2017
14
4
0
168
Cover vol. 14, no. 4, August 2017
2
2
1

0
0
SOME COMPUTATIONAL RESULTS FOR THE FUZZY RANDOM VALUE OF LIFE ACTUARIAL LIABILITIES
2
2
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 wellknown actuarial liabilities structures, namely life insurances, endowments and life annuities.
1

1
25


J.
de AndresSanchez
Social and Business Research Laboratory, Department of Business Management, Rovira i Virgili University, Spain
Social and Business Research Laboratory,
Spain


L. GonzalezVila
Puchades
Department of Mathematics for Economics, Finance and Actuarial Science, University of Barcelona, Spain
Department of Mathematics for Economics,
Spain
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
[[1] 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),##[2] J. AndresSanchez and L. GonzalezVila Puchades, Using fuzzy random variables in life##annuities pricing, Fuzzy sets and Systems, 188 (2012), 2744.##[3] J. AndresSanchez and L. GonzalezVila Puchades, A fuzzy random variable approach to life##insurance pricing, In A. GilLafuente; J. GilLafuente and J.M. Merigo (Eds.), Studies in##Fuzziness and Soft Computing; Soft Computing in Management and Business Economics,##SpringerVerlag, Berlin/Heidelberg, (2012), 111125.##[4] J. AndresSanchez and L. GonzalezVila Puchades, Pricing endowments with soft computing,##Economic Computation and economic cybernetics studies research, 1 (2014), 124142.##[5] J. AndresSanchez and A. Terce~no, Applications of Fuzzy Regression in Actuarial Analysis,##Journal of Risk and Insurance, 70 (2003), 665699.##[6] J. J. Buckley, The fuzzy mathematics of nance, Fuzzy Sets and Systems, 21 (1987), 257273.##[7] J. J. Buckley and Y. Qu, On using cuts to evaluate fuzzy equations, Fuzzy Sets and Systems,##38 (1990), 309312.##[8] L. M. Campos and A. Gonzalez, A subjective approach for ranking fuzzy numbers, Fuzzy##Sets and Systems, 29 (1989), 145153.##[9] I. Couso, D. Dubois, S. Montes and L. Sanchez, On various denitions of the variance of##a fuzzy random variable, 5th International Symposium on Imprecise Probabilities and Their##Applications, Prague, (2007), 135144.##[10] J. D. Cummins and R. A. Derrig, Fuzzy nancial pricing of propertyliability insurance,##North American Actuarial Journal, 1 (1997), 2144.##[11] R. A. Derrig and K. Ostaszewski, Managing the tax liability of a property liability insurance##company, Journal of Risk and Insurance, 64 (1997), 695711.##[12] Y. Feng, L. Hu and H. Shu, The variance and covariance of fuzzy random variables and their##application, Fuzzy Sets and Systems, 120 (2001), 487497.##[13] H. U. Gerber, Life Insurance Mathematics, SpringerVerlag, Berlin/Heidelberg, 1995.##[14] S. Heilpern, The expected value of a fuzzy number, Fuzzy Sets and Systems, 47(1) (1992),##[15] R. Korner, On the variance of fuzzy random variables, Fuzzy Sets and Systems, 92 (1997),##[16] V. Kratschmer, A unied approach to fuzzy random variables, Fuzzy Sets and Systems, 123##(2001), 19.##[17] H. Kwakernaak, Fuzzy random variables I: denitions and theorems, Information Sciences,##15 (1978), 129.##[18] J. Lemaire, Fuzzy insurance, Astin Bulletin, 20 (1990), 3355.##[19] M. Li Calzi, Towards a general setting for the fuzzy mathematics of nance, Fuzzy Sets and##Systems, 35 (1990), 265280.##[20] M. LopezDiaz and M. A. Gil, The average value and the fuzzy expectation of a fuzzy##random variable, Fuzzy Sets and Systems, 99 (1998), 347352.##[21] H. T. Nguyen, A note on the extension principle for fuzzy sets, Journal of Mathematical##Analysis and Applications, 64 (1978), 369380.##[22] K. Ostaszewski, An Investigation Into Possible Applications of Fuzzy Sets Methods in Actu##arial Science, Society of Actuaries, Schaumburg, 1993.##[23] E. Pitacco, Simulation in insurance, In: Goovaerts, M. De Vylder, F. Etienne and J. Haezendonck##(Eds.), Insurance and risk theory, Reidel, Dordretch, (1986), 3777.##[24] M. L. Puri and D. A. Ralescu, Fuzzy random variables, Journal of Mathematical Analysis##and Applications, 114 (1986), 409422.##[25] E. Roventa and T. Spircu, Averaging procedures in defuzzication processes, Fuzzy Sets and##Systems, 136 (2003), 375{385.##[26] A. Shapiro, Modeling future lifetime as a fuzzy random variable, Insurance: Mathematics##and Economics, 53 (2013), 864870.##[27] R. Viertl and D. Hareter, Fuzzy information and stochastics, Iranian Journal of Fuzzy Systems,##1 (2004), 4356.##[28] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353.##[29] C. Zhong and G. Zhou, The equivalence of two denitions of fuzzy random variables, Proceedings##of the 2nd IFSA Congress (1987), Tokyo, 5962,##]
CERTAIN TYPES OF EDGE mPOLAR FUZZY GRAPHS
2
2
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.
1

