IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3332 Research Paper Cover vol. 14, no. 4, August 2017 01 08 2017 14 4 0 0 03 09 2017 03 09 2017 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3332.html

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IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3323 Research Paper SOME COMPUTATIONAL RESULTS FOR THE FUZZY RANDOM VALUE OF LIFE ACTUARIAL LIABILITIES de Andres-Sanchez J. Social and Business Research Laboratory, Department of Business Management, Rovira i Virgili University, Spain Puchades L. Gonzalez-Vila Department of Mathematics for Economics, Finance and Actuarial Science, University of Barcelona, Spain 30 08 2017 14 4 1 25 02 08 2016 02 01 2017 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3323.html

The concept of fuzzy random variable has been applied in several papers to model the present value of life insurance liabilities. It allows the fuzzy uncertainty of the interest rate and the probabilistic behaviour of mortality to be used throughout the valuation process without any loss of information. Using this framework, and considering a triangular interest rate, this paper develops closed expressions for the expected present value and its defuzzified value, the variance and the distribution function of several well-known actuarial liabilities structures, namely life insurances, endowments and life annuities.

Financial pricing Life insurance Endowment Life annuity Stochastic mortality Fuzzy numbers Fuzzy triangular interest rate Fuzzy random variable Fuzzy financial mathematics Fuzzy life insurance mathematics
 A. Alegre and M. Claramunt, Allocation of solvency cost in group of annuities: Actuarial principles and cooperative game theory, Insurance: Mathematics and Economics, 17 (1995),  J. Andres-Sanchez and L. Gonzalez-Vila Puchades, Using fuzzy random variables in life annuities pricing, Fuzzy sets and Systems, 188 (2012), 27-44.  J. Andres-Sanchez and L. Gonzalez-Vila Puchades, A fuzzy random variable approach to life insurance pricing, In A. Gil-Lafuente; J. Gil-Lafuente and J.M. Merigo (Eds.), Studies in Fuzziness and Soft Computing; Soft Computing in Management and Business Economics, Springer-Verlag, Berlin/Heidelberg, (2012), 111-125.  J. Andres-Sanchez and L. Gonzalez-Vila Puchades, Pricing endowments with soft computing, Economic Computation and economic cybernetics studies research, 1 (2014), 124-142.  J. Andres-Sanchez and A. Terce~no, Applications of Fuzzy Regression in Actuarial Analysis, Journal of Risk and Insurance, 70 (2003), 665-699.  J. J. Buckley, The fuzzy mathematics of nance, Fuzzy Sets and Systems, 21 (1987), 257-273.  J. J. Buckley and Y. Qu, On using -cuts to evaluate fuzzy equations, Fuzzy Sets and Systems, 38 (1990), 309-312.  L. M. Campos and A. Gonzalez, A subjective approach for ranking fuzzy numbers, Fuzzy Sets and Systems, 29 (1989), 145-153.  I. Couso, D. Dubois, S. Montes and L. Sanchez, On various de nitions of the variance of a fuzzy random variable, 5th International Symposium on Imprecise Probabilities and Their Applications, Prague, (2007), 135-144.  J. D. Cummins and R. A. Derrig, Fuzzy nancial pricing of property-liability insurance, North American Actuarial Journal, 1 (1997), 21-44.  R. A. Derrig and K. Ostaszewski, Managing the tax liability of a property liability insurance company, Journal of Risk and Insurance, 64 (1997), 695-711.  Y. Feng, L. Hu and H. Shu, The variance and covariance of fuzzy random variables and their application, Fuzzy Sets and Systems, 120 (2001), 487-497.  H. U. Gerber, Life Insurance Mathematics, Springer-Verlag, Berlin/Heidelberg, 1995.  S. Heilpern, The expected value of a fuzzy number, Fuzzy Sets and Systems, 47(1) (1992),  R. Korner, On the variance of fuzzy random variables, Fuzzy Sets and Systems, 92 (1997),  V. Kratschmer, A uni ed approach to fuzzy random variables, Fuzzy Sets and Systems, 123 (2001), 1-9.  H. Kwakernaak, Fuzzy random variables I: de nitions and theorems, Information Sciences, 15 (1978), 1-29.  J. Lemaire, Fuzzy insurance, Astin Bulletin, 20 (1990), 33-55.  M. Li Calzi, Towards a general setting for the fuzzy mathematics of nance, Fuzzy Sets and Systems, 35 (1990), 265-280.  M. Lopez-Diaz and M. A. Gil, The -average value and the fuzzy expectation of a fuzzy random variable, Fuzzy Sets and Systems, 99 (1998), 347-352.  H. T. Nguyen, A note on the extension principle for fuzzy sets, Journal of Mathematical Analysis and Applications, 64 (1978), 369-380.  K. Ostaszewski, An Investigation Into Possible Applications of Fuzzy Sets Methods in Actu- arial Science, Society of Actuaries, Schaumburg, 1993.  E. Pitacco, Simulation in insurance, In: Goovaerts, M. De Vylder, F. Etienne and J. Haezendonck (Eds.), Insurance and risk theory, Reidel, Dordretch, (1986), 37-77.  M. L. Puri and D. A. Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and Applications, 114 (1986), 409-422.  E. Roventa and T. Spircu, Averaging procedures in defuzzi cation processes, Fuzzy Sets and Systems, 136 (2003), 375{385.  A. Shapiro, Modeling future lifetime as a fuzzy random variable, Insurance: Mathematics and Economics, 53 (2013), 864-870.  R. Viertl and D. Hareter, Fuzzy information and stochastics, Iranian Journal of Fuzzy Systems, 1 (2004), 43-56.  L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  C. Zhong and G. Zhou, The equivalence of two de nitions of fuzzy random variables, Proceedings of the 2nd IFSA Congress (1987), Tokyo, 59-62,
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3324 Research Paper CERTAIN TYPES OF EDGE m-POLAR FUZZY GRAPHS Akram Muhammad Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan Waseem Neha Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan Dudek Wieslaw A. Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wyb. Wyspianskiego 27, 50-370, Wroclaw, Poland 30 08 2017 14 4 27 50 02 04 2016 02 02 2017 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3324.html

In this research paper, we present a novel frame work for handling \$m\$-polar information by combining the theory of \$m-\$polar fuzzy  sets with graphs. We introduce certain types of edge regular \$m-\$polar fuzzy graphs and edge irregular \$m-\$polar fuzzy graphs. We describe some useful properties of edge regular, strongly edge irregular and strongly edge totally irregular \$m-\$polar fuzzy graphs. We discuss the relationship between degree of a vertex and degree of an edge in an \$m-\$polar fuzzy graph. We investigate edge irregularity on a path on \$2n\$ vertices and barbell graph \$B_{n,n}.\$We also present an application of \$m-\$polar fuzzy graph to decision making.

