2010
7
2
2
153
Cover Vol. 7, No.2, June 2010
2
2
1

0
0
Exact and approximate solutions of fuzzy LR linear systems: New algorithms using a least squares model and the ABS approach
Exact and approximate solutions of fuzzy LR linear systems: New algorithms using a least squares model and the ABS approach
2
2
We present a methodology for characterization and an approach for computing the solutions of fuzzy linear systems with LR fuzzy variables. As solutions, notions of exact and approximate solutions are considered. We transform the fuzzy linear system into a corresponding linear crisp system and a constrained least squares problem. If the corresponding crisp system is incompatible, then the fuzzy LR system lacks exact solutions. We show that the fuzzy LR system has an exact solution if and only if the corresponding crisp system is compatible (has a solution) and the solution of the corresponding least squares problem is equal to zero. In this case, the exact solution is determined by the solutions of the two corresponding problems. On the other hand, if the corresponding crisp system is compatible and the optimal value of the corresponding constrained least squares problem is nonzero, then we characterize approximate solutions of the fuzzy system by solution of the least squares problem. Also, we characterize solutions by defining an appropriate membership function so that an exact solution is a fuzzy LR vector having the membership function value equal to one and, when an exact solution does not exist, an approximate solution is a fuzzy LR vector with a maximal membership function value. We propose a class of algorithms based on ABS algorithm for solving the LR fuzzy systems. The proposed algorithms can also be used to solve the extended dual fuzzy linear systems. Finally, we show that, when the system has more than one solution, the proposed algorithms are flexible enough to compute special solutions of interest. Several examples are worked out to demonstrate the various possible scenarios for the solutions of fuzzy LR linear systems.
1
We present a methodology for characterization and an approach for computing the solutions of fuzzy linear systems with LR fuzzy variables. As solutions, notions of exact and approximate solutions are considered. We transform the fuzzy linear system into a corresponding linear crisp system and a constrained least squares problem. If the corresponding crisp system is incompatible, then the fuzzy LR system lacks exact solutions. We show that the fuzzy LR system has an exact solution if and only if the corresponding crisp system is compatible (has a solution) and the solution of the corresponding least squares problem is equal to zero. In this case, the exact solution is determined by the solutions of the two corresponding problems. On the other hand, if the corresponding crisp system is compatible and the optimal value of the corresponding constrained least squares problem is nonzero, then we characterize approximate solutions of the fuzzy system by solution of the least squares problem. Also, we characterize solutions by defining an appropriate membership function so that an exact solution is a fuzzy LR vector having the membership function value equal to one and, when an exact solution does not exist, an approximate solution is a fuzzy LR vector with a maximal membership function value. We propose a class of algorithms based on ABS algorithm for solving the LR fuzzy systems. The proposed algorithms can also be used to solve the extended dual fuzzy linear systems. Finally, we show that, when the system has more than one solution, the proposed algorithms are flexible enough to compute special solutions of interest. Several examples are worked out to demonstrate the various possible scenarios for the solutions of fuzzy LR linear systems.
1
18
Reza
Ghanbari
Reza
Ghanbari
Department of Mathematics,
Ferdowsi University of Mashhad,
Mashhad, Iran
Department of Mathematics,
Ferdowsi
Iran
rghanbari@matr.um.ac.ir
Nezam
MahdaviAmiri
Nezam
MahdaviAmiri
Faculty of Mathematical Sciences,
Sharif University of Technology,
Tehran, Iran
Faculty of Mathematical Sciences,
Iran
nezamm@sharif.edu
Rohollah
Yousefpour
Rohollah
Yousefpour
Department of Mathematics,
Mazandaran University,
Babolsar, Iran
Department of Mathematics,
Mazandaran
Iran
yosefpoor@mehr.sharif.edu
Fuzzy linear system
Fuzzy LR solution
ABS algorithm
Least squares approximation
