2017
14
3
0
0
Cover vol. 14, no. 3, June 2017
2
2
1

0
0
ROBUST FUZZY CONTROL DESIGN USING GENETIC ALGORITHM OPTIMIZATION APPROACH: CASE STUDY OF SPARK IGNITION ENGINE TORQUE CONTROL
2
2
In the case of widelyuncertain nonlinear system control design, it was very difficult to design a single controller to overcome control design specifications in all of its dynamical characteristics uncertainties. To resolve these problems, a new design method of robust fuzzy control proposed. The solution offered was by creating multiple softswitching with TakagiSugeno fuzzy model for optimal solution control at all operating points that generate uncertainties. Optimal solution control at each operating point was calculated using genetic algorithm. A case study of engine torque control of spark ignition engine model was used to prove this new method of robust fuzzy control design. From the simulation results, it can be concluded that the controller operates very well for a wide uncertainty.
1

1
13


Aris
Triwiyatno
Department of Electrical Engineering, Diponegoro University,
Semarang, Indonesia
Department of Electrical Engineering, Diponegoro
Indonesia
aristriwiyatno@yahoo.com


Sumardi
Sumardi
Department of Electrical Engineering, Diponegoro University,
Semarang, Indonesia
Department of Electrical Engineering, Diponegoro
Indonesia


Esa
Apriaskar
Department of Electrical Engineering, Diponegoro University, Se
marang, Indonesia
Department of Electrical Engineering, Diponegoro
Indonesia
esaindo@gmail.com
Fuzzy logic
Robust fuzzy control
TakagiSugeno fuzzy model
Genetic Algorithm
Engine torque control
Spark ignition engine
[[1] M. Athans, A tutorial on the LQG/LTR method, in 1986 American Control Conference,##0582(606) (1986), 1289{1296.##[2] S. G. Cao, N. W. Rees and G. Feng, Analysis and design of fuzzy control systems using##dynamic fuzzystate space models, IEEE Trans. Fuzzy Syst., 7(2) (1999), 192200.##[3] M. A. Denai and S. A. Attia, Robust state feedback control of eld oriented induction motor,##in Advanced Motion Control, 2000, (2000), 281{286.##[4] P. Eklund and J. Zhou, Comparison of learning strategies for adaptation of fuzzy controller##parameters, Fuzzy Sets Syst., 106(3) (1999), 321{333.##[5] D. E. Goldberg, Genetic algorithm in search, optimization, and machine learning. Addison##Wesley Publ. Co, (1989), 60{85.##[6] N. Heintz, M. Mews, G. Stier, A. Beaumont and A. Noble, An approach to torquebased##engine management systems, Soc. Automot. Eng., (2001010269) (2001), 95102.##[7] D. M. Lamberson, Torque management of gasoline engines, Thesis, University of California##at Berkeley, (2003), 2740.##[8] Q. Li and T. Chai, Fuzzy adaptive control for a class of nonlinear systems, Fuzzy Sets Syst.,##101(1) (1999), 31{39.##[9] T. Li, S. Tong and G. Feng, A novel robust adaptivefuzzytracking control for a class of##nonlinear multiinput/multioutput systems, Fuzzy Syst. IEEE Trans., 18(1) (2010), 150{##[10] F. Mei, M. Zhihong, X. Yu and T. Nguyen, A robust tracking control scheme for a class of##nonlinear systems with fuzzy nominal models, Appl. Math. Comput. Sci., 8(1:19) (1998),##[11] Y. Shi and M. Mizumoto, A new approach of neurofuzzy learning algorithm for tuning fuzzy##rules, Fuzzy Sets Syst., 112(1) (2000), 99{116.##[12] A. Stefanopoulou, Modeling and control of advanced technology engines, Thesis, The Univer##sity of Michigan, (1996), 1128.##[13] T. Takagi and M. Sugeno, Fuzzy identication of systems and its applications to modeling##and control, IEEE Trans. Syst. Man Cybern., (1) (1985), 116132.##[14] A. Triwiyatno, M. Nuh, A. Santoso and I. N. Sutantra, Engine torque control of spark ignition##engine using robust fuzzy logic control, IACSIT Int. J. Eng. Technol., 3(4) (2011), 352{358.##[15] A. Triwiyatno, M. Nuh, A. Santoso and I. N. Sutantra, Engine torque control of si engine##using linear quadratic integral tracking ( LQIT ) Optimal Control, IPTEK, 22(4) (2011),##[16] A. Triwiyatno, M. Nuh, A. Santoso and I. N. Sutantra, TS fuzzy model design for engine##torque control system of spark ignition engine, Transmisi, 11(4) (2009), 177182.##[17] A. Triwiyatno, M. Nuh, A. Santoso and I. N. Sutantra, A new method of robust fuzzy control?:##case study of engine torque control of spark ignition engine, Int. J. Acad. Res., 3(5) (2011),##]
TIMEVARYING FUZZY SETS BASED ON A GAUSSIAN MEMBERSHIP FUNCTIONS FOR DEVELOPING FUZZY CONTROLLER
2
2
The paper presents a novel type of fuzzy sets, called timeVarying Fuzzy Sets (VFS). These fuzzy sets are based on the Gaussian membership functions, they are depended on the error and they are characterized by the displacement of the kernels to both right and left side of the universe of discourse, the two extremes kernels of the universe are fixed for all time. In this work we focus only on the midpoint movement of the universe, all points of supports (kernels) are shifted by the same distance and in the same direction excepted the two extremes points of supports are always fixed for all computation time. To show the effectiveness of this approach we used these VFS to develop a PDC (Parallel Distributed Compensation) fuzzy controller for a nonlinear and certain system in continuous time described by the TS fuzzy model, the parameters of the functions defining the midpoint movements are optimized by a PSO (Particle Swarm Optimization) approach.
1