Barbell graph \$m-\$polar fuzzy sets \$m-\$polar fuzzy graphs Strongly edge totally irregular \$m-\$polar fuzzy graphs Decision making
 M. Akram, Bipolar fuzzy graphs, Information Sciences, 181 (2011), 5548-5564.  M. Akram and A. Adeel, m-polar fuzzy labeling graphs with application, Math. Computer Sci., 10 (2016), 387-402.  M. Akram and W. A. Dudek, Regular bipolar fuzzy graphs, Neural Computing & Applications, 21 (2012), 197-205.  M. Akram and H. R. Younas, Certain types of irregular m-polar fuzzy graphs, J. Appl. Math. Computing, 53(1) (2017), 365-382.  M. Akram and N. Waseem, Certain metrics in m-polar fuzzy graphs, New Math. Natural Computation, 12 (2016), 135-155.  P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letter, 6 (1987), 297-  J. Chen, S. G. Li, S. Ma and X. Wang, m-polar fuzzy sets, The Scienti c World Journal, Article ID 416530, 2014 (2014), 8 pages.  A. Kau man, Introduction to la Theorie des Sous-emsembles Flous, Masson et Cie, 1 (1973).  S. Mathew and M. S. Sunitha, Types of arcs in a fuzzy graph, Information Sciences, 179 (2009), 1760-1768.  N. R. S. Maheswari and C. Sekar, On strongly edge irregular fuzzy graphs, Kragujevac J. Math., 40 (2016), 125-135.  J. N. Mordeson and P. S. Nair, Fuzzy graphs and fuzzy hypergraphs, Physica Verlag, Heidel- berg, 2nd Edition, 2001.  S. P. Nandhini and E. Nandhini, Strongly irregular fuzzy graphs, Internat. J. Math. Archive, 5 (2014), 110-114.  A. Nagoorgani and K. Radha, Regular property of fuzzy graphs, Bull. Pure Appl. Sci., 27E (2008), 411-419.  K. Radha and N. Kumaravel, On edge regular fuzzy graphs, Internat. J. Math. Archive, 5(9) (2014), 100-112.  A. Rosenfeld, Fuzzy graphs, Fuzzy Sets and Their Applications, Academic Press, New York, (1975), 77-95.  P. K. Singh and Ch. A. Kumar, Bipolar fuzzy graph representation of concept lattice, Infor- mation Sciences, 288 (2014), 437-448.  H. L. Yang, S. G. Li, W. H. Yang and Y. Lu, Notes on bipolar fuzzy graphs, Information Sciences, 242 (2013), 113-121.  L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  L. A. Zadeh, Similarity relations and fuzzy orderings, Information Sciences, 3 (1971), 177-  W. R. Zhang, Bipolar fuzzy sets and relations: a computational framework forcognitive mod- eling and multiagent decision analysis, Proc. of IEEE conf., (1994), 305-309.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3325 Research Paper ARITHMETIC-BASED FUZZY CONTROL Dombi Jozsef Institute of Informatics, University of Szeged, Szeged, Hungary Szepe Tamas Department of Technical Informatics, University of Szeged, Szeged, Hungary 30 08 2017 14 4 51 66 03 12 2015 03 12 2016 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3325.html

Fuzzy control is one of the most important parts of fuzzy theory for which several approaches exist. Mamdani uses \$alpha\$-cuts and builds the union of the membership functions which is called the aggregated consequence function. The resulting function is the starting point of the defuzzification process. In this article, we define a more natural way to calculate the aggregated consequence function via arithmetical operators. Defuzzification is the optimum value of the resultant membership function. The left and right hand sides of the membership function will be handled separately. Here, we present a new ABFC (Arithmetic Based Fuzzy Control) algorithm based on arithmetic operations which use a new defuzzification approach. The solution is much smoother, more accurate, and much faster than the classical Mamdani controller.

