##of thresholding methods, IEEE Transactionson Image Processing, 19(1) (2010), 165173.##[22] R. MedinaCarnicer, F. MadridCuevas, A. CarmonaPoyato and R. M. noz Salinas, On##candidates selection for hysteresis thresholds in edge detection, PatternRecognition, 42(7)##(2009), 12841296.##[23] R. MedinaCarnicer and F. MadridCuevas, Unimodal thresholding for edgeDetection, Pattern##Recognition, 41(7) (2008), 23372346.##[24] R. MedinaCarnicer, F. MadridCuevas, R. MuozSalinas and A. CarmonaPoyato, Solving##the process of hysteresis without determining the optimal thresholds, Pattern Recognition,##43(4) (2010), 12241232.##[25] H. Meng, M. Freeman, N. Pears and C. Bailey, Realtime human action recognition on an em##bedded, recongurable video processing architecture, Journal of RealTime Image Processing,##3 (2008), 163176.##[26] S. Morillas, V. Gregori and Antonio. Hervs, Fuzzy peer groups for reducing mixed gaussian##impulse noise from color images, IEEE Transaction on Image Processing, 18(7) (2009),##14521466.##[27] J. MuseviNiya and A. Aghagolzadeh, Adaptive directional waveletbased edge detection, 2nd##International Symposium on Telecommunications (IST2003), Isfahan, Iran, (2003), 191195.##[28] H. Nezamabadipour, S. Saryazdi and E. Rashedi, Edge detection using ant algorithms, Soft##Computing, 10 (2005), 623628.##[29] E. Pasha, A. Saiedifar and B. Asady, The percentiles of fuzzy numbers and their applications,##Iranian Journal of Fuzzy Systems, 6 (2009), 2744.##[30] W. K. Pratt, Digital image processing, John Wiley and Sons, 2001.##[31] G. Roy Jun and W. Voxman, Topological properties of fuzzy numbers, Fuzzy Sets and Systems,##10(13) (1983), 8799.##[32] X. Ruoning, A linear regression model in fuzzy environment, Adv. Modelling Simulation, 27##(1991), 3140.##[33] F. Russo and G. Ramponi, Edge extraction by FIRE operators , in IEEE World Congress on##Computational Intelligence, 1 (1994), 249253.##[34] J. I. Siddigue and K. E.Barner, Waveletbased multiresolution edge detection utilizing gray##level edge maps, International Conference on Image Processing (ICIP 98), (1998), 550554.##[35] B. Sridevi and R. Nadarajan, Fuzzy similarity measure for generalized fuzzy numbers, Inter##national Journal of Open Problems in Computer Science and Mathematics, 2 (2009), 240253.##[36] P. Terry and D. Vu, Edge detection using neural networks, In IEEE Proceedings of 27th##Asilomar Conference on Signals, Systems and Computers, (1993), 391395.##[37] V. Torre and T. Poggio, On edge detection, Massachusetts Institute of TechnologyArticial##Intelligence Laboratory, 1984.##[38] D. Van De Ville, M. Nachtegael, D. Van Der Weken, E. Kerre, W. Philips and I. Lemahieu,##Noise reduction by fuzzy image ltering, IEEE Transactions on Fuzzy Systems, 11 (2003),##[39] J. Wu, Z. Yin and Y. Xiong, The fast multilevel fuzzy edge detection of blurry images, IEEE##Signal Processing Letters, 14 (2007), 344347. ##[40] R. Xu and C. Li, Multidimensional leastsquares tting with a fuzzy model, Fuzzy Sets and##Systems, 119 (2001), 215223.##[41] F. Yang, S. Wan and Y. Chang, Improved method for gradientthreshold edge detector based##on HVS, Lecture Notes in Computer Science, 3801 (2005), 10511056.##[42] S. Yi, D. Labate, G. R. Easley and H. Krim, A shearlet approach to edge analysis and##detection, Trans. Image Proc,18(15) (2009), 10577149.##[43] X. Zong and W. Liu, Fuzzy edge detection based on wavelets transform, Machine Learning##and Cybernetics, International Conference, (2008), 28692873.##]
SYMMETRIC TRIANGULAR AND INTERVAL
APPROXIMATIONS OF FUZZY SOLUTION TO
LINEAR FREDHOLM FUZZY INTEGRAL
EQUATIONS OF THE SECOND KIND
SYMMETRIC TRIANGULAR AND INTERVAL
APPROXIMATIONS OF FUZZY SOLUTION TO
LINEAR FREDHOLM FUZZY INTEGRAL
EQUATIONS OF THE SECOND KIND
2
2
In this paper a linear Fuzzy Fredholm Integral Equation(FFIE) with arbitrary Fuzzy Function input and symmetric triangular (Fuzzy Interval) output is considered. For each variable, output is the nearest triangular fuzzy number (fuzzy interval) to the exact fuzzy solution of (FFIE).