27
50


Muhammad
Akram
Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan
Department of Mathematics, University of
Pakistan
m.akram@pucit.edu.pk


Neha
Waseem
Department of Mathematics, University of the Punjab, New Campus,
Lahore, Pakistan
Department of Mathematics, University of
Pakistan
neha_waseem@yahoo.com


Wieslaw A.
Dudek
Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wyb. Wyspianskiego 27, 50370, Wroclaw, Poland
Faculty of Pure and Applied Mathematics,
Poland
(wieslaw.dudek@pwr.wroc.pl
Barbell graph
$m$polar fuzzy sets
$m$polar fuzzy graphs
Strongly edge totally irregular $m$polar fuzzy graphs
decision making
[[1] M. Akram, Bipolar fuzzy graphs, Information Sciences, 181 (2011), 55485564.##[2] M. Akram and A. Adeel, mpolar fuzzy labeling graphs with application, Math. Computer##Sci., 10 (2016), 387402.##[3] M. Akram and W. A. Dudek, Regular bipolar fuzzy graphs, Neural Computing & Applications,##21 (2012), 197205.##[4] M. Akram and H. R. Younas, Certain types of irregular mpolar fuzzy graphs, J. Appl. Math.##Computing, 53(1) (2017), 365382.##[5] M. Akram and N. Waseem, Certain metrics in mpolar fuzzy graphs, New Math. Natural##Computation, 12 (2016), 135155.##[6] P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letter, 6 (1987), 297##[7] J. Chen, S. G. Li, S. Ma and X. Wang, mpolar fuzzy sets, The Scientic World Journal,##Article ID 416530, 2014 (2014), 8 pages. ##[8] A. Kauman, Introduction to la Theorie des Sousemsembles Flous, Masson et Cie, 1 (1973).##[9] S. Mathew and M. S. Sunitha, Types of arcs in a fuzzy graph, Information Sciences, 179##(2009), 17601768.##[10] N. R. S. Maheswari and C. Sekar, On strongly edge irregular fuzzy graphs, Kragujevac J.##Math., 40 (2016), 125135.##[11] J. N. Mordeson and P. S. Nair, Fuzzy graphs and fuzzy hypergraphs, Physica Verlag, Heidel##berg, 2nd Edition, 2001.##[12] S. P. Nandhini and E. Nandhini, Strongly irregular fuzzy graphs, Internat. J. Math. Archive,##5 (2014), 110114.##[13] A. Nagoorgani and K. Radha, Regular property of fuzzy graphs, Bull. Pure Appl. Sci., 27E##(2008), 411419.##[14] K. Radha and N. Kumaravel, On edge regular fuzzy graphs, Internat. J. Math. Archive, 5(9)##(2014), 100112.##[15] A. Rosenfeld, Fuzzy graphs, Fuzzy Sets and Their Applications, Academic Press, New York,##(1975), 7795.##[16] P. K. Singh and Ch. A. Kumar, Bipolar fuzzy graph representation of concept lattice, Infor##mation Sciences, 288 (2014), 437448.##[17] H. L. Yang, S. G. Li, W. H. Yang and Y. Lu, Notes on bipolar fuzzy graphs, Information##Sciences, 242 (2013), 113121.##[18] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353.##[19] L. A. Zadeh, Similarity relations and fuzzy orderings, Information Sciences, 3 (1971), 177##[20] W. R. Zhang, Bipolar fuzzy sets and relations: a computational framework forcognitive mod##eling and multiagent decision analysis, Proc. of IEEE conf., (1994), 305309.##]
ARITHMETICBASED FUZZY CONTROL
2
2
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.
1

51
66


Jozsef
Dombi
Institute of Informatics, University of Szeged, Szeged, Hungary
Institute of Informatics, University of Szeged,
Hungary
dombi@@inf.uszeged.hu