Fuzzy controller Mamdani controller Defuzzification Fuzzy arithmetic
 S. Assilian, Arti cial intelligence in the control of real dynamical systems, Ph.D. Thesis, London University, Great Britain, 1974.  J. Dombi, Pliant arithmetics and pliant arithmetic operations, Acta Polytechnica Hungarica, 6(5) (2009), 19{49.  D. Driankov, H. Hellendoorn and M. Reinfrank, An introduction to fuzzy control, Springer Science & Business Media, Berlin, 2013.  D. Dubois and H. Prade, Operations on fuzzy numbers, Int. J. Systems Science, 9 (1978),  D. Dubois and H. Prade, Fuzzy members: An overview, Analysis of Fuzzy Information, Vol. I., CRC Press, Boca Raton, FL, (1987), 3{39.  D. Dubois and H. Prade, Special issue on fuzzy numbers, Fuzzy Sets and System, 24 (3),  D. Filev and R.Yager, A generalized defuzzi cation method via BAD distributions, Internat. J. Intell. Systems, 6 (1991), 689{697.  R. Fuller and R. Mesiar, Special issue on fuzzy arithmetic, Fuzzy Sets and System, 91(2)  R. Jain, Tolerance analysis using fuzzy sets, International Journal of Systems Science, 7(12) (1976), 1393{1401.  T. Jiang and Y. Li, Generalized defuzzi cation strategies and their parameter learning pro- cedures, IEEE Trans. Fuzzy Systems, 4 (1996), 64{71.  A. Kaufmann and M. M. Gupta, Introduction to fuzzy arithmetic: theory and applications, Van Nostrand Reinhold, New York, 1985.  A. Kaufmann and M. M. Gupta, Fuzzy mathematical models in engineering and management science, North-Holland, Amsterdam, 1988.  E. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proc. IEE, 121 (1974), 1585{1588, .  M. Mares, Computation over fuzzy quantities, CRC Press, Boca Raton, FL, 1994.  M. Mizumoto and K. Tanaka, The four operations of arithmetic on fuzzy numbers, Systems Comput. Controls, 7(5) (1976), 73{81.  M. Mizumoto and K. Tanaka, Algebraic properties of fuzzy numbers, Proc. Int. Conf. On Cybernetics and Society, Washington, DC, (1976), 559{563.  S. Nahmias, Fuzy variables, Fuzzy Sets and System, 1 (1978), 97{110.  H. T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl., 64 (1978), 369{380.  A. Patel and B. Mohan, Some numerical aspects of center of area defuzzi cation method, Fuzzy Sets and Systems, 132 (2002), 401{409.  S. Roychowdhury and B.-H.Wang, Cooperative neighbors in defuzzi cation, Fuzzy Sets and Systems,78 (1996), 37{49.  S. Roychowdhury and W. Pedrycz, A survey of defuzzi cation strategies, Internat. J. Intell. Systems, 16 (2001), 679{695.  A. Sakly and M. Benrejeb, Activation of trapezoidal fuzzy subsets with di erent inference methods, International Fuzzy Systems Association World Congress, Springer Berlin Heidel- berg, (2003), 450{457.  Q. Song and R. Leland, Adaptive learning defuzzi cation techniques and applications, Fuzzy Sets and Systems, 81 (1996), 321{329.  M. Sugeno, Industrial Applications of Fuzzy Control, Elsevier Science Publishers, New York,  E. Van Broekhoven and B. De Baets, Fast and accurate centre of gravity defuzzi cation of fuzzy system outputs de ned on trapezoidal fuzzy partitions, Fuzzy Sets and Systems, 157 (2006), 904{918.  W. Van Leekwijck and E. Kerre, Defuzzi cation: criteria and classi cation, Fuzzy Sets and Systems, 108 (1999), 159{178.  R. Yager and D. Filev, SLIDE: a simple adaptive defuzzi cation method, IEEE Trans. Fuzzy Systems, 1 (1993), 69{78.  R. C. Young, The algebra of many-valued quantities, Math. Ann., 104 (1931), 260{290.  L. A. Zadeh, The concept of a linquistic variable and its application to approximate reasoning, Information Sciences, 1(8) (1975), 199{249.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3326 Research Paper AN OPTIMAL FUZZY SLIDING MODE CONTROLLER DESIGN BASED ON PARTICLE SWARM OPTIMIZATION AND USING SCALAR SIGN FUNCTION Chaouech Lotfi Laboratoire d'Ingenierie des Systemes Industriels et des Energies Renouvelables (LISIER), The National Higher Engineering School of Tunis (ENSIT), BP 56, 1008 Tunis, Tunisia Soltani Mo^ez Laboratoire d'Ingenierie des Systemes Industriels et des Energies Renouvelables (LISIER), The National Higher Engineering School of Tunis (ENSIT), BP 56, 1008 Tunis, Tunisia Dhahri Slim Laboratoire d'Ingenierie des Systemes Industriels et des Energies Renouvelables (LISIER), The National Higher Engineering School of Tunis (ENSIT), BP 56, 1008 Tunis, Tunisia Chaari Abdelkader Laboratoire d'Ingenierie des Systemes Industriels et des Energies Renouvelables (LISIER), The National Higher Engineering School of Tunis (ENSIT), BP 56, 1008 Tunis, Tunisia 30 08 2017 14 4 67 85 03 03 2016 03 01 2017 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3326.html

This paper addresses the problems caused by an inappropriate selection of sliding surface parameters in fuzzy sliding mode controllers via an optimization approach. In particular, the proposed method employs the parallel distributed compensator scheme to design the state feedback based control law. The controller gains are determined in offline mode via a linear quadratic regular. The particle swarm optimization is incorporated into the linear quadratic regular technique for determining the optimal weight matrices. Consequently, an optimal sliding surface is obtained using the scalar \$sign\$ function. This latter is used to design the proposed control law. Finally, the effectiveness of the proposed fuzzy sliding mode controller based on parallel distributed compensator and using particle swarm optimization is evaluated by comparing the obtained results with other reported in literature.