1
In this paper a linear Fuzzy Fredholm Integral Equation(FFIE) with arbitrary Fuzzy Function input and symmetric triangular (Fuzzy Interval) output is considered. For each variable, output is the nearest triangular fuzzy number (fuzzy interval) to the exact fuzzy solution of (FFIE).
87
99
Majid
Alavi
Majid
Alavi
Department of Mathematics, Islamic Azad University, Arak Branch,
Arak, Iran
Department of Mathematics, Islamic Azad University
Iran
Malavi@Iauarak.ac.ir
Babak
Asady
Babak
Asady
Department of Mathematics, Islamic Azad University, Arak Branch,
Arak, Iran
Department of Mathematics, Islamic Azad University
Iran
Basay@Iauarak.ac.ir
Fuzzy number
Expected interval
Fuzzy integral equations
Symmetric fuzzy number
Nystrom method
[[1] S. Abbasbandy and T. Allahviranloo, Numerical solution of fuzzy dierential equation by##rungeKutta method, Nonlinear Stud, 11 (2004), 117129.##[2] S. Abbasbandy and B. Asady, The nearest trapezoidal fuzzy number to a fuzzy quantity, Appl##Math Comput, 156 (2004), 381386.##[3] S. Abbasbandy, E. Babolian and M. Alavi, Numerical method for solving linear Fredholm##fuzzy integral equations of the second kind, Chaos, Soliton and Fractals, 31 (2007), 138146.##[4] T. Allahviranloo and M. Otadi, Gaussian quadratures for approximate of fuzzy integrals,##Applied Mathematics and Computation, 170 (2005), 874885.##[5] K. Atkinson, A survey of numerical methods for solving nonlinear integral equations, J. of##Integral Equat. and Appl., 4 (1992), 1546.##[6] E. Babolian, H. Sadeghi Goghary and S. Abbasbandy, Numerical solution of linear Fredholm##fuzzy integral equations of the second kind by Adomian method, Applied Mathematics and##Computation, 161 (2005), 733744##[7] E. Babolian, H. Sadeghi and S. Javadi, Numerically solution of fuzzy dierential equations##by Adomian method, Appl. Math. Comput., 149 (2004), 547557.##[8] K. Balachandran and K. Kanagarajan, Existence of solutions of general nonlinear fuzzy##VolteraFeredholm integral equations, J. Appl. Math. Stochastic Anal, 3 (2005), 333343.##[9] K. Balachandran and P. Prakash, Existence of solution of nonlinear fuzzy VolteraFeredholm##integral equations, Indian J. Pure Appl. Math, 333 (2002), 329343.##[10] B. Bede and S. G. Gal, Quadrature rules for integral of fuzzynumbervalued functions, Fuzzy##Sets and Systems, 145 (2004), 359380.##[11] A. M. Bica, Error estimation in the Approximation of the solution of nonlinear fuzzy Fered##holm integral equations, Information Sciences, 174 (2008), 12791292.##[12] J. J. Buckley and T. Furing, Fuzzy integral equations, J. Fuzzy Math, 10 (2002), 10111024.##[13] S. S. L. Chang and L. Zadeh, On fuzzy mapping and control, IEEE Trans System Man##Cybernet, 2 (1972), 3034.##[14] W. Cingxin and M. Ming, On embedding problem of fuzzy number spaces: part I, Fuzzy Sets##and Systems, 44 (1991), 3338.##[15] W. Cingxin and M. Ming, On embedding problem of fuzzy number spaces: part II, Fuzzy Sets##and Systems, 45 (1992), 189202.##[16] W. Cingxin and M. Ming, On embedding problem of fuzzy number spaces: part III, Fuzzy##Sets and Systems, 44 (1992), 281286.##[17] W. Congxin and M. Ming, On the integrals. series and integral equations of fuzzy setvalued##functions, J. Harbin Inst Technol, 21 (1990), 1119.##[18] L. M. Delves and J. L. Mohemed, Computational methods for bntegral equations, Cambridge##University Press, Cambridge, 1985.##[19] K. Deimling, Multivalued dierential equations, Walter de Gruyter, New York, 1992.##[20] P. Diamond, Stability and periodicity in fuzzy dierential equations, IEEE Trans. Fuzzy Syst,##8 (2000), 583590.##[21] D. Dubois and H. prade, Towards fuzzy dierential calculus, Fuzzy Sets and System, 8 (1982),##[22] M. Friedman, M. Ma and A. Kandel, Numerical solutions of fuzzy dierential and integral##equations, Fuzzy Sets and Systems, 106 (1999), 3548.