2009
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Cove vol.6, no.3, October 2009
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MINIMUM TIME SWING UP AND STABILIZATION OF
ROTARY INVERTED PENDULUM USING PULSE STEP
CONTROL
MINIMUM TIME SWING UP AND STABILIZATION OF
ROTARY INVERTED PENDULUM USING PULSE STEP
CONTROL
2
2
This paper proposes an approach for the minimum time swing upof a rotary inverted pendulum. Our rotary inverted pendulum is supported bya pivot arm. The pivot arm rotates in a horizontal plane by means of a servomotor. The opposite end of the arm is instrumented with a joint whose axisis along the radial direction of the motor. A pendulum is suspended at thejoint. The task is to design a controller that swings up the pendulum, keepsit upright and maintains the arm position. In the general intelligent hybridcontroller, a PD controller with a positive feedback is designed for swinging upand a fuzzy balance controller is designed for stabilization. In order to achievethe swing up in a minimal time, a controller named Minimum Time IntelligentHybrid Controller is proposed which is precisely a PD controller together witha pulse step controller for swinging up along with the fuzzy balance controllerfor stabilization. The impulsive control action is tuned by trial and errorto achieve the minimumtime swingup. An energy based switching controlmethod is proposed to switch over from swing up mode to stabilization mode.Extensive computer simulation results demonstrate that the swing up timeof the proposed minimumtime controller is significantly less than that of theexisting general hybrid nonlinear controller.
1
This paper proposes an approach for the minimum time swing upof a rotary inverted pendulum. Our rotary inverted pendulum is supported bya pivot arm. The pivot arm rotates in a horizontal plane by means of a servomotor. The opposite end of the arm is instrumented with a joint whose axisis along the radial direction of the motor. A pendulum is suspended at thejoint. The task is to design a controller that swings up the pendulum, keepsit upright and maintains the arm position. In the general intelligent hybridcontroller, a PD controller with a positive feedback is designed for swinging upand a fuzzy balance controller is designed for stabilization. In order to achievethe swing up in a minimal time, a controller named Minimum Time IntelligentHybrid Controller is proposed which is precisely a PD controller together witha pulse step controller for swinging up along with the fuzzy balance controllerfor stabilization. The impulsive control action is tuned by trial and errorto achieve the minimumtime swingup. An energy based switching controlmethod is proposed to switch over from swing up mode to stabilization mode.Extensive computer simulation results demonstrate that the swing up timeof the proposed minimumtime controller is significantly less than that of theexisting general hybrid nonlinear controller.
1
15
P
Melba Mary
P
Melba Mary
Anna University, Department of Electrical and Electronics Engineering,
TamilNadu, India
Anna University, Department of Electrical
India
melbance 2k4@yahoo.co.in
.N. S
Marimuthu
N. S.
Marimuthu
Anna University, Department of Electrical and Electronics Engineering,
TamilNadu, India
Anna University, Department of Electrical
India
drnsmphd@yahoo.co.in
Fuzzy logic control (FLC)
Rotary inverted pendulum (RIP)
Swing up control
PD control
Energy based switching control
Hybrid control and Minimumtime control