Sliding mode control Takagi-Sugeno fuzzy model particle swarm optimization Parallel distributed compensator Linear quadratic regulator
 F. Allouani, D. Boukhetala and F. Boudjema, Particle swarm optimization based fuzzy sliding mode controller for the twin rotor mimo system, 16th IEEE Mediterranean Electrotechnical Conference (MELECON), (2012), 1063{1066.  P. P. Bonissone, V. Badami, K. Chaing, P. Khedkar, K. Marcelle and M. Schutten, Industrial applications of fuzzy logic at general electric, Proceedings of the IEEE, 83(3) (1995), 450{  A. Boubaki, F. Boudjema, C. Boubakir and S. Labiod, A fuzzy sliding mode controller using nonlinear sliding surface applied to the coupled tanks system, International Journal of Fuzzy Systems, 10(2) (2008), 112{118.  W. Chang, J. B. Park, Y. H. Joob and G. Chen, Design of robust fuzzy-model- based controller with sliding mode control for siso nonlinear systems, Fuzzy Sets and Systems, 125(1) (2002),  L. Chaouech and A. Chaari, Design of sliding mode control of nonlinear system based on Takagi-Sugeno fuzzy model, World Congress on Computer and Information Technology (WCCIT), (2013), 1{6.  L. Chaouech, M. Soltani, S. Dhahri and A. Chaari, Design of new fuzzy sliding mode con- troller based on parallel distributed compensation controller and using the scalar sign func- tion, Mathematics and Computers in Simulation, 132 (2017), 277{288.  P. C. Chen, C. W. Chen and W. L. Chiang, GA-based fuzzy sliding mode controller for nonlinear systems, Mathematical Problems in Engineering, 2008 (2008), 1{16.  C. M. Dorling and A. S. I. Zinober, Two approaches to hyperplane design in multivariable variable structure control systems, International Journal of Control, 44(1) (1986), 65{82.  P. Durato, C. Abdallah and V. Cerone, Linear quadratic control: An introduction, Prentice Hall, USA, 1995.  S. El Beid and S. Doubabi, DSP-Based implementation of fuzzy output tracking control for a boost converter, IEEE Transactions on Industrial Electronics, 61(1) (2014), 196{209.  P. Guan, X. J. Liu and J. Z. Liu, Adaptive fuzzy sliding mode control for exible satellite, Engineering Applications of Arti cial Intelligence, 18(4) (2005), 451{459.  S. Hong and R. Langari, Robust fuzzy control of a magnetic bearing system subject to har- monic disturbances, IEEE Transactions on Control Systems Technology, 8 (2) (2000), 366{  Z. Hongbing, P. Chengdong, K. Eguchi and G. Jinguang, Euclidean particle swarm opti- mization, Second International Conference on Intelligent Networks and Intelligent Systems, Tianjin (2009), 669{672.  Y. J. Huang and H. K. Wei, Sliding mode control design for discrete multivariable systems with time-delayed input signals, International Journal of Systems Science, 33(10) (2002),  L. Hung, H. Lin and H. Chung, Design of self-tuning fuzzy sliding mode control for TORA system, Expert Systems with Applications, 32 (1) (2007), 201{212.  E. Iglesias, Y. Garcia, M. Sanjuan, O. Camacho and C. Smith, Fuzzy surface-based sliding mode control, ISA Transactions, 43(1) (2007), 73{83.  A. Isidoti, Nonlinear Control Systems, Springer, Berlin, 1989.  K. Jafar, B. M. Mohammad and K. Mansour, Feedback-linearization and fuzzy controllers for trajectory tracking of wheeled mobile robots, Kybernetes, 39(1) (2010), 83{106.  K. Jafar and B. M. Mohammad, From Nonlinear to Fuzzy Approaches in Trajectory Tracking Control of Wheeled Mobile Robots, Asian Journal of Control, 14 (4) ( 2012), 960{973.  A. Khosla, S. Kumar and K. R. Ghosh, A comparison of computational e orts between particle swarm optimization and genetic algorithm for identi cation of fuzzy models, Fuzzy Information Processing Society, (2007), 245{250.  R. J. Lian, Adaptive self-organizing fuzzy sliding-mode radial basis-function neural-network controller for robotic systems, IEEE Transactions on Industrial Electronics, 61(3) (2014), 1493{1503.  C. Liang and J. P. Su, A new approach to the design of a fuzzy sliding mode controller, Fuzzy Sets and Systems, 139(1) (2003), 111{124.  Z. Liang, Y. Yang and Y. Zeng, Eliciting compact T-S fuzzy models using subtractive clus- tering and coevolutionary particle swarm optimization, Neuro-computing, 72(10-12) (2009), 2569{2575.  M. Mohamed, M. Anis, L. Majda, S. N. Ahmed and B. A. Ridha, Fuzzy discontinuous term for a second order asymptotic dsmc: An experimental validation on a chemical reactor, Asian Journal of Control, 13(3) (2010), 369{381.  R. M. Nagarale and B. M. Patre, Decoupled neural fuzzy sliding mode control of nonlinear systems, IEEE International Conference on Fuzzy Systems, (2013), 1{8.  T. Niknam and B. Amiri, An ecient hybrid approach based on PSO, ACO and k-means for cluster analysis, Applied Soft Computing, 10(1) (2010), 183{197.  V. Panchal, K. Harish and K. Jagdeep, Comparative study of particle swarm optimization based unsupervised clustering techniques, International Journal of Computer Science and Network Security, 9(10) (2009), 132{140.  K. Saji and K. Sasi, Fuzzy sliding mode control for a PH process, IEEE International Conference on Communication Control and Computing Technologies, (2010), 276{281.  A. Shahraz and R. B. Boozarjomehry, A fuzzy sliding mode control approach for nonlinear chemical processes, Control Engineering Practice, 17(5) (2009), 541{550.  S. F. Shehu, D. Filev and R. Langari, Fuzzy Control: Synthesis and Analysis, John Wiley and Sons LTD, USA, 1997.  L. Shieh, Y. Tsay and R. Yates, Some properties of matrix sign function derived from contin- ued fractions, IEEE Proceedings of Control Theory and Applications, 130 (1983), 111{118.  M. Singla, L. S. Shieh, G. Song, L. Xie and Y. Zhang, A new optimal sliding mode controller design using scalar sign function, ISA Transactions, 53(2) (2014), 267{279.  M. Soltani and A. Chaari, A PSO-Based fuzzy c-regression model applied to nonlinear data modeling, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 23(6) (2015), 881{892.  M. Sugeno and G. Kang, Fuzzy modeling and control of multilayer incinerator, Fuzzy Sets and Systems, 18 (3) (1986), 329{345.  T. Takagi and M. Sugeno, Fuzzy identi cation of systems and its application to modeling and control, IEEE Transactions on Systems, Man and Cybernetics, 15(1) (1985 ), 116{132.  K. Tanaka and M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems, 45(2)(1992), 135{156.  V. I. Utkin, Variable structure systems with sliding mode, IEEE Transactions on Automatic Control, 26(2) (1977), 212{222.  S. Vishnu Teja, T. N. Shanavas and S. K. Patnaik, Modi ed PSO based sliding-mode controller parameters for buck converter, Conference on Electrical, Electronics and Computer Science (SCEECS), (2012), 1{4.  R. Wai, C. Lin and C. Hsu, Adaptive fuzzy sliding-mode control for electrical servo drive, Fuzzy Sets and Systems, 143(2) (2004), 295{310.  H. O. Wang, K. Tanaka and M. F. Grin, An approach to fuzzy control of nonlinear systems: Stability and design issues, IEEE Transactions on Fuzzy Systems, 4(1) (1996), 14{23.  H. O. Wang, K. Tanaka and M. F. Grin, Parallel distributed compensation of nonlinear systems by Takagi-Sugeno fuzzy model, Proceedings FUZZY- IEEE/IFES, (1995), 531{538.  G. O. Wang, K. Tanaka and T. Ikeda, Fuzzy modeling and control of chaotic systems, IEEE Symposium Circuits and Systems, Atlanta, USA 3 (1996), 209{212.  T. Wang, W. Xie and Y. Zhang, Sliding mode fault tolerant control dealing with modeling uncertainties and actuator faults, ISA Transactions, 51(3) (2012), 386{392.  J. Wu, M. Singla, C. Olmi, L. Shieh and G. Song, Digital controller design for absolute value function constrained nonlinear systems via scalar sign function approach, ISA Transactions, 49(3) (2010), 302{310.  Y. Xinghuo, Z. Man and B. Wu, Design of fuzzy sliding-mode control systems, Fuzzy Sets and Systems, 95 (3) (1998), 295{306.  F. K. Yeh and C. M. Chen, J. J. Huang, Fuzzy sliding-mode control for a MINI-UAV, IEEE International Symposium on Computational Intelligence in Scheduling, (2010), 3317{3323.  K. Young, V. I. Utkin and U. Ozguner, A control engineer's guide to sliding mode control, IEEE Transactions on Control Systems Technology, 7(3) (1999), 328{342.  Y. Zhang, D. Huang, M. Ji and F. Xie, Image segmentation using PSO and PCM with mahalanobis distance, Expert Systems with Applications, 38(7) (2011), 9036{9040.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3327 Research Paper INTERVAL-VALUED INTUITIONISTIC FUZZY SETS AND SIMILARITY MEASURE Pekala Barbara Interdisciplinary Centre for Computational Modelling, Faculty of Mathematics and Natural Sciences, University of Rzeszow, Pigonia 1, 35-310 Rzeszow, Poland Balicki Krzysztof Interdisciplinary Centre for Computational Modelling, Faculty of Mathematics and Natural Sciences, University of Rzeszow, Pigonia 1, 35-310 Rzeszow, Poland 30 08 2017 14 4 87 98 03 04 2016 03 01 2017 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3327.html