[bibitem{ABS} J. Abaffy, C. Broyden and E. Spedicato, {it A class of direct methods for linear##systems}, Numerische Mathematik, {bf45} (1984), 361376.##bibitem{ABSBOOK} J. Abaffy and E. Spedicato, {it ABS projection algorithms: mathematical techniques for##linear and nonlinear equations},##Ellis Horwood, Chichester, 1989.##bibitem{Abbasbandy}S. Abbasbandy and M. Alavi, {it A method for solving fuzzy linear system}, Iranian Journal of Fuzzy Systems, {bf2} (2005), 3743.##bibitem{abramovich}F. Abramovich, M. Wagenknecht and Y. I. Khurgin,##{it Solution of LRtype fuzzy systems of linear algebraic equations},##Busefal, {bf35} (1988), 8699.##bibitem{Store} R. Bulrisch and J. Stoer, {it Introduction to numerical analysis (texts in applied mathematics)}, 2nd edition, Springer, 1992.##bibitem{Buckley} J. J. Buckley and Y. Qu,##{it Solving systems of linear fuzzy equations}, Fuzzy Sets and Systems,##{bf43} (1991), 3343.##bibitem{dub} D. Dubois and H. Prade, {it Fuzzy sets and systems##theory and applications}, Academic Press, New York, 1980.##bibitem{Egrvary} E. Egervary, {it On rankdiminishing operations and their applications to the##solution of linear equations}, ZAMP, {bf9} (1960), 376386.##bibitem{iabs} H. Esmaeili, N. MahdaviAmiri and E. Spedicato,##{it A class of ABS algorithms for Diophantine linear systems}, Numerische##Mathematik, {bf91} (2001), 101ï؟½115.##bibitem{Frie1} M. Friedman, M. Ma and A. Kandel,## {it Fuzzy linear systems}, Fuzzy Sets and Systems, {bf96} (1998), 201209.##bibitem{Fuller}R. Fuller,##{it On stability in possibilistic linear equality systems with##Lipschitzian fuzzy numbers}, Fuzzy Sets and Systems, {bf34} (1990), 347ï؟½353.##bibitem{Khoram} M. Khorramizadeh and N. MahdaviAmiri, {it Integer extended ABS algorithms and possible control of intermediate results for linear Diophantine systems}, 4OR, {bf7(2)} (2009), 145 167.##bibitem{FrieD} M. Ma, M. Friedman and A. Kandel,##{it Duality in fuzzy linear systems}, Fuzzy Sets and Systems, {bf109} (2000), 5558.##bibitem{Golab}##G. H. Golub and C. F. Van Loan, {it Matrix computations}, Baltimore,##MD, 3rd edition, Johns Hopkins University Press, 1996.##bibitem{mah2} N. MahdaviAmiri and S. H. Nasseri, {it Duality results##and a dual simplex method for linear programming problems with##trapezoidal fuzzy variables}, Fuzzy Sets and System, {bf158} (2007), 19611978.##bibitem{mal1} H. R. Maleki, M. Tata and M. Mashinchi,##{it Linear programming with fuzzy variables}, Fuzzy Sets and Systems, {bf109} (2000),##bibitem{Minc} H. Minc, {it Nonnegative matrices}, Wiley, New York, 1988.##bibitem{Muz1} S. Muzzioli and H. Reynaerts, {it Fuzzy linear systems of form##$A_{1}x+b_{1}=A_2x+b_2$}, Fuzzy Sets and Systems, {bf157} ( 2006), 939951.##bibitem{fuzzy logic} H. T. Nguyen and E. A. Walker, {it A first course in fuzzy##logic}, Chapman & Hall, 2000.##bibitem{Nicolai}S. Nicolai and E. Spedicato, {it A bibliography of the ABS methods},##Optimization Methods and Software, {bf8} (1997), 171183.##bibitem{spidicato1} E. Spedicato, E. Bodon, A. D. Popolo and N.##MahdaviAmiri, {it ABS methods and ABSPACK for linear systems and##optimization: a review}, 4OR, {bf1} (2003), 5166.##bibitem{spedicato2} E. Spedicato, Z. Xia and L. Zhang,##{it ABS algorithms for linear equations and optimization}, Journal of##Computational and Applied Mathematics, {bf124} (2000), 155170.##bibitem{vorman} A. Vroman, G. Deschrijver and E. Kerre, {it Solving systems of linear fuzzy equations by parametric## functionsan improved algorithm}, Fuzzy Sets and Systems, {bf158} (2007), 1515##ï؟½1534.##bibitem{Wang} X. Wang, Z. Zhong and M. Ha, {it Iteration##algorithms for solving a system of fuzzy linear equations}, Fuzzy##Sets and Systems, {bf119} (2001), 121128.##bibitem{zim2} H. J. Zimmermann, {it Fuzzy set theory and its applications},##Third ed., Kluwer Academic, Norwell, 1996.##]
Fuzzy linear regression model with crisp coefficients: A goal programming approach
Fuzzy linear regression model with crisp coefficients: A goal programming approach
2
2
The fuzzy linear regression model with fuzzy inputoutput data andcrisp coefficients is studied in this paper. A linear programmingmodel based on goal programming is proposed to calculate theregression coefficients. In contrast with most of the previous works, theproposed model takes into account the centers of fuzzy data as animportant feature as well as their spreads in the procedure ofconstructing the regression model. Furthermore, the model can dealwith both symmetric and nonsymmetric triangular fuzzy data as wellas trapezoidal fuzzy data which have rarely been considered in theprevious works. To show the efficiency of the proposed model, somenumerical examples are solved and a simulation study is performed.The computational results are compared with some earlier methods.
1
The fuzzy linear regression model with fuzzy inputoutput data andcrisp coefficients is studied in this paper. A linear programmingmodel based on goal programming is proposed to calculate theregression coefficients. In contrast with most of the previous works, theproposed model takes into account the centers of fuzzy data as animportant feature as well as their spreads in the procedure ofconstructing the regression model. Furthermore, the model can dealwith both symmetric and nonsymmetric triangular fuzzy data as wellas trapezoidal fuzzy data which have rarely been considered in theprevious works. To show the efficiency of the proposed model, somenumerical examples are solved and a simulation study is performed.The computational results are compared with some earlier methods.
1
153
H
Hassanpour
H
Hassanpour
Department of Mathematics,
University of Birjand,
Birjand, Iran
Department of Mathematics,
University
Iran
hhassanpour@birjand.ac.ir
H. R
Maleki
H. R
Maleki
Faculty of Basic Sciences,
Shiraz University of Technology,
Shiraz, Iran
Faculty of Basic Sciences,
Shiraz
Iran
maleki@sutech.ac.ir
M. A
Yaghoobi