Tamas
Szepe
Department of Technical Informatics, University of Szeged, Szeged,
Hungary
Department of Technical Informatics, University
Hungary
Fuzzy controller
Mamdani controller
Defuzzification
Fuzzy arithmetic
[[1] S. Assilian, Articial intelligence in the control of real dynamical systems, Ph.D. Thesis,##London University, Great Britain, 1974.##[2] J. Dombi, Pliant arithmetics and pliant arithmetic operations, Acta Polytechnica Hungarica,##6(5) (2009), 19{49.##[3] D. Driankov, H. Hellendoorn and M. Reinfrank, An introduction to fuzzy control, Springer##Science & Business Media, Berlin, 2013.##[4] D. Dubois and H. Prade, Operations on fuzzy numbers, Int. J. Systems Science, 9 (1978),##[5] D. Dubois and H. Prade, Fuzzy members: An overview, Analysis of Fuzzy Information, Vol.##I., CRC Press, Boca Raton, FL, (1987), 3{39.##[6] D. Dubois and H. Prade, Special issue on fuzzy numbers, Fuzzy Sets and System, 24 (3),##[7] D. Filev and R.Yager, A generalized defuzzication method via BAD distributions, Internat.##J. Intell. Systems, 6 (1991), 689{697.##[8] R. Fuller and R. Mesiar, Special issue on fuzzy arithmetic, Fuzzy Sets and System, 91(2)##[9] R. Jain, Tolerance analysis using fuzzy sets, International Journal of Systems Science, 7(12)##(1976), 1393{1401. ##[10] T. Jiang and Y. Li, Generalized defuzzication strategies and their parameter learning pro##cedures, IEEE Trans. Fuzzy Systems, 4 (1996), 64{71.##[11] A. Kaufmann and M. M. Gupta, Introduction to fuzzy arithmetic: theory and applications,##Van Nostrand Reinhold, New York, 1985.##[12] A. Kaufmann and M. M. Gupta, Fuzzy mathematical models in engineering and management##science, NorthHolland, Amsterdam, 1988.##[13] E. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proc. IEE,##121 (1974), 1585{1588, .##[14] M. Mares, Computation over fuzzy quantities, CRC Press, Boca Raton, FL, 1994.##[15] M. Mizumoto and K. Tanaka, The four operations of arithmetic on fuzzy numbers, Systems##Comput. Controls, 7(5) (1976), 73{81.##[16] M. Mizumoto and K. Tanaka, Algebraic properties of fuzzy numbers, Proc. Int. Conf. On##Cybernetics and Society, Washington, DC, (1976), 559{563.##[17] S. Nahmias, Fuzy variables, Fuzzy Sets and System, 1 (1978), 97{110.##[18] H. T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl., 64##(1978), 369{380.##[19] A. Patel and B. Mohan, Some numerical aspects of center of area defuzzication method,##Fuzzy Sets and Systems, 132 (2002), 401{409.##[20] S. Roychowdhury and B.H.Wang, Cooperative neighbors in defuzzication, Fuzzy Sets and##Systems,78 (1996), 37{49.##[21] S. Roychowdhury and W. Pedrycz, A survey of defuzzication strategies, Internat. J. Intell.##Systems, 16 (2001), 679{695.##[22] A. Sakly and M. Benrejeb, Activation of trapezoidal fuzzy subsets with dierent inference##methods, International Fuzzy Systems Association World Congress, Springer Berlin Heidel##berg, (2003), 450{457.##[23] Q. Song and R. Leland, Adaptive learning defuzzication techniques and applications, Fuzzy##Sets and Systems, 81 (1996), 321{329.##[24] M. Sugeno, Industrial Applications of Fuzzy Control, Elsevier Science Publishers, New York,##[25] E. Van Broekhoven and B. De Baets, Fast and accurate centre of gravity defuzzication of##fuzzy system outputs dened on trapezoidal fuzzy partitions, Fuzzy Sets and Systems, 157##(2006), 904{918.##[26] W. Van Leekwijck and E. Kerre, Defuzzication: criteria and classication, Fuzzy Sets and##Systems, 108 (1999), 159{178.##[27] R. Yager and D. Filev, SLIDE: a simple adaptive defuzzication method, IEEE Trans. Fuzzy##Systems, 1 (1993), 69{78.##[28] R. C. Young, The algebra of manyvalued quantities, Math. Ann., 104 (1931), 260{290.##[29] L. A. Zadeh, The concept of a linquistic variable and its application to approximate reasoning,##Information Sciences, 1(8) (1975), 199{249.##]
AN OPTIMAL FUZZY SLIDING MODE CONTROLLER DESIGN BASED ON PARTICLE SWARM OPTIMIZATION AND USING SCALAR SIGN FUNCTION
2
2
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.
1

67
85


Lotfi
Chaouech
Laboratoire d'Ingenierie des Systemes Industriels et des Energies
Renouvelables (LISIER), The National Higher Engineering School of Tunis (ENSIT), BP 56, 1008 Tunis, Tunisia
Laboratoire d'Ingenierie des Systemes
Tunisia


Mo^ez
Soltani
Laboratoire d'Ingenierie des Systemes Industriels et des Energies Renouvelables (LISIER), The National Higher Engineering School of Tunis (ENSIT), BP
56, 1008 Tunis, Tunisia
Laboratoire d'Ingenierie des Systemes
Tunisia


Slim
Dhahri
Laboratoire d'Ingenierie des Systemes Industriels et des Energies Renouvelables (LISIER), The National Higher Engineering School of Tunis (ENSIT), BP
56, 1008 Tunis, Tunisia
Laboratoire d'Ingenierie des Systemes
Tunisia