15
39


Salim
Ziani
Department of Electronics, Laboratory of Automatic and Robotics
LARC, University of Mentouri brother's Constantine, Route Ain ElBey, 25000, Constantine , Algeria
Department of Electronics, Laboratory of
Algeria
Fuzzy sets
Fuzzy system
Gaussian Membership function
PDC fuzzy controller
PSO method
TS model and stability
LMI
[[1] A. Baratella Lugli, E. Raimundo Neto, J. Paulo C. Henriques, M. Daniela Arambulo Her##vas, M. Mauro Dias Santos and J. Francisco Justo, Industrial application control with fuzzy##systems, International Journal of Innovative Computing, Information and Control (IJICIC),##12(2) (2016), 665{676.##[2] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear matrix inequalities in systems##and control theory ,SIAM Studies in Applied Mathematics, SIAM, Philadelphia, PA, (1994),##[3] O. Castillo, N. Cazarez, D. Rico and L. T. Aguilar, Intelligent control of dynamic systems##using type2 fuzzy logic and stability issues , International Mathematical Forum, 1(28) (2006),##1371{1382.##[4] S. Chopra, R. Mitra and V. Kumar, Reduction of fuzzy rules and membership functions##and its application to fuzzy pi and pd type controllers, International Journal of Control,##Automation and Systems, 4(2) (2006), 438{447.##[5] Z. Daqing, Z. Qingling and Z. Yan, Stabilization of TS fuzzy systems: an SOS approach,##International Journal of Innovative Computing, Information and Control, 4(9) (2008), 2273{##[6] J. Dong and G. H. Yang, Static output feedback control synthesis for discretetime TS fuzzy##systems, static output feedback control synthesis for discretetime TS fuzzy systems, Inter##national Journal of Control, Automation, and Systems, 5(3) (2007), 349354.##[7] C. H. Fang, Y. S. Liu, S. W. Kau, L. Hong and C. H. Lee, A new LMIbased approach to##relaxed quadratic stabilization of TS fuzzy control systems, IEEE Trans. on Fuzzy Sysems,##14(3) (2006), 386{397.##[8] P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, LMI control toolbox, The Math Works,##Natick, MA, 1 (1995), 175{356.##[9] Z. L. Gaing, A particle swarm optimization approach for optimum design of PID controller##in AVR system, IEEE Trans. Energy Conv., 9(2) (2004), 384{391.##[10] M.R. Ghazali, Z. Ibrahim, M. Helmi Suid, M. Salihin Saealal and M. Zaidi Mohd Tumari,##Single input fuzzy logic controller for ##exible joint manipulator, International Journal of##Innovative Computing, Information and Control (IJICIC), 12(1) (2016), 181{191.##[11] S. J. Huang and W. C. Lin, A selforganizing fuzzy controller for an active suspension system,##Journal of Vibration and Control, 9 (2003), 1023{1040.##[12] S. J. Huang and J. S. Lee, Stable selforganizing fuzzy controller for robotic motion control,##IEEE Transactions on Industrial Electronics, 47(2) (2000), 421{428.##[13] S. Jafarzadeh and al., Stability analysis and control of discrete Type1 and Type2 TSK##fuzzy systems: Part II. control design, IEEE Transactions on fuzzy systems, 19(6) (2011),##[14] S. Kaitwanidvilai, P. Olranthichachart and I. Ngamroo, PSO based automatic weight selec##tion and xed structure robust loop shaping control for power system control applications,##International Journal of Innovative Computing, Information and Control, 7(4) (2011), 1549{##[15] J. Kennedy and R. Eberhart, Particle swarm optimization, from Proc. IEEE Int. Conf. on##Neural Networks (Perth, Australia), IEEE Service Center, Piscataway, NJ, (1995), 1942{1948.##[16] D. W. Kim, J. B. Park, Y. H. Joo and S. H. Kim, Multirate digital control for fuzzy systems##LMIbased design and stability analysis, International Journal of Control, Automation, and##Systems, 4(4) (2006), 506{515.##[17] H. K. Lam, F. H. F. Leung and P. K. S. Tam, A Linear matrix inequality approach for the##control of uncertain fuzzy systems, IEEE Control Systems Magazine, 22(4) (2002), 20{25.##[18] Y. Li, S. Sui and S. Tong, Adaptive fuzzy control design for stochastic nonlinear switched sys##tems with arbitrary switchings and unmodeled dynamics, IEEE Transactions on Cybernetics,##(99) (2016), 1{12.##[19] Y. Li, S. Tong and T. Li, Adaptive fuzzy output feedback dynamic surface control of intercon##nected nonlinear purefeedback systems, IEEE Transactions on Cybernetics, 45(1) (2015),##[20] J. Li, H. O. Wang, D. Niemann and K. Tanaka, Dynamic parallel distributed compensation##for TakagiSugeno fuzzy systems: An LMI approach, Information Sciences, 123(34) (2000),##[21] R. I. Lian, B. F. Lin and W. T. Sie, Selforganizing fuzzy control of active suspension systems,##International Journal of Systems Science, 36(3) (2005), 119{135.##[22] D. Maravall, C. Zhou and J. Alonso, Hybrid fuzzy control of the inverted pendulum via vertical##forces, International journal of intelligent systems, 20 (2005), 195{211.##[23] M. Margaliot and G. Langholz, New approaches to fuzzy modeling and control:design and##analysis, World scientic, Singapore, 11(4) (2000), 486{494.##[24] J. M. Mendel, L. Fellow, Robert I. John and Feilong Liu., Interval type2 fuzzy systems made##simple, IEEE transactions on fuzzy systems, 14(6) (2006), 808{821.##[25] D. Niemann, J. Li, H. O. Wang and K. Tanaka, Parallel distributed compensation for Takagi##Sugeno fuzzy models: New stability conditions and dynamic feedback designs, Proceeding of##14th World Congress of IFAC, (1999), 207{212.##[26] Y. M. Park, U. C. Moon and K. Y. Lee, A selforganizing fuzzy logic for dynamic system##using fuzzy autoregressive moving average (FARMA) model, IEEE Transaction on fuzzy##system, 3(1) (1995), 75{82.##[27] P. A. Phan and T. J. Gale, Direct adaptive fuzzy control with a self structuring algorithm,##Fuzzy Sets Systems, 159(8) (2008), 871{899.##[28] M. Sugeno and G. T. Nishida, Fuzzy control of model car, Fuzzy Sets and Systems, 16(1)##(1985), 103{113.##[29] M. Sugeno, Industrial applications of fuzzy control, Elsevier Science Pub. Co., New York,##(1985), 249{269.##[30] T. Takagi and M. Sugeno, Fuzzy identication of systems and its application to modeling##and control, IEEE Trans. on Syst. Man and Cybernetics, 15 (1985), 116{132.##[31] K. Tanaka, T. Ikeda and H. O. Wang. , Design of fuzzy control systems based on relaxed LMI##stability conditions, 35th IEEE Conference on Decision and Control, 1 (1996), 598{603.##[32] K. Tanaka and H. O. Wang, Fuzzy control systems design and analysis : A linear matrix##inequality approach, Hoboken, NJ: Wiley, 1 (2001), 06{81.##[33] K. Tanaka and M. Sano, Fuzzy stability criterion of a class nonlinear systems, Inform. SC.,##71(2) (1993), 3{26.##[34] T. Tanaka, T. Ikeda and H. O. Wang, An LMI approach to fuzzy controller designs based on##relaxed stability Conditions, Proc. of the IEEE Conf. on Fuzzy Systems, (1997), 171{176.##[35] K. Tanaka, T. Taniguchi and H. O. Wang, Modelbased fuzzy control of TORA system:##fuzzy regulator and fuzzy observer design via LMIs that represent decay rate, disturbance##rejection, robustness, optimality, Seventh IEEE International Conference on Fuzzy Systems,##(1998), 313{318.##[36] K. Tanaka, T. Ikeda and H. O. Wang, Robust stabilization of a class of uncertain nonlinear##systems Via fuzzy control: quadratic stabilizability, control theory and linear matrix inequal##ities, IEEE Trans. on Fuzzy Systems, 4(1) (1996), 1{13. ##[37] C. W. Ting and C. Quek., A novel blood glucose regulation using TSKFCMAC: A fuzzy##CMAC based on the zeroordered TSK fuzzy inference scheme, IEEE Trans. Neural Netw.,##20(5) (2009), 856{871.##[38] H. D. Tuan, P. Apkarian, T. Narikiyo and M. Kanota, New fuzzy control model and dynamic##output feedback parallel distributed compensation, IEEE Trans. on Fuzzy Systems, 12(1)##(2004), 13{21.##[39] H. O. Wang, K. Tanaka and M. F. Grin, An approach to fuzzy control of nonlinear systems:##Stability and design issues, IEEE Transaction on fuzzy systems, 4(1) (1996), 14{23.##[40] D. Wu and J. M. Mendel, Enhanced KarnikMendel algorithms, IEEE Transactions on fuzzy##systems,17(4) (2009).##[41] L. A. Zadeh, The concept of linguistic variable and its application for approximate reasoning,##Information Sciences, 8(3) (1975), 199{249.##[42] L. A. Zadeh, Fuzzy sets, Information Control, 8 (1965), 335{353.##[43] S. Ziani and S. Filali, Timevarying fuzzy sets in adaptive control, 14th International##Conference on Sciences and Techniques of Automatic Control and Computer Engineering##(STA'2013), 1 (2013), 6{13.##[44] S. Ziani, S. Filali and Yunfu Huo, A timevarying fuzzy sets as functions of the error, Interna##tional Journal of Innovative Computing, Information and Control, 6(12) (2010), 5709{5723.##]
MULTI CLASS BRAIN TUMOR CLASSIFICATION OF MRI IMAGES USING HYBRID STRUCTURE DESCRIPTOR AND FUZZY LOGIC BASED RBF KERNEL SVM
2
2
Medical Image segmentation is to partition the image into a set of regions that are visually obvious and consistent with respect to some properties such as gray level, texture or color. Brain tumor classification is an imperative and difficult task in cancer radiotherapy. The objective of this research is to examine the use of pattern classification methods for distinguishing different types of brain tumors, such as primary gliomas from metastases, and also for grading of gliomas. Manual classification results look better because it involves human intelligence but the disadvantage is that the results may differ from one person to another person and takes long time. MRI image based automatic diagnosis method is used for early diagnosis and treatment of brain tumors. In this article, fully automatic, multi class brain tumor classification approach using hybrid structure descriptor and Fuzzy logic based Pair of RBF kernel support vector machine is developed. The method was applied to a population of 102 brain tumors histologically diagnosed as Meningioma (115), Metastasis (120), Gliomas grade II (65) and Gliomas grade II (70). Classification accuracy of proposed system in class 1(Meningioma) type tumor is 98.6%, class 2 (Metastasis) is 99.29%, class 3(Gliomas grade II) is 97.87% and class 4(Gliomas grade III) is 98.6%.
1