M. A
Yaghoobi
Department of Statistics,
Shahid Bahonar University of Kerman,
Kerman, Iran
Department of Statistics,
Shahid
Iran
yaghoobi@mail.uk.ac.ir
Fuzzy linear regression
Goal programming
Linear programming
Fuzzy number
[bibitem{AbAl:AMSFLS} S. Abbasbandy and M. Alavi, {it A method for solving fuzzy linear## systems}, Iranian Journal of Fuzzy Systems, {bf 2}textbf{(2)} (2005), 3744.##bibitem{AbMa:pcPFNRC} S. M. Abu Nayeem and M. Pal, {it The pcenter problem on fuzzy## networks and reduction of cost}, Iranian Journal of Fuzzy Systems,## {bf 5}textbf{(1)} (2008), 126.##bibitem{ArTa:EPFLRM} A. R. Arabpour and M. Tata, {it Estimating the parameters of a fuzzy linear## regression model}, Iranian Journal of Fuzzy Systems, {bf 5}textbf{(2)} (2008), 119.##bibitem{BaJaSh:LPandNF} M. S. Bazaraa, J. J. Jarvis and H. D. Sherali, {it Linear## programming and network flows}, Jhon Wiley and Sons, New York, 1990.##bibitem{CaVe:APFOMM} J. M. Cadenas and J. L. Verdegay, {it A primer on fuzzy optimization models## and methods}, Iranian Journal of Fuzzy Systems, {bf 3}textbf{(1)} (2006), 122.##bibitem{Celmins:LSMFFVD} A. Celmins, {it Least squares model fitting to fuzzy vector## data}, Fuzzy Sets and Systems, {bf 22} (1987), 245269.##bibitem{ChBa:FRLADE} S. H. Choi and J. J. Buckley, {it Fuzzy regression using least## absolute deviation estimators}, Soft Computing, {bf 12}textbf{(3)} (2008), 257263.##bibitem{ChKiPa:NRQE} S. H. Choi, H. K. Kim and K. O. Park, {it Nonlinear regression quantiles estimation}, Journal of the Korean Statistical Society, {bf 29} (2000), 187199.##bibitem{CDuGiSa:LSELRMFR} R. Coppi, P. D'Urso, P. Giordani and A. Santoro,## {it Least squares estimation of a linear regression model with LR fuzzy response},## Computational Statistics & Data Analysis, {bf 51} (2006), 267286.##bibitem{Diam:FLS} P. Diamond, {it Fuzzy least squares}, Information Sciences,## {bf 46} (1988), 141157.##bibitem{Diam:LSFSV} P. Diamond, {it Least squares fitting of several variables},## Preprint of Second IFSA Congress, Tokyo, July (1987), 2025.##bibitem{DiKl:MSFS} P. Diamond and P. Kloeden, {it Metrics spaces of fuzzy sets:## theory and applications}, World Scientific, 1994.##bibitem{D'Urso:LRFCIFCOD} P. D'Urso, {it Linear regression analysis for fuzzy/crisp input and fuzzy/crisp## output data}, Computational Statistics & Data Analysis, {bf 42} (2003), 4772.##bibitem{HaMaYa:AGPAFLR} H. Hassanpour, H. R. Maleki and M. A. Yaghoobi, {it A goal programming## approach to fuzzy linear regression with nonfuzzy input and fuzzy## output data}, APJOR, {bf 26}textbf{(5)} (2009), 587604.##bibitem{HaMaYa:ANEFLRMCMF} H. Hassanpour, H. R. Maleki and M. A. Yaghoobi, {it A note on evaluation## of fuzzy linear regression models by comparing membership functions}, Iranian Journal of Fuzzy Systems, {bf 6}textbf{(2)} (2009), 16.##bibitem{HoBeSm:ASMCFLR} M. Hojati, C. R. Bector and K. Smimou, {it A simple method for computation## of fuzzy linear regression}, EJOR, {bf 166} (2005), 172184.##bibitem{HuYa:AOADOFRM} W. L. Hung and M. S. Yang, {it An omission approach for detecting outliers in## fuzzy regression models}, Fuzzy Sets and Systems, {bf 157} (2006), 31093122.##bibitem{JOZ:OELDTPFLR} S. Jozsef, {it On the effect of linear data transformations in## possibilistic fuzzy linear regression}, Fuzzy Sets and Systems,## {bf 45} (1992), 185188.##bibitem{KaCh:AFLRMBEP} C. Kao and C. L. Chyu, {it A fuzzy linear regression model with better## explanatory power}, Fuzzy Sets and Systems, {bf 126} (2002), 401409.##bibitem{KaCh:LSEFRA} C. Kao and C. L. Chyu, {it Leastsquares## estimates in fuzzy regression analysis}, EJOR, {bf 148} (2003), 426435.##bibitem{KiBi:EFLRMCMF} B. Kim and R. R. Bishu, {it Evaluation of fuzzy linear regression models by## comparing membership functions}, Fuzzy Sets and Systems, {bf 100} (1998), 343352.##bibitem{KiYoLi:APLSEFO} H. K. Kim, J. H. Yoon and Y. Li, {it Asymptotic properties of least squares## estimation with fuzzy observations}, Information Sciences,## {bf 178} (2008), 439451.##bibitem{KoBa} R. Koenker and G. Bassett, {it Regression quantiles}, Econometrica,## {bf 46} (1978), 3350.##bibitem{LINGO} L. Schage, {it LINGO release 6.0}, LINDO System, Inc., 1999.##bibitem{MATLAB} {it MATLAB 7.0}, The Mathworks, (www.mathworks.com).##bibitem{MPL} Maximal software inc., (www.maximalsoftware.com).##bibitem{MiFrKa:GFLS} M. Ming, M. Friedman and A. Kandel, {it General## fuzzy leastsquares}, Fuzzy Sets and Systems, {bf 88} (1997), 107118.##bibitem{MoNaNa:FLRAPVR} M. Modarres, E. Nasrabadi and M. M. Nasrabadi, {it Fuzzy linear regression## analysis from the point of view risk}, International Journal of Uncertainty,## Fuzziness and KnowledgeBased Systems, {bf 12}textbf{(5)} (2004), 635649.##bibitem{MoNaNa:FLRMLSE} M. Modarres, E. Nasrabadi and M. M. Nasrabadi, {it Fuzzy linear regression models with least square errors}, Applied Mathematics and Computation, {bf 163} (2005), 977989.##bibitem{MoTa:PFFLSR} J. Mohammadi and S. M. Taheri, {it Pedomodels fitting with fuzzy least## squares regression}, Iranian Journal of Fuzzy Systems, {bf 1}textbf{(29)} (2004), 4562.