[[1] K. J. Astrom and K. Furuta, Swinging up a pendulum by energy control, Automatica, 36##(2000), 287295.##[2] http://people.msoe.edu/˜saadat/download/3 Inverted Pendulum Project.pdf##[3] H. K. Lam, F. H. Leung and P. K. S. Tam, Design and stability analysis of fuzzy modelbased##nonlinear controller for nonlinear systems using genetic algorithm, IEEE Trans. on Systems,##Man and Cybernetics, Part B, 33(2) (2003), 250257.##[4] Z. G. Li, C. Y. Wen and Y. C. Soh , Analysis and design of impulsive control systems, IEEE##Trans. on Automatic Control, 46(6) (2001), 894897.##[5] C. M. Lin and Y. J. Mon, Decoupling control by hierarchical fuzzy slidingmode controller,##IEEE Trans. on Control Systems Technology, 13(4) (2005), 593598. ##[6] N. Muskinja and B. Tovornik, Swinging up and Stabilization of a real inverted pendulum,##IEEE Trans. on Industry electronics, 53(2) (2006), 631639.##[7] S. Nudrakwang, T. Benjanarasuth, J. Ngamwiwit and N. Komine, Hybrid PDservo state##feedback control algorithm for swing up inverted pendulum system, Proceedings of the ICCAS,##2(5) (2005), 690693.##[8] S. Nudrakwang, T. Benjanarasuth, J. Ngamwiwit and N. Komine, Hybrid controller for##Swinging up Inverted Pendulum System, Proceedings of the IEEE, ICICS (2005), 488492.##[9] G. N. Silva and R. B. Vinter, Necessary conditions for optimal impulsive control problems,##SIAM Journal on Control and Optimization, 35(6) (1997), 18291846.##[10] W. TorresPomales and O. R. Gonzalez, Nonlinear control of swingup inverted pendulum,##Proceedings of the IEEE International Conference on Control Application, (1996), 259264.##[11] A. Wallen and K. J. Astrom, Pulse step control, Proceedings of the 15th IFAC Triennial##World Congress, Barcelona, Spain, 2002.##[12] C. H. Wang, T. C. Lin, T. T. Lee and H. L. Liu, Adaptive hybrid intelligent control for##uncertain nonlinear dynamical systems, IEEE Trans. on Systems, Man and Cybernetics,##Part B, 32 (5) (2002), 583597.##[13] S. Weibel, G. W. Howell and J. Baillieul, Control of singledegreeoffreedom Hamiltonian##systems with impulsive inputs, Proceedings of the 35th IEEE Conference on Decision and##Control, Kobe, Japan, (1996), 46614666.##[14] M. Yamakita, et.al., Robust swing up control of double pendulum, Proceedings of the American##Control Conference, 1 (1995), 290295.##[15] T. Yang, Impulsive control, IEEE Trans. Autom. Contr, 44 (1999), 10811083.##[16] S. Yasunobu and M. Mori, Swing up fuzzy controller for inverted pendulum based on a human##control strategy, Proceedings of the Sixth IEEE International Conference on Fuzzy Systems,##3 (1997), 16211625.##[17] J. Yi, N. Yubazaki and K. Hirota, A new fuzzy controller for stabilization of paralleltype##double inverted pendulum system, Fuzzy Sets and Systems, 126(1) (2002), 105119.##[18] J. Yi, N. Yubazaki and K. Hirota, Upswing and stabilization control of inverted pendulum##system based on the SIRMs dynamically connected fuzzy inference model, Fuzzy Sets and##Systems, 122 (2001), 139152.##[19] K. Yoshida, Swing up control of an inverted pendulum by energy based methods, Proceedings##of the American Control Conference, 6 (1999), 4045 4047.##]
FUZZY LINEAR PROGRAMMING WITH GRADES OF
SATISFACTION IN CONSTRAINTS
FUZZY LINEAR PROGRAMMING WITH GRADES OF
SATISFACTION IN CONSTRAINTS
2
2
We present a new model and a new approach for solving fuzzylinear programming (FLP) problems with various utilities for the satisfactionof the fuzzy constraints. The model, constructed as a multiobjective linearprogramming problem, provides flexibility for the decision maker (DM), andallows for the assignment of distinct weights to the constraints and the objectivefunction. The desired solution is obtained by solving a crisp problemcontrolled by a parameter. We establish the validity of the proposed modeland study the effect of the control parameter on the solution. We also illustratethe efficiency of the model and present three algorithms for solving theFLP problem, the first of which obtains a desired solution by solving a singlecrisp problem. The other two algorithms, interact with the decision maker,and compute a solution which achieves a given satisfaction level. Finally, wepresent an illustrative example showing that the solutions obtained are ofteneven more satisfactory than asked for.