41
54


A.
Jayachandran
Department of CSE, PSN College of Engineering and Technology, Tirunelveli, India
Department of CSE, PSN College of Engineering
India


R.
Dhanasekaran
Department of EEE, Syed Ammal Engineering College,
Ramanathapuram,India
Department of EEE, Syed Ammal Engineering
India
MRI
Classification
Fuzzy support vector machine
Feature selection
Texture
tumor class
Radial Basics Function (RBF)
[[1] S. Bauer S, L. P. Nolte and M. Reyes, Fully automatic segmentation of brain tumor im##ages using support vector machine classication in combination with hierarchical conditional##random eld regularization, Med Image Comput Assist Interv., 14(3) (2011), 354{361.##[2] C. J. C. Burges., A tutorial on support vector machines for pattern recognition, Data Mining##and Knowledge Discovery, 2 (1998), 121{167.##[3] S. Chaplot , L. M. Patnai and N. R. Jagannathan , Classication of magnetic resonance brain##images using wavelets as input to support vector machine and neural network, Biomedical##Signal Processing and Control, 1(1) (2006), 86{92.##[4] J. Chunming Li, R. Huang, Z. Ding and J. Chris Gatenby, A level set method for image##segmentation in the presence of intensity inhomogeneities with application to MRI, IEEE##Trans Image Process, 20 (2011), 2007{2016. ##[5] R. Dhanasekaran and A. Jayachandranm, Brain tumor detection using fuzzy support vector##machine classication based on a texton Cooccurrence matrix, Journal of imaging Science##and Technology, 57(1) (2013), 105071{105077.##[6] S. R. Dubey, S. K. Singh, and R. K. Singh, Rotation and scale invariant hybrid image##descriptor and retrieval, Computers & Electrical Engineering, 46(8) (2015), 288{302.##[7] N. E. Ibrahim, S. Khalid and M. Manaf, SeedBased region growing (SBRG) vs adaptive##networkbased inference system (ANFIS) vs fuzzy cmeans(FCM)  brain abnormalities seg##mentation, World Acad. Sci. Eng. Technol., 68 (2010), 425{435.##[8] A. Jayachandran and R. Dhanasekaran, Automatic detection of brain tumor in magnetic##resonance images using multitexton histogram and support vector machinel, International##Journal of Imaging Systems and Technology, 23(2) (2013), 97{103.##[9] A. Jayachandran and R. Dhanasekaran, Severity analysis of brain tumor in MRI images uses##modied multitexton structure descriptor and kernel SVM, The Arabian Journal of science##and engineering, 39(10) (2014), 7073{7086.##[10] B. Julesz, Textonsthe elements of texture perception and their interactions, Nature, 290##(1981), 91{97.##[11] G. Kharmega Sundararaj and A. Jayachandran, Abnormality segmentation and classica##tion of multiclass brain tumor in MR images using fuzzy logicbased hybrid kernel SVM,##International journal of Fuzzy System, 17(3) (2015), 434{443.##[12] J. S. Lin, K. S. Cheng and C. W. Mao, Segmentation of multispectral magnetic resonance im##age using penalized fuzzy competitive learning network, Journal of Computers and Biomedical##Research, 29(4) (1996), 314{326.##[13] G. H. Liu, L. Zhang, Yingkun Hou, Zuoyong Li and JingYu Yang, Image retrieval based on##multitexton histogram, Pattern Recognition, 43(7) (2010), 2380{2389.##[14] C. L. P. LongChen, Philip Chen and L. U. Mingzhu, A multiplekernel fuzzy Cmeans algo##rithm for image segmentation, IEEE Transactions On Systems, Man, and Cybernetics{Part##B: Cybernetics, 41(5) (2011), 1263{1274.##[15] T. Wang and H. M. Chiang, Fuzzy support vector machine for multiclass text categorization,##Information Processing and Management, 43 (2007), 914{929.##[16] R. J. Young and E. A. Knopp, Brain MRI: tumor evaluation, Journal of Magnetic Resonance##Imaging, 24 (2006), 709{724.##[17] E. I. Zacharaki, S. Wang, S. Chawla, E. R. Melhem and C. Davatzikos, Classication of brain##tumor type and grade using MRI texture in a machine learning technique, Magn. Reson. Med.,##62 (2009), 1609{1618.##[18] K. Zhang, H. X. Cao and H. Yan, Application of support vector machines on network abnor##mal intrusion detection, Application Research of Computers, 5 (2006), 98{100.##[19] C. Zhu and T. Jiang, Multi context fuzzy clustering for separation of brain tissues in magnetic##resonance images, Neuro lmage, 18(3) (2003), 685 {696.##[20] W. Zhu, N. Zeng and N. Wang, Sensitivity, specicity, accuracy, associated condence in##terval and ROC analysis with practical SAS implementations, In: Proceedings of the SAS##Conference, NESUG 210, November 14{17, Baltimore, Maryland, 2010.##]
FUZZY PREORDERED SET, FUZZY TOPOLOGY AND FUZZY AUTOMATON BASED ON GENERALIZED RESIDUATED LATTICE
2
2
This work is towards the study of the relationship between fuzzy preordered sets and Alexandrov (left/right) fuzzy topologies based on generalized residuated lattices here the fuzzy sets are equipped with generalized residuated lattice in which the commutative property doesn't hold. Further, the obtained results are used in the study of fuzzy automata theory.
1