##bibitem{OzDu:MOFRGF} E. C. "{O}zelkan and L. Duckstein, {it Multiobjective fuzzy regression:## a general framework}, Computers & Operations Research, {bf 27} (2000), 635652.##bibitem{Pet:FLRFI} G. Peters, {it Fuzzy linear regression with fuzzy## intervals}, Fuzzy Sets and Systems, {bf 63} (1994), 4555.##bibitem{ReWo:PCFLRM} D. T. Redden and W. H. Woodall, {it Properties of certain fuzzy linear## regression models}, Fuzzy Sets and Systems, {bf 64} (1994), 361375.##bibitem{SaMaZa:AnZMSFLPp} M. R. Safi, H. R. Maleki and E. Zaeimazad, {it A note on zimmermann## method for solving fuzzy linear programming problems}, Iranian Journal of## Fuzzy Systems, {bf 4}textbf{(2)} (2007), 3146.##bibitem{Sak:FSIMO} M. Sakawa, {it Fuzzy sets and interactive multiobjective optimization},## Plenum Press, New York, 1993.##bibitem{SaYa:FLRA} M. Sakawa and H. Yano, {it Fuzzy linear regression and its applications}, In:## Studies in Fuzziness, Fuzzy Regression Analysis,## J. Kacprzyk and M. Fedrizzi, Omnitech Press, Warsaw,## Poland, {bf1} (1992), 6180.##bibitem{SaYa:MFLRAFIOD} M. Sakawa and H. Yano, {it Multiobjective fuzzy linear## regression analysis for fuzzy inputoutput data}, Fuzzy Sets and## Systems, {bf 47} (1992), 173181.##bibitem{SaPe:EFLRM} D. A. Savic and W. Pedrycz, {it Evaluation of fuzzy linear## regression models}, Fuzzy Sets and Systems, {bf 39} (1991), 5163.##bibitem{ShKhGh:OLOFSFRICMC} E. Shivanian, E. Khorram and A. Ghodousian, {it Optimization of linear## objective function subject to fuzzy relation inequalities constraints with maxaverage composition}, Iranian Journal of Fuzzy Systems, {bf 4}textbf{(2)} (2007), 1530.##bibitem{Tan:FDAPLM} H. Tanaka, {it Fuzzy data analysis by possibilistic linear models},## Fuzzy Sets and Systems, {bf 24} (1987), 363375.##bibitem{TaHaWa:PLRAFD}H. Tanaka, I. Hayashi and J. Watada, {it Possibilistic linear## regression analysis for fuzzy data}, EJOR, {bf 40} (1989), 389396.##bibitem{TaUeAs:LRAFM} H. Tanaka, S. Uejima and K. Asai, {it Linear regression analysis with fuzzy## model}, IEEE Transactions on Systems, Man and Cybernetics,## {bf 12} (1982), 903907.##bibitem{VaKa:tBAFNS} S. M. Vaezpour and F. Karimi, {it Tbest approximation in fuzzy normed## spaces}, Iranian Journal of Fuzzy Systems, {bf 5}textbf{(2)} (2008),## 9399.##bibitem{WaZhMe:FNRLLST} N. Wang, W. X. Zhang and C. L. Mei, {it Fuzzy nonparametric regression based on local linear smoothing technique}, Information Sciences, {bf 177} (2007), 38823900. ##bibitem{Wu:FERPLRMIIOD} H. C. Wu, {it Fuzzy estimates of regression parameters in linear regression models for imprecise input and output data}, Computational Statistics & Data## Analysis, {bf 42} (2003), 203  217.##bibitem{Wu:FLRMFSP} H. C. Wu, {it Fuzzy linear regression model based on fuzzy scalar## product}, Soft Computing, {bf 12} (2008), 469477.##bibitem{WuNa:LSFRFRV} A. W"{u}nsche and W. N"{a}ther, {it Leastsquares fuzzy regression with fuzzy## random variables}, Fuzzy Sets and Systems, {bf 130} (2002), 4350.##bibitem{XuLi:MLSFFM} R. Xu and C. Li, {it Multidimensional leastsquares fitting with a fuzzy## model}, Fuzzy Sets and Systems, {bf 119} (2001), 215223.##bibitem{YaKo:OCFCCPFD} M. S. Yang and C. H. Ko, {it On a class of fuzzy cnumbers clustering procedures for fuzzy data}, Fuzzy sets and Systems, {bf 84} (1996), 4960.##bibitem{YaLi:FLSLRAFIOD} M. S. Yang and T. S. Lin, {it Fuzzy least squares linear regression analysis for fuzzy inputoutput data}, Fuzzy Sets and Systems, {bf 126} (2002), 389399.##bibitem{YaTa:ASNRBGPFPVP} M. A. Yaghoobi and M. Tamiz, {it A short note on the relationship between## goal programming and fuzzy programming for vectormaximum problems},## Iranian Journal of Fuzzy Systems, {bf 2}{bf (2)} (2005), 3136.##bibitem{Za:Fs} L. A. Zadeh, {it Fuzzy sets}, Information and Control, {bf 8} (1965),## 338353.##bibitem{Zadeh:FSBTP} L. A. Zadeh, {it Fuzzy sets as a basis for a theory of possibility},## Fuzzy Sets and Systems, {bf 1} (1978), 328.##]
FUZZY CONVEX SUBALGEBRAS OF COMMUTATIVE
RESIDUATED LATTICES
FUZZY CONVEX SUBALGEBRAS OF COMMUTATIVE RESIDUATED LATTICES
2
2
In this paper, we define the notions of fuzzy congruence relations
and fuzzy convex subalgebras on a commutative residuated lattice and we obtain
some related results. In particular, we will show that there exists a one
to one correspondence between the set of all fuzzy congruence relations and
the set of all fuzzy convex subalgebras on a commutative residuated lattice.
Then we study fuzzy convex subalgebras of an integral commutative residuated
lattice and will prove that fuzzy filters and fuzzy convex subalgebras of
an integral commutative residuated lattice coincide.
1
In this paper, we define the notions of fuzzy congruence relations
and fuzzy convex subalgebras on a commutative residuated lattice and we obtain
some related results. In particular, we will show that there exists a one
to one correspondence between the set of all fuzzy congruence relations and
the set of all fuzzy convex subalgebras on a commutative residuated lattice.
Then we study fuzzy convex subalgebras of an integral commutative residuated
lattice and will prove that fuzzy filters and fuzzy convex subalgebras of
an integral commutative residuated lattice coincide.