Abdelkader
Chaari
Laboratoire d'Ingenierie des Systemes Industriels et des Energies Renouvelables (LISIER), The National Higher Engineering School of Tunis (ENSIT), BP 56, 1008 Tunis, Tunisia
Laboratoire d'Ingenierie des Systemes
Tunisia
Sliding mode control
TakagiSugeno fuzzy model
Particle swarm optimization
Parallel distributed compensator
Linear quadratic regulator
[[1] 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.##[2] 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{##[3] 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.##[4] W. Chang, J. B. Park, Y. H. Joob and G. Chen, Design of robust fuzzymodel based controller##with sliding mode control for siso nonlinear systems, Fuzzy Sets and Systems, 125(1) (2002),##[5] L. Chaouech and A. Chaari, Design of sliding mode control of nonlinear system based on##TakagiSugeno fuzzy model, World Congress on Computer and Information Technology (WCCIT),##(2013), 1{6.##[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.##[7] P. C. Chen, C. W. Chen and W. L. Chiang, GAbased fuzzy sliding mode controller for##nonlinear systems, Mathematical Problems in Engineering, 2008 (2008), 1{16.##[8] 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.##[9] P. Durato, C. Abdallah and V. Cerone, Linear quadratic control: An introduction, Prentice##Hall, USA, 1995.##[10] S. El Beid and S. Doubabi, DSPBased implementation of fuzzy output tracking control for##a boost converter, IEEE Transactions on Industrial Electronics, 61(1) (2014), 196{209. ##[11] P. Guan, X. J. Liu and J. Z. Liu, Adaptive fuzzy sliding mode control for ##exible satellite,##Engineering Applications of Articial Intelligence, 18(4) (2005), 451{459.##[12] 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{##[13] 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.##[14] Y. J. Huang and H. K. Wei, Sliding mode control design for discrete multivariable systems##with timedelayed input signals, International Journal of Systems Science, 33(10) (2002),##[15] L. Hung, H. Lin and H. Chung, Design of selftuning fuzzy sliding mode control for TORA##system, Expert Systems with Applications, 32 (1) (2007), 201{212.##[16] E. Iglesias, Y. Garcia, M. Sanjuan, O. Camacho and C. Smith, Fuzzy surfacebased sliding##mode control, ISA Transactions, 43(1) (2007), 73{83.##[17] A. Isidoti, Nonlinear Control Systems, Springer, Berlin, 1989.##[18] K. Jafar, B. M. Mohammad and K. Mansour, Feedbacklinearization and fuzzy controllers##for trajectory tracking of wheeled mobile robots, Kybernetes, 39(1) (2010), 83{106.##[19] 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.##[20] A. Khosla, S. Kumar and K. R. Ghosh, A comparison of computational eorts between##particle swarm optimization and genetic algorithm for identication of fuzzy models, Fuzzy##Information Processing Society, (2007), 245{250.##[21] R. J. Lian, Adaptive selforganizing fuzzy slidingmode radial basisfunction neuralnetwork##controller for robotic systems, IEEE Transactions on Industrial Electronics, 61(3) (2014),##1493{1503.##[22] 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.##[23] Z. Liang, Y. Yang and Y. Zeng, Eliciting compact TS fuzzy models using subtractive clus##tering and coevolutionary particle swarm optimization, Neurocomputing, 72(1012) (2009),##2569{2575.##[24] 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.##[25] 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.##[26] T. Niknam and B. Amiri, An ecient hybrid approach based on PSO, ACO and kmeans for##cluster analysis, Applied Soft Computing, 10(1) (2010), 183{197.##[27] 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.##[28] 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.##[29] A. Shahraz and R. B. Boozarjomehry, A fuzzy sliding mode control approach for nonlinear##chemical processes, Control Engineering Practice, 17(5) (2009), 541{550.##[30] S. F. Shehu, D. Filev and R. Langari, Fuzzy Control: Synthesis and Analysis, John Wiley##and Sons LTD, USA, 1997.##[31] 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.##[32] 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.##[33] M. Soltani and A. Chaari, A PSOBased fuzzy cregression model applied to nonlinear data##modeling, International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems,##23(6) (2015), 881{892. ##[34] M. Sugeno and G. Kang, Fuzzy modeling and control of multilayer incinerator, Fuzzy Sets##and Systems, 18 (3) (1986), 329{345.##[35] T. Takagi and M. Sugeno, Fuzzy identication of systems and its application to modeling##and control, IEEE Transactions on Systems, Man and Cybernetics, 15(1) (1985 ), 116{132.##[36] K. Tanaka and M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets##and Systems, 45(2)(1992), 135{156.##[37] V. I. Utkin, Variable structure systems with sliding mode, IEEE Transactions on Automatic##Control, 26(2) (1977), 212{222.##[38] S. Vishnu Teja, T. N. Shanavas and S. K. Patnaik, Modied PSO based slidingmode controller##parameters for buck converter, Conference on Electrical, Electronics and Computer Science##(SCEECS), (2012), 1{4.##[39] R. Wai, C. Lin and C. Hsu, Adaptive fuzzy slidingmode control for electrical servo drive,##Fuzzy Sets and Systems, 143(2) (2004), 295{310.##[40] H. O. Wang, K. Tanaka and M. F. Grin, An approach to fuzzy control of nonlinear systems:##Stability and design issues, IEEE Transactions on Fuzzy Systems, 4(1) (1996), 14{23.##[41] H. O. Wang, K. Tanaka and M. F. Grin, Parallel distributed compensation of nonlinear##systems by TakagiSugeno fuzzy model, Proceedings FUZZY IEEE/IFES, (1995), 531{538.##[42] 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.##[43] 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.##[44] 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.##[45] Y. Xinghuo, Z. Man and B. Wu, Design of fuzzy slidingmode control systems, Fuzzy Sets##and Systems, 95 (3) (1998), 295{306.##[46] F. K. Yeh and C. M. Chen, J. J. Huang, Fuzzy slidingmode control for a MINIUAV, IEEE##International Symposium on Computational Intelligence in Scheduling, (2010), 3317{3323.##[47] 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.##[48] 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.##]
INTERVALVALUED INTUITIONISTIC FUZZY SETS AND SIMILARITY MEASURE
2
2
In this paper, the problem of measuring the degree of inclusion and similarity measure for two intervalvalued intuitionistic fuzzy sets is considered. We propose inclusion and similarity measure by using order on intervalvalued 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.
1

87
98


Barbara
Pekala
Interdisciplinary Centre for Computational Modelling, Faculty of Mathematics and Natural Sciences, University of Rzeszow, Pigonia 1, 35310 Rzeszow, Poland
Interdisciplinary Centre for Computational
Poland
bpekalaur@gmail.com