55
66


Anupam K.
Singh
Amity Institute of Applied Science (AIAS), Amity University,
Sector125, Noida, Uttar Pradesh201313, India
Amity Institute of Applied Science (AIAS),
India
Generalized residuated lattice
(left/right) Subsystem
Fuzzy automata
Alexandrov (left/right) fuzzy topology
[[1] P. Das, A fuzzy topology associated with a fuzzy nite state machine, Fuzzy Sets and Systems,##105(3) (1999), 469{479.##[2] G. Georgescu and A. Popescu, Noncommutative fuzzy Galois connections, Soft Computing,##7(7) (2003), 458{467.##[3] X. Guo, Grammar theory based on latticeorder monoid, Fuzzy Sets and Systems, 160(8)##(2009), 11521161.##[4] J. Ignjatovic, M. Ciric and S. Bogdanovic, Determinization of fuzzy automata with member##ship values in complete residuated lattices, Information Sciences, 178(1) (2008), 164{180.##[5] J. Ignjatovic, M. Ciric and V. Simovic, Fuzzy relation equations and subsystems of fuzzy##transition systems, KnowledgeBased Systems, 38 (2013), 4861.##[6] Y. B. Jun, Intuitionistic fuzzy nite state machines, J. Appl. Math. Comput., 17(1) (2005),##[7] Y. B. Jun, Intuitionistic fuzzy nite switchboard state machines, J. Appl. Math. Comput.,##20(1) (2006), 315325.##[8] Y.B. Jun, Quotient structures of intuitionistic fuzzy nite state machines, Information Sci##ences, 177(22) (2007), 49774986.##[9] Y. H. Kim, J. G. Kim and S. J. Cho, Products of Tgeneralized state machines and T##generalized transformation semigroups, Fuzzy Sets and Systems, 93(1) (1998), 8797.##[10] H. V. Kumbhojkar and S. R. Chaudhri, On proper fuzzication of fuzzy nite state machines,##Int. J. Fuzzy Math., 4(4) (2000), 10191027.##[11] H. Lai, D. Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157(14)##(2006), 18651885.##[12] Y. Li and W. Pedrycz, Fuzzy nite automata and fuzzy regular expressions with membership##values in latticeorderd monoids, Fuzzy Sets and Systems, 156(1) (2005), 6892. ##[13] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., 56(3)##(1976), 621633.##[14] D. S. Malik, J. N. Mordeson and M. K. Sen, Submachines of fuzzy nite state machine,##J. Fuzzy Math., 2 (1994), 781792.##[15] J. N. Mordeson and D. S. Malik, Fuzzy Automata and Languages: Theory and Applications.##Chapman and Hall/CRC, London/Boca Raton, 2002.##[16] D. Qiu, Automata theory based on complete residuated latticevalued logic(I), Science in##China, 44(6) (2001), 419429.##[17] D. Qiu, Automata theory based on complete residuated latticevalued logic(II), Science in##China, 45(6) (2002), 442452.##[18] D. Qiu, Automata theory based on quantum logic: Some characterizations, Information and##Computation, 190(2) (2004), 179195.##[19] D. Qiu, Characterizations of fuzzy nite automata, Fuzzy Sets and Systems, 141(3) (2004),##[20] D. Qiu, Automata theory based on quantum logic: Reversibilities and pushdown automata,##Theoretical Computer Science, 386(12) (2007), 3856.##[21] E. S. Santos, Maximin automata, Information and Control, 12(4) (1968), 367377.##[22] Y. H. She and G. J. Wang, An axiomatic approach of fuzzy rough sets based on residuated##lattices, Comp. Math. Appl., 58(1) (2009), 189201.##[23] A. K. Srivastava and S. P. Tiwari, A topology for fuzzy automata, In: Proc. AFSS Internat.##Conf. on Fuzzy System, Lecture Notes in Articial Intelligence, Springer, Berlin, 2275 (2002),##[24] A. K. Srivastava and S. P. Tiwari, On relationships among fuzzy approximation operators,##fuzzy topology, and fuzzy automata, Comp. Math. Appl., 138(1) (2003), 191204.##[25] S. P. Tiwari and Anupam K. Singh, Fuzzy preorder, fuzzy topology and fuzzy transition##system, in: Proc. ICLA 2013, Lecture Notes in Computer Science, Springer, Berlin, 7750##(2013), 210219.##[26] C. Y. Wang and B. Q. Hu, Fuzzy rough sets based on generalized residuated lattices, Infor##mation Sciences, 248 (2013), 3149.##[27] W. G.Wee, On generalizations of adaptive algorithm and application of the fuzzy sets concept##to pattern classication, Ph. D. Thesis, Purdue University, Lafayette, IN, 2013.##[28] L. Wu and D. Qiu, Automata theory based on complete residuated latticevalued logic: Re##duction and minimization, Fuzzy Sets and Systems, 161(12) (2010), 16351656.##[29] L. A. Zadeh, Fuzzy Sets, Information and Control, 8(3) (1965), 338353.##]
A COMMON FRAMEWORK FOR LATTICEVALUED, PROBABILISTIC AND APPROACH UNIFORM (CONVERGENCE) SPACES
2
2
We develop a general framework for various latticevalued, probabilistic and approach uniform convergence spaces. To this end, we use the concept of $s$stratified $LM$filter, where $L$ and $M$ are suitable frames. A stratified $LMN$uniform convergence tower is then a family of structures indexed by a quantale $N$. For different choices of $L,M$ and $N$ we obtain the latticevalued, probabilistic and approach uniform convergence spaces as examples. We show that the resulting category $sLMN$$UCTS$ is topological, wellfibred and Cartesian closed. We furthermore define stratified $LMN$uniform tower spaces and show that the category of these spaces is isomorphic to the subcategory of stratified $LMN$principal uniform convergence tower spaces. Finally we study the underlying stratified $LMN$convergence tower spaces.
1