41
54
Shokoofeh
Ghorbani
Shokoofeh
Ghorbani
Department of Mathematics of Bam, Shahid Bahonar University
of Kerman, Kerman, Iran
Department of Mathematics of Bam, Shahid
Iran
sh.ghorbani@mail.uk.ac.ir
Abbas
Hasankhani
Abbas
Hasankhani
Department of Mathematics, Shahid Bahonar University of Kerman,
Kerman, Iran
Department of Mathematics, Shahid Bahonar
Iran
abhasan@mail.uk.ac.ir
(Integral) Commutative residuated lattice
Fuzzy convex subalgebra
Fuzzy congruence relation
Fuzzy filter
[[1] T. S. Blyth and M. F. Janovitz,Residuation theory, Perogamon Press, 1972. ##[2] K. Blount and C. Tsinakies,The structure of residuated lattices, Internat. J. Algebra Comput., ##13(4)(2003), 437461. ##[3] P. S. Das,Fuzzy groups and level subgroups, Math. Anal. Appl., 84 (1981), 264269. ##[4] R. P. Dilworth,Noncommutative residuated lattices, Trans. Amer. Math. Soc., (1939), 426444. ##[5] J. Hart, L. Rafter and C. Tsinakis,The structure of commutative residuated lattices, Internat. ##J. Algebra Comput.,12(4) (2002), 509524. ##[6] A. Hasankhani and A. Saadat,Some quotients on BCKalgebra generated by a fuzzy set, ##Iranian Journal of Fuzzy Systems,1(2) (2004), 3343. ##[7] K. Hur, S. Y. Jang and H. W. Kang,Some intuitionistic fuzzy congruences, Iranian Journal ##of Fuzzy Systems,3(1) (2006), 4557. ##[8] A. Iorgulescu,Classes of BCK algebraspart III, Preprint Series of the Institute of Mathematics ##of the Romanian Academy, preprint,3 (2004), 137. ##[9] T. Kowalski and H. Ono,Residuated lattices: an algebraic glimpse at logic without contraction, ##Japan Advanced Insitute of Science and Technology, 2001. ##[10] L. Lianzhen and L. Kaitai,Fuzzy filters of BL algebras, Information Sciences, 173 (2005),141154. ##[11] V. Murali,Fuzzy equivalence relations, Fuzzy Sets and Systems, 30 (1989), 155163. ##[12] E. Turunen,Mathematics behind fuzzy logic, PhysicaVerlag, 1999. ##[13] M. Ward,Residuated distributive lattices, Duke Math. J., (1940), 641651. ##[14] M. Ward and R. P. Dilworth,Residuated lattices, Trans. Amer. Math. Soc., (1939), 335354. ##[15] L. A. Zadeh,Fuzzy sets, Information and Control ,(1965), 338353. ##[16] J. L. Zhang,Fuzzy filters of the residuated lattices, New Math. Nat. Comput., 2(1) (2006),1128.##]
Ordered semigroups characterized by their intuitionistic fuzzy
biideals
Ordered semigroups characterized by their intuitionistic fuzzy
biideals
2
2
Fuzzy biideals play an important role in the study of ordered semigroupstructures. The purpose of this paper is to initiate and study theintiuitionistic fuzzy biideals in ordered semigroups and investigate thebasic theorem of intuitionistic fuzzy biideals. To provide thecharacterizations of regular ordered semigroups in terms of intuitionisticfuzzy biideals and to discuss the relationships of left(resp. right andcompletely regular) ordered semigroups in terms intuitionistic fuzzybiideals.
1
Fuzzy biideals play an important role in the study of ordered semigroupstructures. The purpose of this paper is to initiate and study theintiuitionistic fuzzy biideals in ordered semigroups and investigate thebasic theorem of intuitionistic fuzzy biideals. To provide thecharacterizations of regular ordered semigroups in terms of intuitionisticfuzzy biideals and to discuss the relationships of left(resp. right andcompletely regular) ordered semigroups in terms intuitionistic fuzzybiideals.
55
69
Asghar
Khan
Asghar
Khan
Department of Mathematics,
COMSATS Institute of IT Abbottabad, Pakistan
Department of Mathematics,
COMSATS Institute
Pakistan
azhar4set@yahoo.com
Young Bae
Jun
Young Bae
Jun
Department of Mathematics Educations and RINS ,
Gyengsang National University ,
Chinju 660701, Korea
Department of Mathematics Educations and
Korea
skywine@gmail.com
Muhammad
Shabir
Muhammad
Shabir
Department of Mathematics QuaidiAzam University,
Islamabad, Pakistan
Department of Mathematics QuaidiAzam University,
Korea
mshabirbhatti@yahoo.co.uk
Intuitionistic fuzzy sets
Intuitionistic fuzzy biideals
Regular
Left (resp. right) regular ordered semigroups
Semilattices of left and right simple ordered semigroups