Krzysztof
Balicki
Interdisciplinary Centre for Computational Modelling, Faculty of Mathematics and Natural Sciences, University of Rzeszow, Pigonia 1, 35310 Rzeszow, Poland
Interdisciplinary Centre for Computational
Poland
kbalicki@ur.edu.pl
Intervalvalued intuitionistic fuzzy sets
Inclusion measure
Similarity measure
[[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87{96.##[2] K. T. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets Syst., 64##(1994), 159{174.##[3] K. T. Atanassov, Intuitionistic Fuzzy Sets. Theory and Applications, PhysicaVerlag, Heidelberg/##New York, 1999.##[4] 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.##[5] K. Bosteels and E. E. Kerre, On a re##exivitypreserving family of cardinalitybased fuzzy##comparison measures, Inform. Sci., 179 (2009), 2342{2352.##[6] H. Bustince, J. Fernandez, R. Mesiar, J. Montero and R. Orduna, Overlap functions, Nonlinear##Anal.: Theory Methods Appl., 72 (2010), 1488{1499.##[7] H. Bustince, M. Pagola, R. Mesiar, E. Hullermeier and F. Herrera, Grouping, overlaps, and##generalized bientropic functions for fuzzy modeling of pairwise comparisons, IEEE Trans.##Fuzzy Syst., 20(3) (2012), 405{415.##[8] H. Bustince, J. Fernandez, A. Kolesarova and R. Mesiar, Generation of linear orders for##intervals by means of aggregation functions, Fuzzy Sets Syst., 220 (2013), 6977.##[9] T. Calvo, A. Kolesarova, M. Komornikova 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), 3104.##[10] B. De Baets and R. Mesiar, Triangular norms on product lattices, Fuzzy Sets Syst., 104##(1999), 61{76.##[11] B. De Baets, H. De Meyer and H. Naessens, A class of rational cardinalitybased similarity##measures, J. Comp. Appl. Math., 132 (2001), 51{69.##[12] B. De Baets and H. De Meyer, Transitivity frameworks for reciprocal relations:cycle##transitivity versus FGtransitivity, Fuzzy Sets Syst., 152 (2005), 249{270.##[13] B. De Baets, S. Janssens and H. De Meyer, On the transitivity of a parametric family of##cardinalitybased similarity measures, Int. J. Appr. Reason., 50 (2009), 104{116.##[14] M. De Cock and E. E. Kerre, Why fuzzy Tequivalence relations do not resolve the Poincar'e##paradox, and related issues, Fuzzy Sets Syst., 133 (2003), 181{192.##[15] L. De Miguel, H. Bustince, J. Fernandez, E. Indurain, A. Kolesarova and R. Mesiar, Con##struction of admissible linear orders for intervalvalued Atanassov intuitionistic fuzzy sets##with an application to decision making, Information Fusion, 27 (2016), 189197.##[16] 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.##[17] B. Jayaram and R. Mesiar, IFuzzy equivalence relations and Ifuzzy partitions, Inf. Sci., 179##(2009), 1278{1297.##[18] D. F. Li, Toposisbased nonlinearprogramming methodology for multiattribute decision mak##ing with intervalvalued intuitionistic fuzzy sets, IEEE Trans. Fuzzy Syst., 18 (2010), 299{311.##[19] X. D. Liu, S. H. Zheng and F. L. Xiong, Entropy and subsethood for general intervalvalued##intuitionistic fuzzy sets, Lecture Notes Artif. Intell., 3613 (2005), 42{52.##[20] N. Madrid, A. Burusco, H. Bustince, J. Fernandez and I. Perlieva, Upper bounding overlaps##by groupings, Fuzzy Sets Syst., 264 (2015), 76{99.##[21] S. Ovchinnikov, Numerical representation of transitive fuzzy relations, Fuzzy Sets Syst., 126##(2002), 225{232. ##[22] D. G. Park, Y. C. Kwun, J. H. Park and I. Y. Park, Correlation coecient of interval##valued intuitionistic fuzzy sets and its application to multiple attribute group decision making##problems, Math. Comput. Modell., 50 (2009), 1279{1293.##[23] Z. Switalski, General transitivity conditions for fuzzy reciprocal preference matrices, Fuzzy##Sets Syst., 137 (2003), 85{100.##[24] L. A. Zadeh, Fuzzy sets, Information Contr., 8 (1965), 338 { 353.##[25] W. Y. Zeng and P. Guo, Normalized distance, similarity measure, inclusion measure and##entropy of intervalvalued fuzzy sets and their relationship, Inf. Sci., 178 (2008), 1334{1342.##[26] 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.##[27] 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.##]
TOPOLOGICAL SIMILARITY OF LRELATIONS
2
2
$L$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 Tsimilarity, and we give intuitive characterization ofTsimilarity. 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.
1

99
115


Jing
Hao
College of Mathematics and Information Science, Henan University of
Economics and Law, Zhengzhou, 450000, China
College of Mathematics and Information Science,
China
haojingzy@gmail.com