In this paper, the problem of measuring the degree of inclusion and similarity measure for two   interval-valued intuitionistic  fuzzy sets is considered. We propose inclusion and similarity measure by using  order on interval-valued intuitionistic fuzzy sets connected with lexicographical order. Moreover, some properties of inclusion and similarity measure and some correlation, between them and aggregations are examined. Finally, we have included example of ranking problem in car showrooms.

Interval-valued intuitionistic fuzzy sets Inclusion measure Similarity measure
 K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87{96.  K. T. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets Syst., 64 (1994), 159{174.  K. T. Atanassov, Intuitionistic Fuzzy Sets. Theory and Applications, Physica-Verlag, Heidelberg/ New York, 1999.  U. Bodenhofer, B. De Baets and J. Fodor, A compendium of fuzzy weak orders: Representa- tions and constructions, Fuzzy Sets Syst., 158 (2007), 811{829.  K. Bosteels and E. E. Kerre, On a re exivity-preserving family of cardinality-based fuzzy comparison measures, Inform. Sci., 179 (2009), 2342{2352.  H. Bustince, J. Fernandez, R. Mesiar, J. Montero and R. Orduna, Overlap functions, Nonlinear Anal.: Theory Methods Appl., 72 (2010), 1488{1499.  H. Bustince, M. Pagola, R. Mesiar, E. Hullermeier and F. Herrera, Grouping, overlaps, and generalized bientropic functions for fuzzy modeling of pairwise comparisons, IEEE Trans. Fuzzy Syst., 20(3) (2012), 405{415.  H. Bustince, J. Fernandez, A. Kolesarova and R. Mesiar, Generation of linear orders for intervals by means of aggregation functions, Fuzzy Sets Syst., 220 (2013), 69-77.  T. Calvo, A. Kolesarova, M. Komornikova and R. Mesiar, Aggregation operators: properties, classes and construction methods, In T. Calvo, G. Mayor, and R. Mesiar (Eds.), Physica- Verlag, New York, Aggregation Operators. Studies in Fuzziness and Soft Computing, 97 (2002), 3-104.  B. De Baets and R. Mesiar, Triangular norms on product lattices, Fuzzy Sets Syst., 104 (1999), 61{76.  B. De Baets, H. De Meyer and H. Naessens, A class of rational cardinality-based similarity measures, J. Comp. Appl. Math., 132 (2001), 51{69.  B. De Baets and H. De Meyer, Transitivity frameworks for reciprocal relations:cycle- transitivity versus FG-transitivity, Fuzzy Sets Syst., 152 (2005), 249{270.  B. De Baets, S. Janssens and H. De Meyer, On the transitivity of a parametric family of cardinality-based similarity measures, Int. J. Appr. Reason., 50 (2009), 104{116.  M. De Cock and E. E. Kerre, Why fuzzy T-equivalence relations do not resolve the Poincar'e paradox, and related issues, Fuzzy Sets Syst., 133 (2003), 181{192.  L. De Miguel, H. Bustince, J. Fernandez, E. Indurain, A. Kolesarova and R. Mesiar, Con- struction of admissible linear orders for interval-valued Atanassov intuitionistic fuzzy sets with an application to decision making, Information Fusion, 27 (2016), 189-197.  S. Freson, B. De Baets and H. De Meyer, Closing reciprocal relations w.r.t. stochastic tran- sitivity, Fuzzy Sets Syst., 241 (2014), 2{26.  B. Jayaram and R. Mesiar, I-Fuzzy equivalence relations and I-fuzzy partitions, Inf. Sci., 179 (2009), 1278{1297.  D. F. Li, Toposis-based nonlinear-programming methodology for multiattribute decision mak- ing with interval-valued intuitionistic fuzzy sets, IEEE Trans. Fuzzy Syst., 18 (2010), 299{311.  X. D. Liu, S. H. Zheng and F. L. Xiong, Entropy and subsethood for general interval-valued intuitionistic fuzzy sets, Lecture Notes Artif. Intell., 3613 (2005), 42{52.  N. Madrid, A. Burusco, H. Bustince, J. Fernandez and I. Per lieva, Upper bounding overlaps by groupings, Fuzzy Sets Syst., 264 (2015), 76{99.  S. Ovchinnikov, Numerical representation of transitive fuzzy relations, Fuzzy Sets Syst., 126 (2002), 225{232.  D. G. Park, Y. C. Kwun, J. H. Park and I. Y. Park, Correlation coecient of interval- valued intuitionistic fuzzy sets and its application to multiple attribute group decision making problems, Math. Comput. Modell., 50 (2009), 1279{1293.  Z.  Switalski, General transitivity conditions for fuzzy reciprocal preference matrices, Fuzzy Sets Syst., 137 (2003), 85{100.  L. A. Zadeh, Fuzzy sets, Information Contr., 8 (1965), 338 { 353.  W. Y. Zeng and P. Guo, Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship, Inf. Sci., 178 (2008), 1334{1342.  H. Y. Zhang and W. X. Zhang, Hybrid monotonic inclusion measure and its use in measuring similarity and distance between fuzzy sets, Fuzzy Sets Syst., 160 (2009), 107{118.  Q. Zhang, H. Xing, F. Liu and J. Ye, P. Tang, Some new entropy measures for interval- valued intuitionistic fuzzy sets based on distances and their relationships with similarity and inclusion measures, Inf. Sci., 283 (2014), 55{69.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3328 Research Paper TOPOLOGICAL SIMILARITY OF L-RELATIONS Hao Jing College of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450000, China Huang Shasha College of Mathematics and Information, North China University of Water Resources and Electric Power, Zhengzhou, 450045, China 30 08 2017 14 4 99 115 03 06 2016 03 12 2017 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3328.html

\$L\$-fuzzy rough sets are extensions of the classical rough sets by relaxing theequivalence relations to \$L\$-relations. The topological structures induced by\$L\$-fuzzy rough sets have opened up the way for applications of topological factsand methods in granular computing. In this paper, we firstly prove thateach arbitrary \$L\$-relation can generate an Alexandrov \$L\$-topology.Based on this fact, we introduce the topological similarity of \$L\$-relations,denote it by T-similarity, and we give intuitive characterization ofT-similarity. Then we introduce the variations of a given \$L\$-relation andinvestigate the relationship among them. Moreover, we prove that each\$L\$-relation is uniquely topological similar to an \$L\$-preorder. Finally,we investigate the related algebraic structures of different sets of\$L\$-relations on the universe.