[bibitem{1} K. T. Atanassov, textit{Intuitionistic fuzzy sets}, Fuzzy Sets##and Systems, textbf{20} (1986), 8796. ##bibitem{2} K. T. Atanassov, textit{New operations defined over the##intuitionistic fuzzy sets}, Fuzzy Sets and Systems, $mathbf{61}$ $ (1994)$, $%##137 $$142$. ##bibitem{3} K. T. Atanassov, textit{Intuitionistic fuzzy sets, theory and##applications}, Studies in Fuzziness and Soft Computing, Heidelberg,##PhysicaVerlag, $mathbf{35}$ $ (1999)$. ##bibitem{4} R. A. Borzooei and Y. B. Jun, textit{Intuitionistic fuzzy hyper##BCKideals of hyper BCKalgebras, }Iranian Journal of Fuzzy Systems, textbf{%##1}textbf{(1)} (2006), 2329. ##bibitem{5} P. Burillo and H. Bustince, textit{Vague sets are##intuitionistic fuzzy sets}, Fuzzy Sets and Systems textbf{79} (1996),##bibitem{6} B. Davvaz, W. A. Dudek and Y. B. Juntextit{, Intuitionistic fuzzy }$%##H_{v}$textit{submodules}, Information Sciences, $mathbf{176}$ $(2006)$, $%##285 $$300$. ##bibitem{7} S. K. De, R. Biswas and A. R. Roy, textit{An application of##intuitionistic fuzzy sets in medical diagnosis}, Fuzzy Sets and Systems,##textbf{117} (2001), 209213. ##bibitem{8} L. Dengfeng and C. Chunfian, textit{New similarity measures of##intuitionistic fuzzy sets and applications to pattern recognitions}, Pattern##Reconit. Lett., textbf{23} (2002), 221225. ##bibitem{9} W. L. Gau and D. J. Buehre, textit{Vague sets}, IEEE Trans Syst##Man Cybern, textbf{23} (1993), 610614. ##bibitem{10} S. B. Hosseini, D. Oregan and R. Saadati, textit{Some results on##intuitionistic fuzzy spaces}, Iranian Journal of Fuzzy Systems, textbf{4}textbf{(1)} (2007), 5364. ##bibitem{11} K. Hur, S. Y. Jang and H. W. Kang, textit{Some intuitionistic##fuzzy congruences}, Iranian Journal of Fuzzy Systems, textbf{3}textbf{(1)} (2006),##bibitem{12} Y. B. Jun, textit{Intuitionistic fuzzy biideals of ordered##semigroups}, KYUNGPOOK Math. J., $mathbf{45}$ $(2005)$, $527$$537$. ##bibitem{13} N. Kehayopulu, textit{On regular duo ordered semigroups},##Math. Japonica, $mathbf{37}$$textbf{(6)}$ $(1990)$, $1051$$1056$. ##bibitem{14} N. Kehayopulu and M. Tsingelis, textit{Fuzzy sets in ordered##groupoids}, Semigroup Forum $mathbf{65}$ $(2002)$, $128$$132$. ##bibitem{15} N. Kehayopulu and M. Tsingelis, textit{Fuzzy biideals in##ordered semigroups}, Information Sciences, $mathbf{171}$ $(2005)$, $13$$28$. ##bibitem{16} N. Kehayopulu and M. Tsingelis, textit{Regular ordered##semigroups in terms of fuzzy subset}, Information Sciences, $mathbf{176}$ $%##(2006)$, $65$$71$. ##bibitem{17} N. Kehayopulu and M. Tsingelis, textit{Left and intraregular##ordered semigroups in terms fuzzy sets}, Quasigroups and Related Systems,##textbf{14} (2006), 169178. ##bibitem{18} N. Kehayopulu and M. Tsingelis, textit{Fuzzy ideals in ordered##semigroups}, Quasigroups and Related Systems, textbf{15} (2007), 279289. ##bibitem{19} K. H. Kim and Y. B. Jun, textit{Intuitionistic fuzzy interior##ideals of semigroups,} Int. J. Math. Math. Sci., $mathbf{{27}(5)}$ $ (2001)$, $%##261 $$267.$ ##bibitem{20} K. H. Kim and Y. B. Jun, textit{Intuitionistic fuzzy ideals of##semigroups}, Indian J. Pure Appl. Math., $mathbf{{33}(4)}$ $(2002)$, $443$$449$. ##bibitem{21} K. H. Kim, W. A. Dudek and Y. B. Jun, textit{Intuitionistic##fuzzy subquasigroups of quasigroups}, Quasigroups Related Systems, textbf{7} (2000), 1528. ##bibitem{22} M. Rafi and M. S. M. Noorani, textit{Fixed point theorem on##intuitionistic fuzzy metric spaces}, Iranian Journal of Fuzzy Systems,##textbf{{3}(1)} (2006), 2329. ##bibitem{23} M. Shabir and A. Khan, textit{Characterizations of ordered##semigroups by the properties of their fuzzy generalized biideals}, New##Math. Natural Comput., $mathbf{{4}(2)}$ ($2008$), $237$$250$. ##bibitem{24} M. Shabir and A. Khan, textit{Intuitionistic fuzzy interior##ideals of ordered semigroups}, to appear in J. Applied Math. and##Informatics. ##bibitem{25} M. Shabir and A. Khan, textit{Ordered semigroups characterized##by their intuitionistic fuzzy generalized biideals}, to appear in Fuzzy##Systems and Mathematics. ##bibitem{26} E. Szmidt and J. Kacprzyk, textit{Entropy for intuitionistic##fuzzy sets}, Fuzzy Sets and Systems, textbf{118} (2001), 467477. ##bibitem{27} L. A. Zadeh, textit{Fuzzy sets}, Information and Control, $mathbf{8}$ $(1965)$, $338$$353$.##]
MFUZZIFYING DERIVED OPERATORS AND DIFFERENCE
DERIVED OPERATORS
MFUZZIFYING DERIVED OPERATORS AND DIFFERENCE
DERIVED OPERATORS
2
2
This paper presents characterizations of Mfuzzifying matroids bymeans of two kinds of fuzzy operators, called Mfuzzifying derived operatorsand Mfuzzifying difference derived operators.
1
This paper presents characterizations of Mfuzzifying matroids bymeans of two kinds of fuzzy operators, called Mfuzzifying derived operatorsand Mfuzzifying difference derived operators.