67
81


Gunther
Jager
School of Mechanical Engineering, University of Applied Sciences
Stralsund, 18435 Stralsund, Germany
School of Mechanical Engineering, University
Germany
g.jager@ru.ac.za, gunther.jaeger@fhstralsund.de
Stratified latticevalued uniformity
Stratified latticevalued uniform convergence space
Probabilistic uniform convergence space
Approach uniform convergence space
Stratified $LM$filter
[[1] J. Adamek, H. Herrlich and G. E. Strecker, Abstract and concrete categories, Wiley, New##York 1989.##[2] T. M. G. Ahsanullah and G. Jager, Probabilistic uniform convergence spaces redened, Acta##Math. Hungar., 146 (2015), 376 { 390.##[3] N. Bourbaki, General topology, Chapters 1 { 4, Springer Verlag, Berlin  Heidelberg  New##York  London  Paris  Tokyo, 1990. ##[4] M. H. Burton, M. A. de Prada Vicente and J. Gutierrez Garca, Generalized uniform spaces,##J. Fuzzy Math., 4 (1996), 363 { 380.##[5] C. H. Cook and H. R. Fischer, Uniform convergence structures, Math. Ann. 173 (1967), 290##[6] A. Craig and G. Jager, A common framework for latticevalued uniform spaces and proba##bilistic uniform limit spaces, Fuzzy Sets and Systems, 160(2009), 1177 { 1203.##[7] J. Fang, Latticevalued semiuniform convergence spaces, Fuzzy Sets and Systems, 195 (2012),##[8] R. C. Flagg, Quantales and continuity spaces, Algebra Univers., 37 (1997), 257 { 276.##[9] L. C. Florescu, Probabilistic convergence structures, Aequationes Math., 38 (1989), 123 {##[10] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, A com##pendium of continuous lattices, SpringerVerlag Berlin Heidelberg, 1980.##[11] J. Gutierrez Garca, A unied approach to the concept of fuzzy Luniform space, Thesis,##Universidad del Pais Vasco, Bilbao, Spain, 2000.##[12] J. Gutierrez Garca, M. A. de Prada Vicente and A. P. Sostak, A unied approach to the##concept of fuzzy Luniform space, In: S. E. Rodabaugh, E. P. Klement (Eds.), Topological##and algebraic structures in fuzzy sets, Kluwer, Dordrecht, (2003), 81 { 114.##[13] U. Hohle, Characterization of Ltopologies by Lvalued neighborhoods, In: U. Hohle, S.E.##Rodabauch (Eds.), Mathematics of Fuzzy Sets. Logic, Topology and Measure Theory, Kluwer,##Boston/Dordrecht/London 1999, 389 { 432.##[14] U. Hohle and A. P. Sostak, Axiomatic foundations of xedbasis fuzzy topology, In: U. Hohle,##S. E. Rodabauch (Eds.), Mathematics of Fuzzy Sets. Logic, Topology and Measure Theory,##Kluwer, Boston/Dordrecht/London 1999, 123 { 272.##[15] G. Jager, A category of Lfuzzy convergence spaces, Quaestiones Math., 24 (2001), 501 { 518.##[16] G. Jager, Fischer's diagonal condition for latticevalued convergence spaces, Quaestiones##Math., 31 (2008), 11 { 25.##[17] G. Jager, A note on stratied LMlters, Iranian Journal of Fuzzy Systems, 10(4) (2013),##135 { 142.##[18] G. Jager, Stratied LMNconvergence tower spaces, Fuzzy Sets and Systems, 282 (2016), 62##[19] G. Jager, Uniform connectedness and uniform local connectedness for latticevalued uniform##convergence spaces, Iranian Journal of Fuzzy Systems, 13(3) (2016), 95 { 111.##[20] G. Jager and M. H. Burton, Stratied Luniform convergence spaces, Quaestiones Math., 28##(2005), 11 { 36.##[21] Y. J. Lee and B. Windels, Transitivity in uniform approach theory, Int. J. Math. and Math.##Sci., 32 (2002), 707 { 720.##[22] E. Lowen, R. Lowen and P. Wuyts, The categorical topology approach to fuzzy topology and##fuzzy convergence, Fuzzy Sets and Systems, 40 (1991), 347 { 373.##[23] R. Lowen and B. Windels, AUnif: A commmon supercategory of pMET and Unif, Int. J.##Math. and Math. Sci., 21 (1998), 1 { 18.##[24] H. Nusser, A generalization of probabilistic uniform spaces, Appl. Cat. Structures, 10 (2002),##[25] G. Preuss, Foundations of topology  An Approach to Convenient Topology, Kluwer, Dordrecht##[26] B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, New York, 1983.##[27] O. Wyler, Filter space monads, regularity, completions, In: TOPO 1972  General Topology##and its Applications, Lecture Notes in Mathematics, Vol.378, Springer, Berlin, Heidelberg,##New York, (1974), 591 { 637.##]
GRADED DIUNIFORMITIES
2
2
Graded ditopological texture spaces have been presented and discussed in categorical aspects by Lawrence M. Brown and Alexander {v S}ostak in cite{BS}. In this paper, the authors generalize the structure of diuniformity in ditopological texture spaces defined in cite{OB} to the graded ditopological texture spaces and investigate graded ditopologies generated by graded diuniformities. The autors also compare the properties of diuniformities and graded diuniformities.
1

83
103


Ramazan
Ekmekci
Department of Mathematics, Canakkale Onsekiz Mart University,
Canakkale, TURKEY
Department of Mathematics, Canakkale Onsekiz
Turkey


Rza
Erturk
Department of Mathematics, Hacettepe University, Ankara, TURKEY
Department of Mathematics, Hacettepe University,
Turkey
Texture
Graded ditopology
Graded diuniformity
Fuzzy topology
[[1] J. Adamek, H. Herrlich and G. E. Strecer, Abstract and concrete categories, New York,##Chichester, Brisbane, Toronto, Singapore: John Wiley & Sons, Inc., 1990.##[2] L. M. Brown and M. Diker, Ditopological texture spaces and intuitionistic sets, Fuzzy Sets##and Systems, 98 (1998), 217{224.##[3] L. M. Brown and R. Erturk, Fuzzy sets as texture spaces, I. Representation theorems, Fuzzy##Sets and Systems, 110(2) (2000), 227{236.##[4] L. M. Brown, R. Erturk and S. Dost, Ditopological texture spaces and fuzzy topology, I. Basic##concepts, Fuzzy Sets and Systems, 147(2) (2004), 171{199.##[5] L. M. Brown, R. Erturk and S. Dost, Ditopological texture spaces and fuzzy topology, II.##Topological considerations, Fuzzy Sets and Systems, 147(2) (2004), 201{231.##[6] L. M. Brown, R. Erturk and S. Dost, Ditopological texture spaces and fuzzy topology, III.##Separation Axioms, Fuzzy Sets and Systems, 157(14) (2006), 1886{1912.##[7] L. M. Brown and A. Sostak, Categories of fuzzy topology in the context of graded ditopologies##on textures, Iranian Journal of Fuzzy Systems, 11(6) (2014), 1{20.##[8] C. L. Chang, Fuzzy topological spaces, Math. Anal. Appl., 24 (1968), 182{190.##[9] R. Ekmekci and R. Erturk, Neighborhood structures of graded ditopological texture spaces,##Filomat, 29(7) (2015), 1445{1459.##[10] R. Erturk, Separation axioms in fuzzy topology characterized by bitopologies, Fuzzy Sets and##Systems, 58 (1993), 206{209.##[11] T. Kubiak, On fuzzy topologies, PhD Thesis, A. Mickiewicz University Poznan, Poland (1985).##[12] S. Ozcag, Uniform texture spaces, PhD Thesis, Hacettepe University, Ankara, Turkey (2004).##[13] S. Ozcag and L. M. Brown, Diuniform texture spaces, Appiled General Topology, 4(1)##(2003), 157{192.##[14] S. Ozcag, L. M. Brown and K. Biljana, Diuniformities and Hutton uniformities, Fuzzy Sets##and Systems, 195 (2012), 58{74.##[15] S. Ozcag and S. Dost, A categorical view of diuniform texture spaces, Bol. Soc. Mat. Mexicana,##3(15) (2009), 63{80.##[16] A. Sostak, On a fuzzy topological structure, Rend. Circ. Matem. Palermo, Ser. II, 11 (1985),##[17] A. Sostak, Two decates of fuzzy topology: basic ideas, notions and results, Russian Math.##Surveys, 44(6) (1989), 125{186.##[18] G. Yldz, Ditopological spaces on texture spaces, MSc Thesis, Hacettepe University, Ankara,##Turkey (2005).##]
STABILITY OF THE JENSEN'S FUNCTIONAL EQUATION IN MULTIFUZZY NORMED SPACES
2
2
In this paper, we define the notion of (dual) multifuzzy normedspaces and describe some properties of them. We then investigate UlamHyers stability of Jensen's functional equation for mappings from linear spaces into multifuzzy normed spaces. We establish an asymptotic behavior of the Jensen equation in the framework of multifuzzy normed spaces.
1