##[23] M. Fridman, M. Ma and A. Kandel, On fuzzy integral equations, Fundam. Inform, 37 (1999),##[24] M. Fridman, M. Ming and A. Kandel, Solution to fuzzy integral equations with arbitrary##kernels, Internat. J. Approx. Reason, 20 (1999), 249262.##[25] D. N. Georgion and I. E. Kougias, Bounded solutions for fuzzy integral equations, Int. j.##Math.sci, 312 (2002), 109 114.##[26] D. N. Georgion and I. E. Kougias, On fuzzy fredholm and Voltera integral equations, J. Fuzzy##Math, 94 (2001), 943951. ##[27] R. Goetschel and W. Voxman, Elementary calculus, Fuzzy Sets and Systems, 18 (1986),##[28] P. Grzegorzewski, Metricsand orders in space of fuzzy numbers, Fuzzy Sets and Systems, 97##(1987), 8394##[29] P. Grzegorzewski, Nearst interval approximation of a fuzzy number, Fuzzy Sets ans Systems,##130 (2002), 321330.##[30] P. Grzegorzewski, Trapezoidal approximations of fuzzy numbers preserving the expected in##terval algorithms and properties, Fuzzy Sets and Systems, 159 (2008), 13541364.##[31] H. Hochstadt, Integral equations, Wiley, New York, 1973.##[32] O. Kaleva, Fuzzy dierential equations, Fuzzy Sets and Systems, 24 (1987), 301317.##[33] W. V. Lovitt, Linear integral equation, Dover, New York, 1950.##[34] M. Ma, M. Friedman and A. Kandel, Numerical solution of fuzzy dierential equations, Fuzzy##Sets and Systems, 105 (1999), 133138.##[35] M. Matloka, On fuzzy integrals Proc, 2nd Polish Symp. on Interval and Fuzzy Mathematics,##Politechnika Poznnsk, (1987), 167170.##[36] A. Maturo, On some structure of fuzzy numbers, Iranian Journal of Fuzzy Systems, 6 (2009),##[37] A. Molabahrami, A. Shidfar and A. Ghyasi, An analytical method for solving linear Feredholm##fuzzy integral equations of the second kind, Computers and Mathematics with Applications,##61 (2011), 27542761.##[38] J. Mordeson and W. Newman, Fuzzy integral equations, Information Sciences, 814 (1995),##[39] J. J. Nieto and R. RodriguezLopaz, Bounded solution for fuzzy dierential and integral##equations, Choas Solitons and Fractals, 275 (2006), 13761386.##[40] N. Parandin and M. A. Fariborzi Araghi, The approximate solution of linear fuzzy Fered##holm integral equations of the second kind by using iterative interpolation, Word Academy##of science, Engineering and Technology, 49 (2009), 425431.##[41] E. Pasha, A. saiedifar and B. Asady, The presentation on fuzzy numbers and their applica##tions, Iranian Journal of Fuzzy Systems, 6 (2009), 2744.##[42] J. Y. Perk and J. U. Jeong, On the existence and uniquenes of solutions of fuzzy Voltera##Feredholm, integral equations, Fuzzy Sets and Systems, 115 (2000), 425431.##[43] O. Solaymani and A. Vahidian kamyad, Modied Kstep method for solving fuzzy initial value##problems, Iranian Journal of Fuzzy Systems, 8 (2011), 4959.##[44] J. Vrba, A note on inverse in arithwith fuzzy numbers, Fuzzy Sets and Systems, 50 (1992),##]
THE RELATIONSHIP BETWEEN LFUZZY PROXIMITIES AND
LFUZZY QUASIUNIFORMITIES
THE RELATIONSHIP BETWEEN LFUZZY PROXIMITIES AND
LFUZZY QUASIUNIFORMITIES
2
2
In this paper, we investigate the Lfuzzy proximities and the relationships betweenLfuzzy topologies, Lfuzzy topogenous order and Lfuzzy uniformity. First, we show that the category offuzzy topological spaces can be embedded in the category of Lfuzzy quasiproximity spaces as a coreective full subcategory. Second, we show that the category of L fuzzy proximity spaces is isomorphic to the category of Lfuzzy topogenous order spaces. Finally,we obtain that the category of Lfuzzy proximity spaces can be embeddedin the category of Lfuzzy uniform spaces as a bireective full subcategory.