1
We present a new model and a new approach for solving fuzzylinear programming (FLP) problems with various utilities for the satisfactionof the fuzzy constraints. The model, constructed as a multiobjective linearprogramming problem, provides flexibility for the decision maker (DM), andallows for the assignment of distinct weights to the constraints and the objectivefunction. The desired solution is obtained by solving a crisp problemcontrolled by a parameter. We establish the validity of the proposed modeland study the effect of the control parameter on the solution. We also illustratethe efficiency of the model and present three algorithms for solving theFLP problem, the first of which obtains a desired solution by solving a singlecrisp problem. The other two algorithms, interact with the decision maker,and compute a solution which achieves a given satisfaction level. Finally, wepresent an illustrative example showing that the solutions obtained are ofteneven more satisfactory than asked for.
17
35
Nikbakhsh
Javadian
Nikbakhsh
Javadian
Department of Industrial Engineering, Mazandaran University
of Science and Technology, Babol, Iran
Department of Industrial Engineering, Mazandaran
Iran
nijavadian@ustmb.ac.ir
Yashar
Maali
Yashar
Maali
Department of Industrial Engineering, Mazandaran University of
Science and Technology, Babol, Iran
Department of Industrial Engineering, Mazandaran
Iran
y_maali@ustmb.ac.ir
Nezam
MahdaviAmiri
Nezam
MahdaviAmiri
Department of Mathematical Sciences, Sharif University of
Technology, P.O. Box 113659415, Tehran, Iran
Department of Mathematical Sciences, Sharif
Iran
nezamm@sharif.edu
Fuzzy linear programming
Fuzzy constraints
Multiobjective linear programming
[[1] R. E. Bellman and L. A. Zadeh, Decision making in fuzzy environment, Management Science,##17 (1970), 141164. ##[2] V. J. Bowman, On the relationship of the Tchebycheff norm and the efficient frontier of multicriteria##objectives, In: H. Thiriez and S. Zionts (Eds.), Multiple Criteria Decision Making,##SpringerVerlag, Berlin, 1976, 7686.##[3] J. M. Cadenas and J. L. Verdegay, A primer on fuzzy optimization models and methods,##Iranian Journal of Fuzzy Systems, 3(1) (2006), 121.##[4] M. Delgado, J. L. Verdegay and M. A. Vila, A general model for fuzzy linear programming,##Fuzzy Sets and Systems, 29 (1989), 2129.##[5] S. C. Fang and C. F. Hu, Linear programming with fuzzy coefficients in constraints, Computers##and Mathematics with Applications, 37 (1999), 6376.##[6] S. M. Guu and Y. K. Wu, Twophase approach for solving the fuzzy linear programming##problems, Fuzzy Sets and Systems, 107 (1999), 191195.##[7] U. Keymak and J. M. Sousa, Weighting of constraint in fuzzy optimization, Proceedings of##the 10th IEEE International Conference on Fuzzy Systems, 3 (2001), 11311134.##[8] K. Kosaka, H. Nonaka, M. F. Kawaguchi and T. Datet, The application of fuzzy linear##programming to flow control of crossing gate network, Proceeding of the 9th Fuzzy System##Symposium: Sapporo, (1993), 189182.##[9] Y. J. Lai and C. L. Hwang, Fuzzy mathematical programming, Springer, Berlin, 1992.##[10] F. Li, M. Liu, C. Wu and S. Lou, Fuzzy optimization problems based on inequality degree,##Proceedings of the First International Conference on Machine Learning and Cybernetics:##Beijing, (2002), 15661570.##[11] X. Q. Li, B. Zhang and H. Li, Computing efficient solutions to fuzzy multiple objective linear##programming problems, Fuzzy Sets and Systems, 157 (2006), 13281332.##[12] N. MahdaviAmiri and S. H. Nasseri, Duality results and a dual simplex method for linear##programming problems with trapezoidal variables, Fuzzy Sets and Systems, 158 (2007), 1961##[13] H. R. Maleki, M. Tata and M. Mashinchi, Linear programming with fuzzy variables, Fuzzy##Sets and Systems, 109 (2000), 2133.