71
81
Xiu
Xin
Xiu
Xin
Department of Mathematics, Tianjin University of Technology, Tianjin,300384, P.R.China
Department of Mathematics, Tianjin University
China
xinxiu518@163.com
FuGui
Shi
FuGui
Shi
Department of Mathematics, Beijing Institute of Technology, Beijing,
100081, P.R.China
Department of Mathematics, Beijing Institute
China
fuguishi@bit.edu.cn
ShengGang
Li
ShengGang
Li
College of Mathematics and Information Science, Shaanxi Normal
University, Xi’an, 710062, P.R.China
College of Mathematics and Information Science,
China
shenggangli@yahoo.com.cn
Mfuzzifying matroid
Mfuzzifying closure operator
Mfuzzifying derived operator
Mfuzzifying difference derived operator
[[1] A. Borumand Saeid, Intervalvalued fuzzy Balgebras, Iranian Journal of Fuzzy Systems, 3##(2006), 6374.##[2] P. Dwinger, Characterizations of the complete homomorphic images of a completely distributive##complete lattice I, Indagationes Mathematicae (Proceedings), 85 (1982), 403414.##[3] R. Goetschel and W. Voxman, Fuzzy matroids, Fuzzy Sets and Systems, 27 (1988), 291302.##[4] R. Goetschel and W. Voxman, Bases of fuzzy matroids, Fuzzy Sets and Systems, 31 (1989),##[5] R. Goetschel and W. Voxman, Fuzzy matroids and a greedy algorithm, Fuzzy Sets and Systems,##37 (1990), 201214.##[6] R. Goetschel and W. Voxman, Fuzzy rank functions, Fuzzy Sets and Systems, 42 (1991),##[7] I. C. Hsueh, On fuzzication of matroids, Fuzzy Sets and Systems, 53 (1993), 319327.##[8] H. L. Huang and F. G. Shi, Lfuzzy numbers and their properties, Information Sciences, 178##(2008), 11411151.##[9] S. P. Li, Z. Fang and J. Zhao, P2connectedness in Ltopological spaces, Iranian Journal of##Fuzzy Systems, 2 (2005), 2936.##[10] L. A. Novak, A comment on ‘Bases of fuzzy matroids’, Fuzzy Sets and Systems, 87 (1997),##[11] L. A. Novak, On fuzzy independence set systems, Fuzzy Sets and Systems, 91 (1997), 365374.##[12] L. A. Novak, On goetschel and voxman fuzzy matroids, Fuzzy Sets and Systems, 117 (2001),##[13] J. G. Oxley, Matroid theory, Oxford University Press, New York, 1992.##[14] F. G. Shi, Theory of Lnested sets and Lnested sets and its applications, Fuzzy Systems##and Mathematics, in Chinese, 4 (1995), 6572.##[15] F. G. Shi, Lfuzzy relation and Lfuzzy subgroup, J. Fuzzy Math., 8 (2000), 491499.##[16] F. G. Shi, A new approach to the fuzzification of matroids, Fuzzy Sets and Systems, 160##(2009), 696705.##[17] F. G. Shi, (L,M)fuzzy matroids, Fuzzy Sets and Systems, 160 (2009), 23872400.##[18] G. J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems, 47 (1992),##[19] L. Wang and F. G. Shi, Characterization of Lfuzzifying matroids by Lfuzzifying closure##operators, Iranian Journal of Fuzzy Systems, 7 (2010), 4758.##]
LOCAL BASES WITH STRATIFIED STRUCTURE IN $I$TOPOLOGICAL VECTOR SPACES
LOCAL BASES WITH STRATIFIED STRUCTURE IN
$I$TOPOLOGICAL VECTOR SPACES
2
2
In this paper, the concept of {sl local base with stratifiedstructure} in $I$topological vector spaces is introduced. Weprove that every $I$topological vector space has a balanced localbase with stratified structure. Furthermore, a newcharacterization of $I$topological vector spaces by means of thelocal base with stratified structure is given.
1
In this paper, the concept of {sl local base with stratifiedstructure} in $I$topological vector spaces is introduced. Weprove that every $I$topological vector space has a balanced localbase with stratified structure. Furthermore, a newcharacterization of $I$topological vector spaces by means of thelocal base with stratified structure is given.
83
93
JinXuan
Fang
JinXuan
Fang
School of Mathematical Science,
Nanjing Normal University,
Nanjing, Jiangsu 210097,
P. R. China
School of Mathematical Science,
China
jxfang@njnu.edu.cn
$I$topological vector spaces
$Q$neighborhood base
$W$neighborhood base
Local base with stratified structure
[bibitem{Chang} C. L. Chang, {it Fuzzy topological spaces}, J. Math. Anal. Appl.,##{bf 24} (1968), 182190.##bibitem{Fang1} J. X. Fang, {it On local bases of fuzzy topological vector##spaces}, Fuzzy Sets and Systems, {bf 87} (1997), 341347.##bibitem{Fang2} J. X. Fang, {it Fuzzy linear orderhomomorphism and its##structures}, J. Fuzzy Math., {bf 4}textbf{(1)} (1996), 93102.##bibitem{Fe} C. Felbin, {it Finite dimensional fuzzy normed linear space},##Fuzzy Sets and Systems, {bf 48} (1992), 239248.##bibitem{HR} U. H"{o}hle and S. E. Rodabaugh (Eds.), {it Mathematics of##fuzzy sets: logic, topology and measure theory}, The Handbooks of##Fuzzy Sets Series, Kluwer Academic Publishers, Dordrecht,##{bf 3} (1999).##bibitem{JY} S. Q. Jing and C. H. Yan, {it Fuzzy bounded sets and totally fuzzy bounded##sets in $I$topological vector spaces}, Iranian Journal of Fuzzy##Systems, {bf 4}textbf{(1)} (2009).##bibitem{KL} A. K. Katsaras and D. B. Liu, {it Fuzzy vector spaces and fuzzy topological vector spaces}, J.##Math. Anal. Appl., {bf 58} (1997), 135146.##bibitem{Ka} A. K. Katsaras, {it Fuzzy topological vector spaces I}, Fuzzy##Sets and Systems, {bf 6} (1981), 8595.##bibitem{PL} P. M. Pu and Y. M. Liu, {it Fuzzy topology I, neighborhood##structures of a fuzzy points and MooreSmith convergence}, J.##Math. Anal. Appl., {bf 76} (1980), 571599.##bibitem{Wan} G. J. Wang, {it Orderhomomorphism of fuzzes}, Fuzzy Sets and##Systems, {bf 12} (1982), 281288.##bibitem{War} R. H. Warren, {it Neighborhoods bases and continuity in fuzzy##topological spaces}, Rocky Mountain J. Math., {bf 8} (1978),##bibitem{WF} C. X. Wu and J. X. Fang, {it Redefine of fuzzy topological##vector space}, Since Exploration, in Chinese, {bf 2}textbf{(4)} (1982),##bibitem{XF} G. H. Xu and J. X. Fang, {it A new $I$vector topology##generated by a fuzzy norm}, Fuzzy Sets and Systems, {bf 158}##(2007), 23752385.##bibitem{YF} C. H. Yan and J. X. Fang, {it $L$fuzzy bilinear operator and its##continuity}, Iranian Journal of Fuzzy Systems, {bf 4}textbf{(1)}##(2007), 6573.##]
About the fuzzy grade of the direct product of two hypergroupoids
About the fuzzy grade of the direct product of two hypergroupoids
2
2
The aim of this paper is the study of the sequence of join spacesand fuzzy subsets associated with a hypergroupoid. In thispaper we give some properties of the membership function$widetildemu_{otimes}$ corresponding to the direct product oftwo hypergroupoids and we determine the fuzzy grade of thehypergroupoid $langle Htimes H, otimesrangle$ in a particularcase.