Shasha
Huang
College of Mathematics and Information, North China University of
Water Resources and Electric Power, Zhengzhou, 450045, China
College of Mathematics and Information, North
China
s_s_huang@163.com
$L$fuzzy rough set
$L$relation
Alexandrov $L$topology
$L$preorder
Topological similarity
[[1] K. Blount and C. Tsinakis, The structure of residuated lattices, International Journal of##Algebra and Computation, 13(4) (2003), 437{461.##[2] D. Boixader, J. Jacas and J. Recasens, Upper and lower approximations of fuzzy sets, International##Journal of General System, 29(4) (2000), 555{568. ##[3] C. L. Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications,##24(1) (1968), 182{190.##[4] X. Chen and Q. Li, Construction of rough approximations in fuzzy setting, Fuzzy sets and##Systems, 158(23) (2007), 2641{2653.##[5] 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.##[6] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of##General System, 17(23) (1990), 191{209.##[7] J. A. Goguen, Lfuzzy sets, Journal of Mathematical Analysis and Applications, 18(1) (1967),##[8] J. Hao and Q. Li, The relationship between Lfuzzy rough set and Ltopology, Fuzzy Sets and##Systems, 178(1) (2011), 74{83.##[9] J. Jarvinen and J. Kortelainen, A unifying study between modallike operators, topologies and##fuzzy sets, Fuzzy Sets and Systems, 158(11) (2007), 1217{1225.##[10] H. Lai and D. Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157(14)##(2006), 1865{1885.##[11] Z. Li and R. Cui, Tsimilarity of fuzzy relations and related algebraic structures, Fuzzy Sets##and Systems, 275 (2015), 130{143.##[12] G. Liu, Generalized rough sets over fuzzy lattices, Information Sciences, 178(6) (2008), 1651{##[13] G. Liu and W. Zhu, The algebraic structures of generalized rough set theory, Information##Sciences, 178(21) (2008), 4105{4113.##[14] R. Lowen, Fuzzy topological spaces and fuzzy compactness, Journal of Mathematical Analysis##and Applications, 56(3) (1976), 621{633.##[15] Z. M. Ma and B. Q. Hu, Topological and lattice structures of Lfuzzy rough sets determined##by lower and upper sets, Information Sciences, 218 (2013), 194{204.##[16] N. N. Morsi and M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy sets and Systems, 100(1)##(1998), 327{342.##[17] Z. Pawlak, Rough sets, International Journal of Computer & Information Sciences, 11(5)##(1982), 341{356.##[18] K. Qin and Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems,##151(3) (2005), 601{613.##[19] A. M. Radzikowska and E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and##Systems, 126(2) (2002), 137{155.##[20] 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.##[21] 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.##[22] 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.##[23] A. Skowron and J. Stepaniuk, Tolerance approximation spaces, Fundamenta Informaticae,##27(23) (1996), 245{253.##[24] H. Thiele, On axiomatic characterization of fuzzy approximation operators II, the rough fuzzy##set based case, Proceedings of the 31st IEEE International Symposium on MultipleValued##Logic, (2001), 330{335.##[25] 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.##[26] D. Vanderpooten, Similarity relation as a basis for rough approximations, Advances in Machine##Intelligence and Soft Computing, 4 (1997), 17{33.##[27] C. Y. Wang and B. Q. Hu, Fuzzy rough sets based on generalized residuated lattices, Information##Sciences, 248 (2013), 31{49.##[28] M.Ward and R. P. Dilworth, Residuated lattices, Transactions of the American Mathematical##Society, 45(3) (1939), 335{354. ##[29] 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.##[30] W. Z. Wu, J. S. Mi and W. X. Zhang, Generalized fuzzy rough sets, Information Sciences,##151 (2003), 263{282.##[31] Y. Yao, Constructive and algebraic methods of the theory of rough sets, Information Sciences,##109(1) (1998), 21{47.##[32] W. Zhu, Topological approaches to covering rough sets, Information Sciences, 177(6) (2007),##1499{1508.##]
FUZZY INCLUSION LINEAR SYSTEMS
2
2
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.
1

117
137


Mojtaba
Ghanbari
Department of Mathematics, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran
Department of Mathematics, Aliabad Katoul
Iran
ghanbari@aliabadiau.ac.ir
Fuzzy linear system
Fuzzy inclusion linear system
Fuzzy solution set
Lower $r$boundary
Upper $r$boundary
[[1] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Applied Mathematics##and Computation, 155 (2004), 493–502.##[2] T. Allahviranloo and M. Ghanbari, On the algebraic solution of fuzzy linear systems based##on interval theory, Applied Mathematical Modelling, 36 (2012), 5360–5379.##[3] T. Allahviranloo and M. Ghanbari, Solving Fuzzy Linear Systems by Homotopy Perturbation##Method, International Journal of Computational Cognition, 8(2) (2010), 24–30.##[4] 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.##[5] T. Allahviranloo and M. Ghanbari, A new approach to obtain algebraic solution of interval##linear systems, Soft Computing, 16 (2012), 121–133.##[6] 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.##[7] T. Allahviranloo and S. Salahshour, Fuzzy symmetric solution of fuzzy linear systems, Journal##of Computational and Applied Mathematics, 235(16) (2011), 4545–4553.##[8] T. Allahviranloo, R. Nuraei, M. Ghanbari, E. Haghi and A. A. Hosseinzadeh, A new metric##for LR fuzzy numbers and its application in fuzzy linear systems, Soft Computing, 16 (2012),##17431754. ##[9] B. Bede and S. G. Gal, Generalizations of the differentiability of fuzzynumbervalued functions##with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151 (2005),##581–599.##[10] R. Ezzati, Solving fuzzy linear systems, Soft Computing, 15(2010), 193197.##[11] M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems,##96(1998), 201–209.##[12] 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.##[13] M. Ghanbari and R. Nuraei, Convergence of a semianalytical method on the fuzzy linear##systems, Iranian Journal of Fuzzy Systems, 11(4) (2014), 45–60.##[14] 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.##[15] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984),##215–229.##[16] 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.##[17] C. Wu and M. Ming, On embedding problem of fuzzy number space: Part 1, Fuzzy Sets and##Systems, 44 (1991), 33–38.##]
STONE DUALITY FOR R0ALGEBRAS WITH INTERNAL STATES
2
2
$Rsb{0}$algebras, which were proved to be equivalent to Esteva and Godo's NMalgebras modelled by Fodor's nilpotent minimum tnorm, are the equivalent algebraic semantics of the leftcontinuous tnorm based fuzzy logic firstly introduced by Guojun Wang in the mid 1990s.In this paper, we first establish a Stone duality for the category of MVskeletons of $Rsb{0}$algebras and the category of threevalued Stone spaces.Then we extend FlaminioMontagna internal states to $Rsb{0}$algebras.Such internal states must be idempotent MVendomorphisms of $Rsb{0}$algebras.Lastly we present a Stone duality for the category of MVskeletons of $Rsb{0}$algebras with FlaminioMontagna internal states and the category of threevalued 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.
1