##[14] Y. Nakahara, User oriented ranking criteria and its application to fuzzy mathematical programming##problems, Fuzzy Sets and Systems, 94 (1998), 275286.##[15] S. Nakamura, K. Kosakat, M. F. Kawaguchi, H. Nonaka and T. Datet, Fuzzy linear programming##with grade of satisfaction in each constraint, Proceedings of the Joint Fourth IEEE##International Conference on Fuzzy Systems and the Second International Fuzzy Engineering##Symposium, (1995), 781786.##[16] J. R. Ramik and H. Rommelfanger, Fuzzy mathematical programming based on some new##inequality relations, Fuzzy Sets and Systems, 81 (1996), 7787.##[17] H. Rommelfanger, Fuzzy linear programming and applications, European Journal of Operational##Research, 92 (1996), 512527.##[18] M. Sakawa, Fuzzy sets and interactive multiobjective optimization, Plenum Press: New York##and London, 1993.##[19] H. Tanaka, T. Okuda and K. Asai, On fuzzy mathematical programming, Journal of Cybernetics,##3 (1974), 3746.##[20] J. L. Verdegay, Fuzzy mathematical programming, In : M. M. Gupta and E. Sanchez (Eds.),##Approximate Reasoning in Decision Analysis, NorthHolland: Amsterdam, 1982, 231236.##[21] B. Werners, An interactive fuzzy programming system, Fuzzy Sets and Systems, 23 (1987),##[22] B. Werners, Interactive multiple objective programming subject to flexible constraints, European##Journal of Operational Research, 31 (1987), 342349.##[23] B. Werners, Aggregation models in mathematical programming, In: G. Mitra (Ed.), Mathematical##Models for Decision Support, Springer, Berlin, (1988), 295305.##[24] Y. K. Wu, On the manpower allocation within matrix organization: a fuzzy linear programming##approach, European Journal of Operational Research, 183(1) (2007), 384393.##[25] L. Xinwang, Measuring the satisfaction of constraints in fuzzy linear programming, Fuzzy##Sets and Systems, 122 (2001), 263275.##[26] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353. ##[27] H. J. Zimmermann, Description and optimization of fuzzy systems, International Journal of##General Systems, 2 (1976), 209215.##]
FUZZY TRANSPOSITION HYPERGROUPS
FUZZY TRANSPOSITION HYPERGROUPS
2
2
In this paper we introduce the notions of fuzzy transposition hypergroupsand fuzzy regular relations and investigate their basic properties.We also study fuzzy quotient hypergroups of a fuzzy transposition hypergroup.
1
In this paper we introduce the notions of fuzzy transposition hypergroupsand fuzzy regular relations and investigate their basic properties.We also study fuzzy quotient hypergroups of a fuzzy transposition hypergroup.
37
52
Goutam
Chowdhury
Goutam
Chowdhury
Department of Mathematics, Derozio Memorial College
Rajarhat Road, Kolkata700136, India
Department of Mathematics, Derozio Memorial
India
chowdhurygoutam@yahoo.com
Semihypergroup
Fuzzy semihypergroup
Fuzzy hypergroup
Fuzzy transposition hypergroup
Fuzzy subsemihypergroup
Fuzzy subhypergroup
Fuzzy closed set
Fuzzy reflexive subsemihypergroup
Fuzzy normal subsemihypergroup
Fuzzy regular relation
Fuzzy quotient hypergroup