105
119


Mahnaz
Khanehgir
Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
Department of Mathematics, Mashhad Branch,
Iran
mkhanehgir@gmail.com
Fuzzy normed space
UlamHyers stability
Jensen's functional equation
Multinormed space
[[1] A. Alotaibi, M. Mursaleen, H. Dutta and S. A. Mohiuddine, On the Ulam stability of Cauchy##functional equation in IFNspaces, Appl. Math. Inf. Sci. 8(3) (2014), 1135{1143.##[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan,##2 (1950), 64{66.##[3] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math.##11(3) (2003), 687{705.##[4] T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Syst. 151 (2005),##[5] S. C. Cheng and J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces,##Bull. Calcutta Math. Soc., 86(5) (1994), 429{436.##[6] H. G. Dales and M. E. Polyakov, Multinormed spaces and multiBanach algebras, University##of Leeds, preprint, (2008), 1{156.##[7] C. Felbin, Finitedimensional fuzzy normed linear space, Fuzzy Sets and Syst., 48(2) (1992),##[8] A. Ghaari and A. Alinejad, Stabilities of cubic mappings in fuzzy normed spaces, Adv.##Dierence Equ., 2010 (2010), 1{15.##[9] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A.,##27 (1941), 222{224. ##[10] S. M. Jung, HyersUlamRassias stability of Jensens equation and its application, Proc.##Amer. Math. Soc., 126 (1998), 3137{3143.##[11] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Syst., 12(2) (1984), 143{##[12] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11(5)##(1975), 336344.##[13] S. V. Krishna and K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and##Syst., 63(2) (1994), 207{217.##[14] L. Li, J. Chung and D. Kim, Stability of Jensen equations in the space of generalized func##tions, J. Math. Anal. Appl., 299 (2004), 578{586.##[15] T. Li, A. Zada and S. Faisal, HyersUlam stability of nth order linear dierential equations,##J. Nonlinear Sci. Appl., 9(5) (2016), 2070{2075.##[16] A. K. Mirmostafaee, Perturbation of generalized derivations in fuzzy Menger normed alge##bras, Fuzzy Sets and Syst., 195 (2012), 109117.##[17] A. K. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen##functional equation, Fuzzy Sets and Syst., 159(6) (2008), 730{738.##[18] M. S. Moslehian and H. M. Sirvastava, Jensen's functional equation in multinormed spaces,##Taiwanese J. Math., 14(2) (2010), 453{462.##[19] E. Movahednia, S. Eshtehar and Y. Son, Stability of quadratic functional equations in fuzzy##normed spaces, Int. J. Math. Anal., 6(48) (2012), 2405{2412.##[20] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math.##Soc., 72 (1978), 297{300.##[21] Y. Shen, An integrating factor approach to the HyersUlam stability of a class of exact##dierential equations of second order, J. Nonlinear Sci. Appl., 9(5) (2016), 2520{2526.##[22] S. M. Ulam, A collection of mathematical problems, Interscience Publ., New York, 1960.##[23] S. M. Vaezpour and F. Karimi, tBest approximation in fuzzy normed spaces, Iranian Journal##of Fuzzy Systems, 5(2) (2008), 93{99.##[24] J. Z. Xiao and X. H. Zhu, Fuzzy normed space of operators and its completeness, Fuzzy Sets##and Syst., 133(3) (2003), 389{399.##[25] B. Yood, On the nonexistence of norms for some algebras of functions, Stud. Math., 111(1)##(1994), 97101.##]
THE CATEGORY OF TCONVERGENCE SPACES AND ITS CARTESIANCLOSEDNESS
2
2
In this paper, we define a kind of latticevalued convergence spaces based on the notion of $top$filters, namely $top$convergence spaces, and show the category of $top$convergence spaces is Cartesianclosed. Further, in the lattice valued context of a complete $MV$algebra, a close relation between the category of$top$convergence spaces and that of strong $L$topological spaces is established. In details, we show that the category of strong $L$topological spaces is concretely isomorphic to that of strong $L$topological $top$convergence spaces categorically and bireflectively embedded in that of $top$convergence spaces.
1

121
138


Qian
Yu
Department of Mathematics, Ocean University of China, 238 Songling Road,
266100, Qingdao, P.R. China
Department of Mathematics, Ocean University
China
yuqian198436@sina.com