\$L\$-fuzzy rough set \$L\$-relation Alexandrov \$L\$-topology \$L\$-preorder Topological similarity
 K. Blount and C. Tsinakis, The structure of residuated lattices, International Journal of Algebra and Computation, 13(4) (2003), 437{461.  D. Boixader, J. Jacas and J. Recasens, Upper and lower approximations of fuzzy sets, International Journal of General System, 29(4) (2000), 555{568.  C. L. Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 24(1) (1968), 182{190.  X. Chen and Q. Li, Construction of rough approximations in fuzzy setting, Fuzzy sets and Systems, 158(23) (2007), 2641{2653.  M. De Cock, C. Cornelis and E. Kerre, Fuzzy rough sets: beyond the obvious, Proceedings of 2004 IEEE International Conference on Fuzzy Systems, 1 (2004), 103{108.  D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General System, 17(2-3) (1990), 191{209.  J. A. Goguen, L-fuzzy sets, Journal of Mathematical Analysis and Applications, 18(1) (1967),  J. Hao and Q. Li, The relationship between L-fuzzy rough set and L-topology, Fuzzy Sets and Systems, 178(1) (2011), 74{83.  J. Jarvinen and J. Kortelainen, A unifying study between modal-like operators, topologies and fuzzy sets, Fuzzy Sets and Systems, 158(11) (2007), 1217{1225.  H. Lai and D. Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157(14) (2006), 1865{1885.  Z. Li and R. Cui, T-similarity of fuzzy relations and related algebraic structures, Fuzzy Sets and Systems, 275 (2015), 130{143.  G. Liu, Generalized rough sets over fuzzy lattices, Information Sciences, 178(6) (2008), 1651{  G. Liu and W. Zhu, The algebraic structures of generalized rough set theory, Information Sciences, 178(21) (2008), 4105{4113.  R. Lowen, Fuzzy topological spaces and fuzzy compactness, Journal of Mathematical Analysis and Applications, 56(3) (1976), 621{633.  Z. M. Ma and B. Q. Hu, Topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets, Information Sciences, 218 (2013), 194{204.  N. N. Morsi and M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy sets and Systems, 100(1) (1998), 327{342.  Z. Pawlak, Rough sets, International Journal of Computer & Information Sciences, 11(5) (1982), 341{356.  K. Qin and Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems, 151(3) (2005), 601{613.  A. M. Radzikowska and E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and Systems, 126(2) (2002), 137{155.  A. M. Radzikowska and E. E. Kerre, Fuzzy rough sets based on residuated lattices, In: Transactions on Rough Sets II, LNCS 3135, (2004), 278{296.  A. M. Radzikowska and E. E. Kerre, Characterisation of main classes of fuzzy relations using fuzzy modal operators, Fuzzy Sets and Systems, 152(2) (2005), 223{247.  Y. H. She and G. J. Wang, An axiomatic approach of fuzzy rough sets based on residuated lattices, Computers & Mathematics with Applications, 58(1) (2009), 189{201.  A. Skowron and J. Stepaniuk, Tolerance approximation spaces, Fundamenta Informaticae, 27(2-3) (1996), 245{253.  H. Thiele, On axiomatic characterization of fuzzy approximation operators II, the rough fuzzy set based case, Proceedings of the 31st IEEE International Symposium on Multiple-Valued Logic, (2001), 330{335.  H. Thiele, On axiomatic characterization of fuzzy approximation operators III, the fuzzy dia- mond and fuzzy box based cases, The 10th IEEE International Conference on Fuzzy Systems, 2 (2001), 1148{1151.  D. Vanderpooten, Similarity relation as a basis for rough approximations, Advances in Machine Intelligence and Soft Computing, 4 (1997), 17{33.  C. Y. Wang and B. Q. Hu, Fuzzy rough sets based on generalized residuated lattices, Information Sciences, 248 (2013), 31{49.  M.Ward and R. P. Dilworth, Residuated lattices, Transactions of the American Mathematical Society, 45(3) (1939), 335{354.  W. Z.Wu, Y. Leung and J. S. Mi, On characterizations of (I;T )-fuzzy rough approximation operators, Fuzzy Sets and Systems, 154(1) (2005), 76{102.  W. Z. Wu, J. S. Mi and W. X. Zhang, Generalized fuzzy rough sets, Information Sciences, 151 (2003), 263{282.  Y. Yao, Constructive and algebraic methods of the theory of rough sets, Information Sciences, 109(1) (1998), 21{47.  W. Zhu, Topological approaches to covering rough sets, Information Sciences, 177(6) (2007), 1499{1508.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3329 Research Paper FUZZY INCLUSION LINEAR SYSTEMS Ghanbari Mojtaba Department of Mathematics, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran 30 08 2017 14 4 117 137 03 05 2016 03 12 2016 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3329.html

In this manuscript, we introduce  a new class of fuzzy problems, namely ``fuzzy inclusion linear systems" and   propose a fuzzy solution set for it. Then, we present a theoretical discussion about the relationship between  the fuzzy solution set of a  fuzzy inclusion linear system and the algebraic solution of a fuzzy linear system. New necessary and sufficient conditions are derived for obtaining the unique algebraic solution for a fuzzy linear system. Also, all new concepts are illustrated by numerical examples.