[[1] R. Ameri and M. M. Zahedi, Hypergroup and join spaces induced by a fuzzy subset, Pure##Mathematics and Applications, 8 (1997), 155168.##[2] R. Ameri, Fuzzy (transposition) hypergroups, Italian Journal of Pure and Applied mathematics##(to appear).##[3] P. Corsini, Prolegomena of hypergroup theory, Aviani Editore, 1979.##[4] P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publications,##ISBN 1402012225 (2003).##[5] P. Corsini and V. Leoreanu, Fuzzy sets and join spaces associated with rough sets, Rend.##Circ. Mat., Palermo, 51 (2002), 527536. ##[6] P. Corsini and I. Tofan, On fuzzy hypergroups, Pure Mathematics and Applications, 8 (1997),##[7] B. Davvaz, Fuzzy Hvgroups, Fuzzy Sets and Systems, 101 (1999), 191195.##[8] J. Jantosciak, Transposition hypergroups: noncommutative join spaces, Journal of Algebra,##187 (1997), 97119.##[9] A. Kehagias, Lfuzzy join and meet hyperoperations and the associated Lfuzzy hyperalgebras,##Rend. Circ. Mat., Palermo, 51 (2002), 503526.##[10] V. Leoreanu, Direct limit and inverse limit of join spaces associated with fuzzy sets, Pure##Math. Appl., 11 (2000), 509512.##[11] F. Marty, Sur une generalization de la notion de groupe, 8ieme Congres des Mathematiciens##Scandinaves, Stockholm, (1934), 4549.##[12] V. Murali, Fuzzy equivalence relations, Fuzzy Sets and System, 30 (1989), 155163.##[13] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517.##[14] M. K. Sen, R. Ameri and G. Chowdhury, Fuzzy hypersemigroups, Soft Computing, 12(9)##(2008), 891900.##[15] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338353.##[16] M. M. Zahedi, M. Bolurian and A. Hasankhani, On polygroups and fuzzy subpolygroups, J.##of Fuzzy Mathematics, 1 (1995), 115.##[17] M. M. Zahedi and R. Ameri, On the prime, primary and maximal subhypermodules, Italian##Journal of Pure and Applied Mathematics, 5 (1999), 6180.##]
Some types of $(in,ivq)$intervalvalued fuzzy
ideals of BCI algebras
Some types of $(in,ivq)$intervalvalued fuzzy
ideals of BCI algebras
2
2
In this paper, we introduce the notions of intervalvalued and $(in,ivq)$intervalvalued fuzzy ($p$,$q$ and $a$) ideals of BCI algebras and investigate some of their properties. We then derive characterization theorems for these generalized intervalvalued fuzzy ideals and discuss their relationship.
1
In this paper, we introduce the notions of intervalvalued and $(in,ivq)$intervalvalued fuzzy ($p$,$q$ and $a$) ideals of BCI algebras and investigate some of their properties. We then derive characterization theorems for these generalized intervalvalued fuzzy ideals and discuss their relationship.
53
63
Xueling.
Ma
Xueling.
Ma
Department of Mathematics, Hubei Institute for Nationalities,
Enshi, Hubei Province,445000, P. R. China
Department of Mathematics, Hubei Institute
China
Jianming.
Zhan
Jianming.
Zhan
Department of Mathematics, Hubei Institute for Nationalities,
Enshi, Hubei Province,445000, P. R. China
Department of Mathematics, Hubei Institute
China
zhanjianming@hotmail.com
Young Bea
Jun
Young Bea
Jun
Department of Mathematics Education, Gyeongsang National University,
Chinju 660701, Korea
Department of Mathematics Education, Gyeongsang
Korea
skywine@gmail.com
[[1] S. K. Bhakat, (2, 2_ q)fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and##Systems, 112 (2000), 299312.##[2] S. K. Bhakat and P. Das, (2, 2 _ q)fuzzy subgroups, Fuzzy Sets and Systems, 80 (1996),##[3] R. Biswas, Rosenfeld’s fuzzy subgroups with interval valued membership functions, Fuzzy##Sets and Systems, 63 (1994), 8790.##[4] B. Davvaz, (2, 2 _q)fuzzy subnearrings and ideals, Soft Computing, 10 (2006), 206211.##[5] B. Davvaz and P. Corsini, Redefined fuzzy Hvsubmodules and many valued implications,##Inform. Sci., 177 (2007), 865875.##[6] B. Davvaz and P. Corsini, On (, )fuzzy Hvideals of Hvrings, Iranian J. Fuzzy Systems,##5(2) (2008), 3548.##[7] W. A. Dudek, On grouplike BCIalgebras, Demonstratio Math., 21 (1998), 369376.##[8] W. A. Dudek and J. Thomys, On decompositions of BCHalgebras, Math. Japon., 35 (1990),##11311138.##[9] G. Deschrijver, Arithmetric operators in intervalvalued fuzzy theory, Inform. Sci., 177##(2007), 29062924.