139
161


Hongjun
Zhou
School of Mathematics and Information Science, Shaanxi Normal
University, Xi'an, 710062, CHINA
School of Mathematics and Information Science,
China


HuiXian
Shi
School of Mathematics and Information Science, Shaanxi Normal University
School of Mathematics and Information Science,
China
rubyshi@163.com
$Rsb{0}$algebra
Nilpotent minimum algebra
MVskeleton
internal state
Stone duality
[[1] 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.##[2] L. P. Belluce, Semisimple algebras of innite valued logic and bold fuzzy set theory, Can. J.##Math., 38 (1986), 1356{1379.##[3] W. Blok and D. Pigozzi, Algebraizable logics, Merm. Math. Soc., 77 (1989), 189.##[4] M. Botur and A. Dvurecenskij, Statemorphism algebras{general approach, Fuzzy Sets Syst.,##218 (2013), 90{102.##[5] M. Busaniche, Free nilpotent minimum algebras, Math. Logic Quart., 52 (2006), 219{236.##[6] C. C. Chang, Algebraic analysis of manyvalued logics, Trans. Amer. Math. Soc., 88 (1958),##[7] R. Cignoli and D. Mundici, Stone duality for Dedekind complete `groups with order unit,##J. Algebra, 302 (2006), 848{861.##[8] L. C. Ciungu, Noncommutative MultipleValued Logic Algebras, Springer, New York, 2014.##[9] L. C. Ciungu, G. Georgescu and C. Muresan, Generalized Bosbach states: part I, Arch. Math.##Logic, 52 (2013), 335{376.##[10] L. C. Ciungu, G. Georgescu and C. Muresan, Generalized Bosbach states: part II, Arch.##Math. Logic, 52 (2013), 707{732.##[11] A. Di Nola and A. Dvurecenskij, Statemorphism MValgebras, Ann. Pure Appl. Logic, 161##(2009), 161{173.##[12] A. Di Nola, A. Dvurecenskij and A. Lettieri, On varieties of MValgebras with internal states,##Inter. J. Approx. Reason., 51 (2010), 680{694.##[13] A. Di Nola, A. Dvurecenskij and A. Lettieri, Stone duality type theorems for MValgebras##with internal states, Comm. Algebra, 40 (2012), 327{342.##[14] A. Dvurecenskij, J. Rachunek and D. Salounova, State operators on generalizations of fuzzy##structures, Fuzzy Sets Syst., 187 (2012), 58{76.##[15] C. Elkan, The paradoxical success of fuzzy logic, IEEE Expert, 9 (1994), 3{8.##[16] F. Esteva and L. Godo, Monoidal tnorm based logic: towards a logic for leftcontinuous##tnorms, Fuzzy Sets Syst., 124 (2001), 271{288.##[17] T. Flaminio and F. Montagna, MValgebras with internal states and probabilistic fuzzy logic,##Inter. J. Approx. Reason., 50 (2009), 138{152.##[18] J. Fodor, Nilpotent minimum and related connectives for fuzzy logic, in: Proc. of the 4th##Inter. Conf. on Fuzzy Syst., March 2024, Yokohama, (1995), 2077{2082.##[19] P. F. He, X. L. Xin and Y.W. Yang, On state residuated lattices, Soft Comput., 19 (2015),##2083{2094.##[20] A. Iorgulescu, Algebras of Logic as BCKalgebras, Editura ASE, Bucarest, 2008.##[21] H. W. Liu and G. J. Wang, Unied forms of fully implication restriction methods for fuzzy##reasoning, Inf. Sci., 177(3) (2007), 956{966.##[22] L. Liu and K. Li, Involutive monoidal tnorm based logic and R0logic, Inter. J. Intelligent##Syst., 199 (2004), 491{497.##[23] Y. M. Liu and M. K. Luo, Fuzzy topology, World Scientic, Hong Kong, (1997), 15{68.##[24] D. Mundici, Advanced Lukasiewicz Calculus and MValgebras, Springer, New York, (2011),##[25] D. Mundici, Averaging the truthvalue in Lukasiewicz sentential logic, Stud. Logica, 55##(1995), 113{127.##[26] Z. M. Ma and Z. W. Fu, Algebraic study to generalized Bosbach states on residuated lattices,##Soft Comput., 19 (2015), 2541{2550.##[27] D. W. Pei, On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets Syst., 138##(2003), 187{195. ##[28] D. W. Pei, R0implication: characteristics and applications, Fuzzy Sets Syst., 131 (2002),##[29] D. W. Pei and G. J. Wang, The completeness and applications of the formal system L, Sci.##China F, 45 (2002), 40{50.##[30] M. H. Stone, The theory of representation for Boolean algebras, Trans. Amer. Math. Soc.,##40 (1936), 37{111.##[31] G. J. Wang, A formal deductive system for fuzzy propositional calculus, Chin. Sci. Bull., 42##(1997), 1521{1526.##[32] G. J. Wang, Fuzzy logic and fuzzy reasoning, In: Proc. of the 7th National ManyValued and##Fuzzy Logic Conf., November 1013, Xi'an, (1996), 82{96.##[33] G. J. Wang, Implication lattices and their fuzzy implication space representation theorem,##Acta Math. Sin., (in Chinese), 42 (1999), 133{140.##[34] G. J. Wang, LFuzzy Topological Spaces, Shaanxi Normal Univ. Press, Xi'an, (in Chinese),##(1988), 18{56.##[35] G. J. Wang, X. J. Hui and J. S. Song, The R0type fuzzy logic metric space and an algorithm##for solving fuzzy modus ponens, Comput. Math. Appl., 55(9) (2008), 1974{1987.##[36] G. J. Wang and H. J. Zhou, Introduction to Mathematical Logic and Resolution Principle,##Science Press, Beijing, (2009), 156{298.##[37] S. M. Wang, B. S. Wang and X. Y. Wang, A characterization of truthfunctions in the##nilpotent minimum logic, Fuzzy Sets Syst., 145 (2004), 253{266.##[38] D. X. Zhang and Y. M. Liu, Lfuzzy version of Stone's representation theorem for distributive##lattices, Fuzzy Sets Syst., 76 (1995), 259{270.##[39] H. J. Zhou, Probabilistically Quantitative Logic and its Applications, Science Press, Beijing,##(in Chinese), 2015.##[40] H. J. Zhou, G. J. Wang and W. Zhou, Consistency degrees of theories and methods of graded##reasoning in nvalued R0logic (NMlogic), Inter. J. Approx. Reason., 43 (2006), 117{132.##[41] H. J. Zhou and B. Zhao, Characterizations of maximal consistent theories in the formal##deductive system L (NMlogic) and Cantor space, Fuzzy Set Syst., 158 (2007), 2591{2604.##[42] H. J. Zhou and B. Zhao, Generalized Bosbach and Riecan states based on relative negations##in residuated lattices, Fuzzy Sets Syst., 187 (2012) 3357.##[43] H. J. Zhou and B. Zhao, Stonelike representation theorems and threevalued lters in R0##algebras (nilpotent minimum algebras), Fuzzy Sets Syst., 162 (2011), 1{26.##]
REDUNDANCY OF MULTISET TOPOLOGICAL SPACES
2
2
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.
1