Fuzzy linear system Fuzzy inclusion linear system Fuzzy solution set Lower \$r\$-boundary Upper \$r\$-boundary
 T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation, 155 (2004), 493–502.  T. Allahviranloo and M. Ghanbari, On the algebraic solution of fuzzy linear systems based on interval theory, Applied Mathematical Modelling, 36 (2012), 5360–5379.  T. Allahviranloo and M. Ghanbari, Solving Fuzzy Linear Systems by Homotopy Perturbation Method, International Journal of Computational Cognition, 8(2) (2010), 24–30.  T.Allahviranloo, M. Ghanbari, A.A. Hosseinzadeh, E. Haghi and R. Nuraei, A note on “Fuzzy linear systems”, Fuzzy Sets and Systems, 177 (2011), 87–92.  T. Allahviranloo and M. Ghanbari, A new approach to obtain algebraic solution of interval linear systems, Soft Computing, 16 (2012), 121–133.  T. Allahviranloo, E. Haghi and M. Ghanbari, The nearest symmetric fuzzy solution for a symmetric fuzzy linear system, An. St. Univ. Ovidius Constanta, 20(1) (2012), 151–172.  T. Allahviranloo and S. Salahshour, Fuzzy symmetric solution of fuzzy linear systems, Journal of Computational and Applied Mathematics, 235(16) (2011), 4545–4553.  T. Allahviranloo, R. Nuraei, M. Ghanbari, E. Haghi and A. A. Hosseinzadeh, A new metric for L-R fuzzy numbers and its application in fuzzy linear systems, Soft Computing, 16 (2012), 1743-1754.  B. Bede and S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151 (2005), 581–599.  R. Ezzati, Solving fuzzy linear systems, Soft Computing, 15(2010), 193-197.  M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, 96(1998), 201–209.  M. Ghanbari, T. Allahviranloo and E. Haghi, Estimation of algebraic solution by limiting the solution set of an interval linear system, Soft Computing, 16(12) (2012), 2135–2142.  M. Ghanbari and R. Nuraei, Convergence of a semi-analytical method on the fuzzy linear systems, Iranian Journal of Fuzzy Systems, 11(4) (2014), 45–60.  M. Ghanbari and R. Nuraei, Note on new solutions of LR fuzzy linear systems using ranking functions and ABS algorithms, Fuzzy Inf. Eng., 5(3) (2013), 317–326.  O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984), 215–229.  R. Nuraei, T. Allahviranloo and M. Ghanbari, Finding an inner estimation of the solution set of a fuzzy linear system, Applied Mathematical Modelling, 37 (2013), 5148–5161.  C. Wu and M. Ming, On embedding problem of fuzzy number space: Part 1, Fuzzy Sets and Systems, 44 (1991), 33–38.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3330 Research Paper STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES Zhou Hongjun School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, CHINA Shi Hui-Xian School of Mathematics and Information Science, Shaanxi Normal University 30 08 2017 14 4 139 161 03 04 2016 03 01 2017 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3330.html

\$Rsb{0}\$-algebras, which were proved to be equivalent to Esteva and Godo's NM-algebras modelled by Fodor's nilpotent minimum t-norm, are the equivalent algebraic semantics of the left-continuous t-norm based fuzzy logic firstly introduced by Guo-jun Wang in the mid 1990s.In this paper, we first establish a Stone duality for the category of MV-skeletons of \$Rsb{0}\$-algebras and the category of three-valued Stone spaces.Then we extend Flaminio-Montagna internal states to \$Rsb{0}\$-algebras.Such internal states must be idempotent MV-endomorphisms of \$Rsb{0}\$-algebras.Lastly we present a Stone duality for the category of MV-skeletons of \$Rsb{0}\$-algebras with Flaminio-Montagna internal states and the category of three-valued Stone spaces with Zadeh type idempotent continuous endofunctions.These dualities provide a topological viewpoint for better understanding of the algebraic structures of \$Rsb{0}\$-algebras.