##[10] F. Esteva and L. Godo, Monoidal tnorm based logic: towards a logic for leftcontinuous##tnorms, Fuzzy Sets and Systems, 124 (2001), 271 288.##[11] P. H´ajek, Metamathematics of fuzzy logic, Kluwer Academic Press, Dordrecht, 1998.##[12] Y. Imai and K. Is´eki, On axiom system of propositional calculus, Proc. Japan Acad., 42##(1966), 1922.##[13] A. Iorgulescu, Some direct ascendents of Wajsberg and MV algebras, Sci. Math. Japon., 57##(2003), 583647.##[14] A. Iorgulescu, Iseki algebras, connection with BL algebras, Soft Computing, 8 (2004), 449##[15] K. Is´eki, An algebra related with a propositional calculus, Proc. Japan Acad., 42 (1966),##[16] K. Is´eki, On BCIalgebras, Math. Seminar Notes (now Kobe Math J.), 8 (1980), 125130.##[17] K. Is´eki and S. Tanaka, Ideal theory of BCKalgebras, Math. Japon., 21 (1966), 351366.##[18] K. Is´eki and S. Tanaka, An introduction to the theory of BCK algebras, Math. Japon., 23##(1978), 126. ##[19] Y. B. Jun, Intervalvalued fuzzy subalgebras/ideals in BCKalgebras, Sci. Math., 3 (2000),##[20] Y. B. Jun, Intervalvalued fuzzy ideals in BCI algebras, J. Fuzzy Math., 9 (2001), 807814.##[21] Y. B. Jun, On (, )fuzzy ideals of BCK/BCI algebras, Sci. Math. Japon., 60 (2004),##[22] Y. B. Jun, On (, )fuzzy subalgebras of BCK/BCIalgebras, Bull. Korean Math. Soc., 42##(2005), 703711.##[23] Y. B. Jun and J. Meng, Fuzzy pideals in BCIalgebras, Math. Japon, 40 (1994), 271282.##[24] Y. B. Jun and J. Meng, Fuzzy commutative ideals in BCIalgebras, Comm. Korean Math.##Soc., 9 (1994), 1925.##[25] Y. L. Liu and J. Meng, Fuzzy qideals of BCIalgebras, J. Fuzzy Math., 8 (2000), 873881.##[26] Y. L. Liu and J. Meng, Fuzzy ideals in BCIalgebras, Fuzzy Sets and Systems, 123 (2001),##[27] Y. L. Liu, J. Meng, X. H. Zhang and Z. C. Yue, qideals and aideals in BCIalgebras, SEA##Bull. Math., 24 (2000), 243253.##[28] Y. L. Liu, Y. Xu and J. Meng, BCIimplicative ideals of BCIalgebras, Inform. Sci., 177##(2007), 49874996.##[29] Y. L. Liu and X. H. Zhang, Fuzzy aideals in BCIalgebras, Adv. in Math. (China), 31 (2002),##[30] X. Ma, J. Zhan, B. Davvaz and Y. B. Jun, Some kinds of (2, 2 _ q)intervalvalued fuzzy##ideals of BCIalgebras, Inform. Sci., 178 (2008), 37383754.##[31] P. M. Pu and Y. M. Liu, Fuzzy topology I: Neighourhood structure of a fuzzy point and##MooreSmith convergence, J. Math. Anal. Appl., 76 (1980), 571599.##[32] A. B. Saeid and Y. B. Jun, Redefined fuzzy subalgebras of BCK/BCIalgebras, Iranian J.##Fuzzy Systems, 5(2) (2008), 6370.##[33] L. Torkzadeh, M. Abbasi and M. M. Zahedi, Some results of intuitionistic fuzzy weak dual##hyper Kideals, Iranian J. Fuzzy Systems, 5(1) (2008), 6578.##[34] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338353.##[35] J. Zhan and Z. Tan, Intuitionistic fuzzy aideals in BCI algebras, Soochow Math. J., 30##(2004), 207216.##]
EMBEDDING OF THE LATTICE OF IDEALS OF A RING INTO
ITS LATTICE OF FUZZY IDEALS
EMBEDDING OF THE LATTICE OF IDEALS OF A RING INTO
ITS LATTICE OF FUZZY IDEALS
2
2
We show that the lattice of all ideals of a ring $R$ can be embedded in the lattice of all its fuzzyideals in uncountably many ways. For this purpose, we introduce the concept of the generalizedcharacteristic function $chi _{s}^{r} (A)$ of a subset $A$ of a ring $R$ forfixed $r , sin [0,1] $ and show that $A$ is an ideal of $R$ if, and only if, its generalizedcharacteristic function $chi _{s}^{r} (A)$ is a fuzzy ideal of $R$. We alsoshow that the set of all generalized characteristic functions $C_{s}^{r}(I(R))$ of the members of $I(R)$ for fixed $r , sin [0,1] $ is acomplete sublattice of the lattice of all fuzzy ideals of $R$ and establishthat this latter lattice is generated by the union of allits complete sublattices $C_{s}^{r} (I(R))$.