Jinming
Fang
Department of Mathematics, Ocean University of China, 238 Songling
Road, 266100, Qingdao, P.R. China
Department of Mathematics, Ocean University
China
jiningfang@163.com
Tlter
Tconvergence
Cartesianclosedness
Topological category
Reflection
Strong Ltopology
[[1] G. Choquet, Convergences, Ann. Univ. Grenoble, 23 (1948), 57{112.##[2] J. M. Fang, Stratied Lordered convergence structures, Fuzzy Sets and Systems, 161 (2010),##2130{2149.##[3] J. M. Fang, Relationships between Lordered convergence structures and strong Ltopologies,##Fuzzy Sets and Systems, 161 (2010), 2923{2944.##[4] J. A. Goguen, Lfuzzy sets, J. Math. Anal. Appl, 18 (1967), 145{174.##[5] J. Gutierrez Garca and M. A. De Prada Vicente, Characteristic values of >lter, Fuzzy##Sets and Systems, 156 (2005), 55{67.##[6] J. Gutierrez Garca and M. A. De Prada Vicente, A unied approach to the concept of fuzzy##Luniform space, In: Topological and Algebraic Structures in Fuzzy Sets{A Handbook of##Recent Devellopments in the Mathematics of Fuzzy Sets, (S.E. Rodabaugh, E.P. Klement,##ed.), Kluwer Academic Publishers, Boston, Dordrecht, London, (2003), 79{114,.##[7] U. Hohle, Commutative residuated `monoids, In: Nonclassical Logics and Their Applications##to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Set Theory,##(U. Hohle, E.P. Klement, Eds.), Klumer Academic Publishers, Dordrecht, Boston, London##(1995), 53{106.##[8] U. Hohle, Propbabilistische topologien, Manuscripta Math., 26 (1978), 223{245.##[9] U. Hohle, Probabilistic topologies induced by Lfuzzy uniformities, Manuscripta Math., 38##(1982), 289{323.##[10] G. Jager, A category of Lfuzzy convergence spaces, Quaestiones Mathematicae, 24 (2001),##[11] G. Jager, Subcategories of latticevalued convergence spaces, Fuzzy Sets and Systems, 156##(2005), 1{24.##[12] G. Jager, Latticevalued convergence spaces and regularity, Fuzzy Sets and Systems, 159##(2008), 2488{2502.##[13] R. Lowen, Convergence in fuzzy topological spaces, Gen. Top. Appl., 10 (1979), 147{160.##[14] E. Lowen and R. Lowen, A topological universe extension of FTS, In: Applications of category##theory to fuzzy sets, (S.E. Rodabaugh, E.P. Klement, U. Hohle, Eds.), Kluwer, Dordrecht##(1992), 153{176. ##[15] E. Lowen, R. Lowen and P. Wuyts, The categorical topology approach to fuzzy topology and##fuzzy convergence, Fuzzy Sets and Systems, 40 (1991), 347{343.##[16] L. Li, Q. Jin and K. Hu, On stratied Lconvergence spaces: Fischer's diagonal axiom, Fuzzy##Sets and Systems, 267 (2015), 31{40.##[17] G. Preuss, Foundations of topology, Kluwer Academic Publishers, Dordrecht, Boston, London##(2002), 30{92.##[18] B. Pang and J. M. Fang, Lfuzzy Qconvergence structures, Fuzzy Sets and System, 182##(2011), 53{65.##[19] B. Pang, On (L;M)fuzzy convergence spaces, Fuzzy Sets and Systems, 238 (2014), 46{70.##[20] W. Yao, On manyvalued stratied Lfuzzy convergence spaces, Fuzzy Sets and Systems, 159##(2008), 2503{2519.##[21] W. Yao, MooreSmith convergence in (L;M)fuzzy topology, Fuzzy Sets and Systems, 190##(2012), 47{62.##[22] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems,##158 (2007), 349{366.##]
MFUZZIFYING MATROIDS INDUCED BY MFUZZIFYING CLOSURE OPERATORS
2
2
In this paper, the notion of closure operators of matroids is generalized to fuzzy setting which is called $M$fuzzifying closure operators, and some properties of $M$fuzzifying closure operators are discussed. The $M$fuzzifying matroid induced by an $M$fuzzifying closure operator can induce an $M$fuzzifying closure operator. Finally, the characterizations of $M$fuzzifying acyclic matroids are given.
1

139
149


Xiu
Xin
Department of Mathematics, Tianjin University of Technology, Tianjin
300384, P.R.China
Department of Mathematics, Tianjin University
China
xinxiu518@163.com


ShaoJun
Yang
School of Mathematics and Statistics, Beijing Institute of Tech
nology, Beijing 100081, P.R.China and Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, P.R.China
School of Mathematics and Statistics, Beijing
China
shaojunyang@outlook.com
M$fuzzifying matroids
$M$fuzzifying closure operators
$M$fuzzifying exchange law
[[1] H. L. Huang and F. G. Shi, Lfuzzy numbers and their properties, Information Sciences, 178##(2008), 1141{1151.##[2] H. Lian and X. Xin, The nullities for Mfuzzifying matroids, Applied Mathematics Letters,##25(3) (2012), 279{286.##[3] J. Oxley, Matroid Theory, Oxford University Press, 1992.##[4] F. G. Shi, A new approach to fuzzication of matroids, Fuzzy Sets and Systems, 160(5)##(2009), 696{705.##[5] F. G. Shi, (L,M)fuzzy matroids, Fuzzy Sets and Systems, 160(16) (2009), 2387{2400.##[6] F. G. Shi and B. Pang, Categories isomorphic to the category of Lfuzzy closure system##spaces, Iranian Journal of Fuzzy Systems, 10(5) (2013), 127{146.##[7] F. G. Shi and L. Wang, Characterizations and applications of Mfuzzifying matroids, Journal##of Intelligent and Fuzzy Systems, 25 (2013), 919{930.##[8] G. J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems, 47(3) (1992),##[9] L. Wang and F. G. Shi, Characterization of Lfuzzifying matroids by Mfuzzifying families##ats, Advances of Fuzzy Sets and Systems, 2 (2009), 203{213.##[10] L. Wang and F. G. Shi, Characterization of Lfuzzifying matroids by Lfuzzifying closure##operators, Iranian Journal of Fuzzy Systems, 7(1) (2010), 47{58.##[11] L. Wang and Y. P. Wei, Mfuzzifying Pclosure operators, Advances in Intelligent and Soft##Computing, 62 (2009), 547{554.##[12] X. Xin and F. G. Shi, Mfuzzifying bases, Proyecciones, 28(3) (2009), 271{283.##[13] X. Xin and F. G. Shi, Categories of bifuzzy prematroids, Computer and Mathematics with##Applications, 59 (2010), 1548{1558.##[14] X. Xin and F. G. Shi, Rank functions for closed and perfect [0,1]matroids, Hacettepe Journal##of Mathematics and Statistics, 39(1) (2010), 31{39.##[15] X. Xin, F. G. Shi and S. G. Li, Mfuzzifying derived operators and dierence derived operators,##Iranian Journal of Fuzzy Systems, 7(2) (2010), 71{81.##[16] Z. Y. Xiu and F. G. Shi, Mfuzzifying submodular functions, Journal of Intelligent and Fuzzy##Systems, 27 (2014), 1243{1255.##[17] W. Yao and F. G. Shi, Base axioms and circuits axioms for fuzzifying matroids, Fuzzy Sets##and Systems, 161 (2010), 3155{3165.##]
SELECTIVE GROUPOIDS AND FRAMEWORKS INDUCED BY FUZZY SUBSETS
2
2
In this paper, we show that every selective groupoid induced by a fuzzy subset is a pogroupoid, and we discuss several properties in quasi ordered sets by introducing the notion of a framework.
1

151
160


Young Hee
Kim
Department of Mathematics, Chungbuk National University, Cheongju, 28644, Korea
Department of Mathematics, Chungbuk National
Korea
yhkim@cbnu.ac.kr


Hee Sik
Kim
Research Institute for Natural Sci., Department of Mathematics,
Hanyang University, Seoul, 04763, Korea
Research Institute for Natural Sci., Department
Korea
heekim@hanyang.ac.kr