\$Rsb{0}\$-algebra Nilpotent minimum algebra MV-skeleton internal state Stone duality
 S. Aguzzoli, M.Busaniche and V. Marra, Spectral duality for nitely generated nilpotent min- imum algebras with applications, J. Logic Comput., 17 (2007), 749{765.  L. P. Belluce, Semisimple algebras of in nite valued logic and bold fuzzy set theory, Can. J. Math., 38 (1986), 1356{1379.  W. Blok and D. Pigozzi, Algebraizable logics, Merm. Math. Soc., 77 (1989), 1-89.  M. Botur and A. Dvurecenskij, State-morphism algebras{general approach, Fuzzy Sets Syst., 218 (2013), 90{102.  M. Busaniche, Free nilpotent minimum algebras, Math. Logic Quart., 52 (2006), 219{236.  C. C. Chang, Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc., 88 (1958),  R. Cignoli and D. Mundici, Stone duality for Dedekind -complete `-groups with order unit, J. Algebra, 302 (2006), 848{861.  L. C. Ciungu, Non-commutative Multiple-Valued Logic Algebras, Springer, New York, 2014.  L. C. Ciungu, G. Georgescu and C. Muresan, Generalized Bosbach states: part I, Arch. Math. Logic, 52 (2013), 335{376.  L. C. Ciungu, G. Georgescu and C. Muresan, Generalized Bosbach states: part II, Arch. Math. Logic, 52 (2013), 707{732.  A. Di Nola and A. Dvurecenskij, State-morphism MV-algebras, Ann. Pure Appl. Logic, 161 (2009), 161{173.  A. Di Nola, A. Dvurecenskij and A. Lettieri, On varieties of MV-algebras with internal states, Inter. J. Approx. Reason., 51 (2010), 680{694.  A. Di Nola, A. Dvurecenskij and A. Lettieri, Stone duality type theorems for MV-algebras with internal states, Comm. Algebra, 40 (2012), 327{342.  A. Dvurecenskij, J. Rachunek and D. Salounova, State operators on generalizations of fuzzy structures, Fuzzy Sets Syst., 187 (2012), 58{76.  C. Elkan, The paradoxical success of fuzzy logic, IEEE Expert, 9 (1994), 3{8.  F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets Syst., 124 (2001), 271{288.  T. Flaminio and F. Montagna, MV-algebras with internal states and probabilistic fuzzy logic, Inter. J. Approx. Reason., 50 (2009), 138{152.  J. Fodor, Nilpotent minimum and related connectives for fuzzy logic, in: Proc. of the 4th Inter. Conf. on Fuzzy Syst., March 20-24, Yokohama, (1995), 2077{2082.  P. F. He, X. L. Xin and Y.W. Yang, On state residuated lattices, Soft Comput., 19 (2015), 2083{2094.  A. Iorgulescu, Algebras of Logic as BCK-algebras, Editura ASE, Bucarest, 2008.  H. W. Liu and G. J. Wang, Uni ed forms of fully implication restriction methods for fuzzy reasoning, Inf. Sci., 177(3) (2007), 956{966.  L. Liu and K. Li, Involutive monoidal t-norm based logic and R0-logic, Inter. J. Intelligent Syst., 199 (2004), 491{497.  Y. M. Liu and M. K. Luo, Fuzzy topology, World Scienti c, Hong Kong, (1997), 15{68.  D. Mundici, Advanced  Lukasiewicz Calculus and MV-algebras, Springer, New York, (2011),  D. Mundici, Averaging the truth-value in  Lukasiewicz sentential logic, Stud. Logica, 55 (1995), 113{127.  Z. M. Ma and Z. W. Fu, Algebraic study to generalized Bosbach states on residuated lattices, Soft Comput., 19 (2015), 2541{2550.  D. W. Pei, On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets Syst., 138 (2003), 187{195.  D. W. Pei, R0-implication: characteristics and applications, Fuzzy Sets Syst., 131 (2002),  D. W. Pei and G. J. Wang, The completeness and applications of the formal system L, Sci. China F, 45 (2002), 40{50.  M. H. Stone, The theory of representation for Boolean algebras, Trans. Amer. Math. Soc., 40 (1936), 37{111.  G. J. Wang, A formal deductive system for fuzzy propositional calculus, Chin. Sci. Bull., 42 (1997), 1521{1526.  G. J. Wang, Fuzzy logic and fuzzy reasoning, In: Proc. of the 7th National Many-Valued and Fuzzy Logic Conf., November 10-13, Xi'an, (1996), 82{96.  G. J. Wang, Implication lattices and their fuzzy implication space representation theorem, Acta Math. Sin., (in Chinese), 42 (1999), 133{140.  G. J. Wang, L-Fuzzy Topological Spaces, Shaanxi Normal Univ. Press, Xi'an, (in Chinese), (1988), 18{56.  G. J. Wang, X. J. Hui and J. S. Song, The R0-type fuzzy logic metric space and an algorithm for solving fuzzy modus ponens, Comput. Math. Appl., 55(9) (2008), 1974{1987.  G. J. Wang and H. J. Zhou, Introduction to Mathematical Logic and Resolution Principle, Science Press, Beijing, (2009), 156{298.  S. M. Wang, B. S. Wang and X. Y. Wang, A characterization of truth-functions in the nilpotent minimum logic, Fuzzy Sets Syst., 145 (2004), 253{266.  D. X. Zhang and Y. M. Liu, L-fuzzy version of Stone's representation theorem for distributive lattices, Fuzzy Sets Syst., 76 (1995), 259{270.  H. J. Zhou, Probabilistically Quantitative Logic and its Applications, Science Press, Beijing, (in Chinese), 2015.  H. J. Zhou, G. J. Wang and W. Zhou, Consistency degrees of theories and methods of graded reasoning in n-valued R0-logic (NM-logic), Inter. J. Approx. Reason., 43 (2006), 117{132.  H. J. Zhou and B. Zhao, Characterizations of maximal consistent theories in the formal deductive system L (NM-logic) and Cantor space, Fuzzy Set Syst., 158 (2007), 2591{2604.  H. J. Zhou and B. Zhao, Generalized Bosbach and Riecan states based on relative negations in residuated lattices, Fuzzy Sets Syst., 187 (2012) 33-57.  H. J. Zhou and B. Zhao, Stone-like representation theorems and three-valued lters in R0- algebras (nilpotent minimum algebras), Fuzzy Sets Syst., 162 (2011), 1{26.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3331 Research Paper REDUNDANCY OF MULTISET TOPOLOGICAL SPACES Ghareeb A. Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt 30 08 2017 14 4 163 168 03 05 2016 03 12 2016 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3331.html

In this paper, we show the redundancies of multiset topological spaces. It is proved that \$(P^star(U),sqsubseteq)\$ and \$(Ds(varphi(U)),subseteq)\$ are isomorphic. It follows that multiset topological spaces are superfluous and unnecessary in the theoretical view point.

Multiset Multiset topology Isomorphism
 K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87{96.  W. D. Blizard, Multiset theory, Notre Dame J. Formal Logic, 30(1) (1988), 36{66.  S. A. El-Sheikh, R. A. K. Omar and M. Raafat, -operation in m-topological space, Gen. Math. Notes, 7(1) (2015), 40{54.  S. A. El-Sheikh, R. A. K. Omar and M. Raafat, Separation axioms on multiset topological space, Journal of New Theory, 7 (2015), 11{21.  K. P. Girish and J. S. Jacob, On multiset topologies, Theory and Applications of Mathematics and Computer Science, 2(1) (2012), 37{52.  K. P. Girish and S. J. John, Transactions on rough sets XIV, Springer Berlin Heidelberg, Berlin, Heidelberg, (2011), Ch. Rough Multiset and Its Multiset Topology, 62{80.  K. P. Girish and S. J. John, Multiset topologies induced by multiset relations, Information Sciences, 188 (2012), 298{313.  A. Kandil, O. Tantawy, S. El-Sheikh and A. Zakaria, Multiset proximity spaces, Journal of Egyptian Mathematical Society, 24(4) (2016), 562-567.  P. M. Mahalakshmi and P. Thangavelu, m-connectedness in m-topology, International Journal of Pure and Applied Mathematics, 106(8) (2016), 21-25.  J. Mahanta and D. Das, Boundary and exterior of a multiset topology, ArXiv e-prints, arXiv:1501.07193.  D. Molodtsov, Soft set theory- rst results, Computers and Mathematics with Applications, 37 (45) (1999), 19{31.  F. G. Shi and B. Pang, Redundancy of fuzzy soft topological spaces, Journal of Intelligent and Fuzzy Systems, 27 (4) (2014), 1757{1760.  F. G. Shi and B. Pang, A note on soft topological spaces, Iranian Journal of Fuzzy Systems, 12 (5) (2015), 149{155.  R. R. Yager, On the theory of bags, International Journal of General Systems, 13 (1986),  L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338{353.
IJFS University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 University of Sistan and Baluchestan 21 10.22111/ijfs.2017.3333 Research Paper Persian-translation vol. 14, no. 4, August 2017 01 08 2017 14 4 171 179 03 09 2017 03 09 2017 Copyright © 2017, University of Sistan and Baluchestan. 2017 http://ijfs.usb.ac.ir/article_3333.html

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