1
In this paper, we investigate the Lfuzzy proximities and the relationships betweenLfuzzy topologies, Lfuzzy topogenous order and Lfuzzy uniformity. First, we show that the category offuzzy topological spaces can be embedded in the category of Lfuzzy quasiproximity spaces as a coreective full subcategory. Second, we show that the category of L fuzzy proximity spaces is isomorphic to the category of Lfuzzy topogenous order spaces. Finally,we obtain that the category of Lfuzzy proximity spaces can be embeddedin the category of Lfuzzy uniform spaces as a bireective full subcategory.
101
111
EunSeok
Kim
EunSeok
Kim
Department of Mathematics, Chonnam National University, 300 Yongbong
dong, Bukgu, 500757, Gwangju, Korea
Department of Mathematics, Chonnam National
Korea
manmunje@hanmail.net
SeungHo
Ahn
SeungHo
Ahn
Department of Mathematics, Chonnam National University, 300 Yongbong
dong, Bukgu, 500757, GwangJu, Korea
Department of Mathematics, Chonnam National
Korea
shahn@chonnam.ac.kr
Dae Heui
Park
Dae Heui
Park
Department of Mathematics, Chonnam National University, 300 Yongbong
dong, Bukgu, 500757, GwangJu, Korea
Department of Mathematics, Chonnam National
Korea
dhpark3331@chonnam.ac.kr
LFuzzy topology
Lfuzzy proximity
Lfuzzy uniformity
Lfuzzy topogenous order
Fuzzy remote neighborhood systems
[[1] J. Adamek J, H. Herrlich and G. E. Strecker, Abstract and concrete categories, J. Wiley and##Sons, New York, 1990.##[2] G. Artico and R. Moresco, Fuzzy proximities and totally bounded fuzzy uniformities, J. Math.##Anal. Appl., 9 (1984), 3201337.##[3] G. Artico and R. Moresco, Fuzzy proximities compatible with Lowen uniformities, Fuzzy Sets##and Systems, 21 (1987), 8599.##[4] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182190.##[5] A. Csaszar, Foundations of general topology, Pergamon Press, 1963.##[6] D. Dzajanbajev and A. Sostak, On a fuzzy uniform structure, Acta Comm. Univ. Tartu, 940##(1992), 3136. ##[7] J. M. Fang and Y. L. Yue, Base and subbase in Ifuzzy topological spaces, J. Math. Res.##Expositon, 26 (2006), 8995.##[8] M. H. Ghanim, O. A. Tantawy and F. M. Selim, On Sfuzzy quasiproximity spaces, Fuzzy##Sets and Systems, 109 (2000), 285290.##[9] F. Gierz, et al., A compendium of continuous lattices, Berlin: Springer Verlag, 1980.##[10] J. Gutierrez Garcia, M. A. de Prada Vicente and A. Sostak, A unied approach to the##concept of fuzzy Luniform space, In: Topological and Algebraic Strutures in Fuzzy Sets,##A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, S. E. Rodabaugh##and E. P. Klement eds. Kluwer Acad. Publ., Dordrecht, Boston, London, Chapter 3, (2003),##[11] U. Hohle, Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl., 78 (1980),##[12] B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl., 58 (1977), 557571.##[13] A. K. Katsaras, On fuzzy syntopogenous structures, J. Math. Anal. Appl., 99 (1983), 219236.##[14] A. K. Katsaras, On fuzzy uniform spaces, J. Math. Anal. Appl., 101 (1984), 97114.##[15] T. Kubiak, On fuzzy topologies, PhD Thesis, Adam Mickiewicz, Poznan (Poland), 1985.##[16] W. J. Liu, Fuzzy proximity spaces redened, Fuzzy Sets and Systems, 15 (1985), 241248.##[17] Y. M. Liu and M. K. Luo, Fuzzy topology, Singapore: World Scientic Press, 1997.##[18] S. Markin and A.Sostak , Another approach to the concept of a fuzzy proximity, Suppl. Rend.##Mat. Palermo, 29 (1992), 530551.##[19] G. Preuss, Theory of topological structures: an approch to categorical topology, D. Reidel##Publishing Company, 1987.