J.
Neggers
Department of Mathematics, University of Alabama, Tuscaloosa, AL
354870350, U. S. A.
Department of Mathematics, University of
United States
jneggers@gp.as.ua.edu
Fuzzy subset
$d/BCK$algebra
Framework
Selective groupoid
Pogroupoid
Poset
[[1] R. K. Bandaru, K. P. Shum and N. Ra, Fuzzy ideals of implication groupoids, Italian J.##Pure and Appl. Math., 34 (2015), 277{290.##[2] G. Gratzer, General lattice theory, Springer, New York, 1978.##[3] J. S. Han, H. S. Kim and J. Neggers, Strong and ordinary dalgebras, J. Mult.Valued Logic##& Soft Computing, 16 (2010), 331{339.##[4] D. Kelly and I. Rival, Planar lattices, Canad. J. Math., 27 (1975), 636{665.##[5] M. Khan, M. Shakeel, M. Gulistan and S. Rashid, Generalized fuzzy biideals of order right##modular groupoids, Int. J. Algebra and Statistics, 4 (2015), 46{56.##[6] H. S. Kim and J. Neggers, The semigroups of binary systems and some perspectives, Bull.##Korean Math. Soc., 45 (2008), 651{661.##[7] J. Neggers, Partially ordered sets and groupoids, Kyungpook Math. J., 16 (1976), 7{20.##[8] J. Neggers and H. S. Kim, Modular posets and semigroups, Semigroup Forum, 53 (1996),##[9] J. Neggers and H. S. Kim, On dalgebras, Math. Slovaca, 49 (1999), 19{26.##[10] J. Neggers and H. S. Kim, Algebras associated with posets, Demonstratio Math., 34 (2001),##[11] J. Neggers and H. S. Kim, Fuzzy posets on sets, Fuzzy Sets and Sys., 117 (2001), 391{402.##[12] J. Neggers and H. S. Kim, Fuzzy pogroupoids, Information Sci., 175 (2005), 108{119.##[13] S. J. Shin, H. S. Kim and J. Neggers, On Abelian and related fuzzy subsets of groupoids, The##Scientic World J., Article ID 476057, 2013 (2013), 5 pages.##[14] S. J. Shin, H. S. Kim and J. Neggers, The intersection between fuzzy subsets and groupoids,##The Scientic World J., Article ID 246285, 2014 (2014), 6 pages.##[15] L. Zadeh, Fuzzy Sets, Inform. and Control, 8 (1965), 338{353.##]
SOME FIXED POINT RESULTS FOR ADMISSIBLE GERAGHTY CONTRACTION TYPE MAPPINGS IN FUZZY METRIC SPACES
2
2
In this paper, we introduce the notions of fuzzy $alpha$Geraghty contraction type mapping and fuzzy $beta$$varphi$contractive mapping and establish some interesting results on the existence and uniqueness of fixed points for these two types of mappings in the setting of fuzzy metric spaces and nonArchimedean fuzzy metric spaces. The main results of our work generalize and extend some known comparable results in the literature. Furthermore, several illustrative examples are given to support the usability of our obtained results.
1

161
177


Mina
Dinarvand
Faculty of Mathematics, K. N. Toosi University of Technology,
P.O. Box 163151618, Tehran, Iran
Faculty of Mathematics, K. N. Toosi University
Iran
dinarvand_mina@yahoo.com
Fixed point
Fuzzy $alpha$Geraghty contraction type mapping
Fuzzy $beta$$varphi$contractive mapping
Fuzzy metric space
NonArchimedean fuzzy metric space
[[1] I. Altun and D. Mihet, Ordered nonArchimedean fuzzy metric spaces and some xed point##results, Fixed Point Theory Appl., Article ID 782680, 2010 (2010), 1{11.##[2] M. Amini and R. Saadati, Topics in fuzzy metric spaces, J. Fuzzy Math., 11(4) (2003),##[3] A. AminiHarandi and H. Emami, A xed point theorem for contraction type maps in partially##ordered metric spaces and application to ordinary dierential equations, Nonlinear Anal., 72##(2010), 2238{2242.##[4] S. H. Cho, J. S. Bae and E. Karapinar, Fixed point theorems for Geraghty contraction type##maps in metric spaces, Fixed Point Theory Appl., Article ID 329, 2013 (2013), 1{11. ##[5] C. Di Bari and C. Vetro, A xed point theorem for a family of mappings in a fuzzy metric##space, Rend. Circ. Mat. Palermo, 52(2) (2003), 315{321.##[6] C. Di Bari and C. Vetro, Fixed points, attractors and weak fuzzy contractive mappings in a##fuzzy metric space, J. Fuzzy Math., 13(4) (2005), 973{982.##[7] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64##(1994), 395{399.##[8] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets##Syst., 90 (1997), 365{368.##[9] M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604{608.##[10] D. Gopal and C. Vetro, Some new xed point theorems in fuzzy metric spaces, Iranian J.##Fuzzy Syst., 11(3) (2014), 95{107.##[11] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst., 27 (1988), 385{389.##[12] V. Gregori and A. Sapena, On xed point theorems in fuzzy metric spaces, Fuzzy Sets Syst.,##125 (2002), 245{253.##[13] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetica, 11(5)##(1975), 336{344.##[14] V. La Rosa and P. Vetro, Fixed points for Geraghtycontractions in partial metric spaces, J.##Nonlinear Sci. Appl., 7(1) (2014), 1{10.##[15] D. Mihet, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets Syst., 144 (2004),##[16] D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets Syst., 158 (2007),##[17] D. Mihet, Fuzzy contractive mappings in nonArchimedean fuzzy metric spaces, Fuzzy Sets##Syst., 159 (2008), 739{744.##[18] D. Mihet, A class of contractions in fuzzy metric spaces, Fuzzy Sets Syst., 161 (2010),##1131{1137.##[19] P. P. Murthy, U. Mishra, Rashmi and C. Vetro, Generalized ('; )weak contractions involv##ing (f; g)reciprocally continuous maps in fuzzy metric spaces, Ann. Fuzzy Math. Inform.,##5(1) (2013), 45{57.##[20] R. Saadati, S. M. Vaezpour and Y. J. Cho, Quicksort algorithm: Application of a xed point##theorem in intuitionistic fuzzy quasimetric spaces at a domain of words, J. Comput. Appl.##Math., 228(1) (2009), 219{225.##[21] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for  contractive type mappings,##Nonlinear Anal., 75 (2012), 2154{2165.##[22] B. Schweizer and A. Sklar, Statistical metric spaces, Pacic J. Math., 8(3) (1965), 338{353.##[23] C. Vetro, D. Gopal and M. Imdad, Common xed point theorems for (; )weak contractions##in fuzzy metric spaces, Indian J. Math., 52(3) (2010), 573{590.##[24] C. Vetro and P. Vetro, Common xed points for discontinuous mappings in fuzzy metric##spaces, Rend. Circ. Mat. Palermo, 57(2) (2008), 295{303.##[25] L. A. Zadeh, Fuzzy Sets, Inform. Control, 10(1) (1960), 385{389.##]
Persiantranslation vol. 14, no. 3, June 2017
2
2
1

181
191