163
168


A.
Ghareeb
Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt
Department of Mathematics, Faculty of Science,
Egypt
nasserfuzt@hotmail.com
Multiset
Multiset topology
Isomorphism
[[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87{96.##[2] W. D. Blizard, Multiset theory, Notre Dame J. Formal Logic, 30(1) (1988), 36{66.##[3] S. A. ElSheikh, R. A. K. Omar and M. Raafat, ##operation in mtopological space, Gen.##Math. Notes, 7(1) (2015), 40{54.##[4] S. A. ElSheikh, R. A. K. Omar and M. Raafat, Separation axioms on multiset topological##space, Journal of New Theory, 7 (2015), 11{21.##[5] K. P. Girish and J. S. Jacob, On multiset topologies, Theory and Applications of Mathematics##and Computer Science, 2(1) (2012), 37{52.##[6] 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.##[7] K. P. Girish and S. J. John, Multiset topologies induced by multiset relations, Information##Sciences, 188 (2012), 298{313.##[8] A. Kandil, O. Tantawy, S. ElSheikh and A. Zakaria, Multiset proximity spaces, Journal of##Egyptian Mathematical Society, 24(4) (2016), 562567.##[9] P. M. Mahalakshmi and P. Thangavelu, mconnectedness in mtopology, International Journal##of Pure and Applied Mathematics, 106(8) (2016), 2125.##[10] J. Mahanta and D. Das, Boundary and exterior of a multiset topology, ArXiv eprints,##arXiv:1501.07193.##[11] D. Molodtsov, Soft set theoryrst results, Computers and Mathematics with Applications,##37 (45) (1999), 19{31.##[12] F. G. Shi and B. Pang, Redundancy of fuzzy soft topological spaces, Journal of Intelligent and##Fuzzy Systems, 27 (4) (2014), 1757{1760.##[13] F. G. Shi and B. Pang, A note on soft topological spaces, Iranian Journal of Fuzzy Systems,##12 (5) (2015), 149{155.##[14] R. R. Yager, On the theory of bags, International Journal of General Systems, 13 (1986),##[15] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338{353.##]
Persiantranslation vol. 14, no. 4, August 2017
2
2
1

171
179