1
We show that the lattice of all ideals of a ring $R$ can be embedded in the lattice of all its fuzzyideals in uncountably many ways. For this purpose, we introduce the concept of the generalizedcharacteristic function $chi _{s}^{r} (A)$ of a subset $A$ of a ring $R$ forfixed $r , sin [0,1] $ and show that $A$ is an ideal of $R$ if, and only if, its generalizedcharacteristic function $chi _{s}^{r} (A)$ is a fuzzy ideal of $R$. We alsoshow that the set of all generalized characteristic functions $C_{s}^{r}(I(R))$ of the members of $I(R)$ for fixed $r , sin [0,1] $ is acomplete sublattice of the lattice of all fuzzy ideals of $R$ and establishthat this latter lattice is generated by the union of allits complete sublattices $C_{s}^{r} (I(R))$.
65
71
Iffat
Jahan
Iffat
Jahan
Department of Mathematics, Ramjas College, University Of Delhi,
Delhi, India
Department of Mathematics, Ramjas College,
India
ij.umar@yahoo.com
Algebra
Ideal of a ring
Morphism
Embedding
Lattice
[[1] N. Ajmal and K. V. K. Thomas, The lattice of fuzzy subgroups and fuzzy normal subgroups,##Information Sci., 76 (1994), 111.##[2] T. Head, A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and##Systems, 73 (1995), 349358.##[3] L. Wangjin, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8 (1982),##[4] D. S. Malik and J. N. Mordeson, Radicals of fuzzy ideals, Information Sci., 65 (1992) 23,##[5] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517.##[6] A. Weinberger, Embedding lattices of fuzzy subalgebras into lattices of crisp subalgebras,##Information Sci., 108 (1998), 5170.##]
FUZZY BOUNDED SETS AND TOTALLY FUZZY BOUNDED
SETS IN ITOPOLOGICAL VECTOR SPACES
FUZZY BOUNDED SETS AND TOTALLY FUZZY BOUNDED
SETS IN ITOPOLOGICAL VECTOR SPACES
2
2
In this paper, a new definition of fuzzy bounded sets and totallyfuzzy bounded sets is introduced and properties of such sets are studied. Thena relation between totally fuzzy bounded sets and Ncompactness is discussed.Finally, a geometric characterization for fuzzy totally bounded sets in I topologicalvector spaces is derived.
1
In this paper, a new definition of fuzzy bounded sets and totallyfuzzy bounded sets is introduced and properties of such sets are studied. Thena relation between totally fuzzy bounded sets and Ncompactness is discussed.Finally, a geometric characterization for fuzzy totally bounded sets in I topologicalvector spaces is derived.
73
90
ShenQing
Jiang
ShenQing
Jiang
Institute of Math., school of Math. and Computer Sciences, Nanjing
Normal University, Nanjing Jiangsu 210046, People0 s Republic of China
Institute of Math., school of Math. and Computer
China
jsq288@163.com
Conghua
Yan
Conghua
Yan
Institute of Math., school of Math. and Computer Sciences, Nanjing
Normal University, Nanjing Jiangsu 210046, People0 s Republic of China
Institute of Math., school of Math. and Computer
China
chyan@njnu.edu.cn
Itopological vector space
Fuzzy bounded set
Totally fuzzy bounded set
Ncompact set
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