##[20] F. G. Shi, The category of pointwise Sproximity spaces, Fuzzy Sets and Systems, 152 (2005),##[21] A. Sostak, On a fuzzy topological structure, Rend. Cire. Matem. Palermo, Ser. II, 11 (1985),##[22] A. Sostak, Basic structures of fuzzy topology, J. Math. Sciences, 78 (1996), 662697.##[23] A. Sostak, Fuzzy syntopogenous structures, Quaestiones Math., 20 (1997), 431461.##[24] Y. Yue and J. Fang, Categories isomorphic to the KubiakSostak extension of TML, Fuzzy##Sets and Systems, 157 (2006), 832842.##[25] Y. Yue and F. G. Shi, Generalized quasiproximities, Fuzzy Sets and Systems, 158 (2007),##]
Uniquely Remotal Sets in $c_0$sums and $ell^infty$sums of Fuzzy Normed Spaces
Uniquely Remotal Sets in $c_0$sums and $ell^infty$sums of Fuzzy Normed Spaces
2
2
Let $(X, N)$ be a fuzzy normed space and $A$ be a fuzzy boundedsubset of $X$. We define fuzzy $ell^infty$sums and fuzzy $c_0$sums offuzzy normed spaces. Then we will show that in these spaces, all fuzzyuniquely remotal sets are singletons.
1
Let $(X, N)$ be a fuzzy normed space and $A$ be a fuzzy boundedsubset of $X$. We define fuzzy $ell^infty$sums and fuzzy $c_0$sums offuzzy normed spaces. Then we will show that in these spaces, all fuzzyuniquely remotal sets are singletons.
113
122
Alireza
Kamel Mirmostafaee
Alireza
Kamel Mirmostafaee
Center of Excellence in Analysis on Algebraic Struc
tures, Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box
1159, Mashhad 91775, Mashhad, Iran
Center of Excellence in Analysis on Algebraic
Iran
mirmostafaei@ferdowsi.um.ac.ir
Madjid
Mirzavaziri
Madjid
Mirzavaziri
Center of Excellence in Analysis on Algebraic Structures, De
partment of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mash
had 91775, Mashhad, Iran
Center of Excellence in Analysis on Algebraic
Iran
mirzavaziri@gmail.com
Fuzzy normed spaces
Fuzzy remotal set
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(IC)LMFUZZY TOPOLOGICAL SPACES
(IC)LMFUZZY TOPOLOGICAL SPACES
2
2
The aim of the present paper is to define and study (IC)$LM$fuzzytopological spaces, a generalization of (weakly) induced $LM$fuzzytopological spaces. We discuss the basic properties of(IC)$LM$fuzzy topological spaces, and introduce the notions ofinterior (IC)fication and exterior (IC)fication of $LM$fuzzytopologies and prove that {bf ICLMFTop} (the category of(IC)$LM$fuzzy topological spaces) is an isomorphismclosed fullproper subcategory of {bf LMFTop} (the category of $LM$fuzzytopological spaces) and {bf ICLMFTop} is a simultaneouslybireflective and bicoreflective full subcategory of {bf LMFTop}.
1
The aim of the present paper is to define and study (IC)$LM$fuzzytopological spaces, a generalization of (weakly) induced $LM$fuzzytopological spaces. We discuss the basic properties of(IC)$LM$fuzzy topological spaces, and introduce the notions ofinterior (IC)fication and exterior (IC)fication of $LM$fuzzytopologies and prove that {bf ICLMFTop} (the category of(IC)$LM$fuzzy topological spaces) is an isomorphismclosed fullproper subcategory of {bf LMFTop} (the category of $LM$fuzzytopological spaces) and {bf ICLMFTop} is a simultaneouslybireflective and bicoreflective full subcategory of {bf LMFTop}.
123
133
HaiYang
Li
HaiYang
Li
School of Science, Xi'an Polytechnic University, Xi'an 710048, P. R.
China
School of Science, Xi'an Polytechnic University,
China
fplihiayang@126.com
LMfuzzy topology
(IC) LMfuzzy topological spaces
(IC)fication of LMfuzzy topology
Category
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