1
The aim of this paper is the study of the sequence of join spacesand fuzzy subsets associated with a hypergroupoid. In thispaper we give some properties of the membership function$widetildemu_{otimes}$ corresponding to the direct product oftwo hypergroupoids and we determine the fuzzy grade of thehypergroupoid $langle Htimes H, otimesrangle$ in a particularcase.
95
108
Irina
Cristea
Irina
Cristea
DIEA, University of Udine,
Via delle Scienze 2008, 33100 Udine, Italy
DIEA, University of Udine,
Via delle
Italy
irinacri@yahoo.co.uk
Fuzzy set
Hypergroup
Join space
Fuzzy grade. } newlineindent{footnotesize This work was partially supported by the Grant no.88/2008 of the Romanian Academy
A new perspective to the MazurUlam problem in $2$fuzzy $2$normed linear spaces
A new perspective to the MazurUlam problem in $2$fuzzy $2$normed linear spaces
2
2
In this paper, we introduce the concepts of $2$isometry, collinearity, $2$%Lipschitz mapping in $2$fuzzy $2$normed linear spaces. Also, we give anew generalization of the MazurUlam theorem when $X$ is a $2$fuzzy $2$%normed linear space or $Im (X)$ is a fuzzy $2$normed linear space, thatis, the MazurUlam theorem holds, when the $2$isometry mapped to a $2$%fuzzy $2$normed linear space is affine.
1
In this paper, we introduce the concepts of $2$isometry, collinearity, $2$%Lipschitz mapping in $2$fuzzy $2$normed linear spaces. Also, we give anew generalization of the MazurUlam theorem when $X$ is a $2$fuzzy $2$%normed linear space or $Im (X)$ is a fuzzy $2$normed linear space, thatis, the MazurUlam theorem holds, when the $2$isometry mapped to a $2$%fuzzy $2$normed linear space is affine.
109
119
Cihangir
Alaca
Cihangir
Alaca
Department of Mathematics,
Faculty of Science and Arts, Sinop University,
57000 Sinop, Turkey
Department of Mathematics,
Faculty
Turkey
cihangiralaca@yahoo.com.tr
$alpha $$2$Norm
$2$Fuzzy $2$Normed linear spaces
$2$Isometry
$2$Lipschitz mapping
Regular ordered semigroups and intraregular ordered
semigroups in terms of fuzzy subsets
Regular ordered semigroups and intraregular ordered
semigroups in terms of fuzzy subsets
2
2
Let $S$ be an ordered semigroup. A fuzzy subset of $S$ is anarbitrary mapping from $S$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. In this paper, the concept of fuzzygeneralized biideals of an ordered semigroup $S$ is introduced.Regular ordered semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) biideals.Finally, two main theorems which characterize regular orderedsemigroups and intraregular ordered semigroups in terms of fuzzyleft ideals, fuzzy right ideals, fuzzy biideals or fuzzyquasiideals are given. The paper shows that one can pass fromresults in terms of fuzzy subsets in semigroups to orderedsemigroups. The corresponding results of unordered semigroups arealso obtained.
1
Let $S$ be an ordered semigroup. A fuzzy subset of $S$ is anarbitrary mapping from $S$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. In this paper, the concept of fuzzygeneralized biideals of an ordered semigroup $S$ is introduced.Regular ordered semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) biideals.Finally, two main theorems which characterize regular orderedsemigroups and intraregular ordered semigroups in terms of fuzzyleft ideals, fuzzy right ideals, fuzzy biideals or fuzzyquasiideals are given. The paper shows that one can pass fromresults in terms of fuzzy subsets in semigroups to orderedsemigroups. The corresponding results of unordered semigroups arealso obtained.
121
140
XiangYun
Xie
XiangYun
Xie
Department of Mathematics and Physics,
Wuyi University ,
Jiangmen, Guangdong, 529020, P.R.China
Department of Mathematics and Physics,
China
xyxie@wyu.edu.cn
Jian
Tang
Jian
Tang
Jian Tang\
School of
Mathematics and Computational Science,
Fuyang Normal College,
Fuyang, Anhui, 236041, P.R.China
Jian Tang\
School of
Mathematics
China
tangjian0901@126.com
Ordered semigroup
Regular ordered semigroup
Intraregular ordered semigroup
Fuzzy left (right) ideal
Fuzzy (generalized) biideal
Fuzzy quasiideal
Actions, Norms, Subactions and Kernels of (Fuzzy) Norms
Actions, Norms, Subactions and Kernels of (Fuzzy) Norms
2
2
In this paper, we introduce the notion of an action $Y_X$as a generalization of the notion of a module,and the notion of a norm $vt: Y_Xto F$, where $F$ is a field and $vartriangle(xy)vartriangle(y') =$ $ vartriangle(y)vartriangle(xy')$ as well as the notion of fuzzy norm, where $vt: Y_Xto [0, 1]subseteq {bf R}$, with $bf R$ the set of all real numbers. A great many standard mappings on algebraic systems can be modeled on norms as shown in the examples and it is seen that $mathrm{Ker}vt ={yvt(y)=0}$ has many useful properties. Some are explored, especially in the discussion of fuzzy norms as they relate to the complements of subactions $N_X$ of $Y_X$.
1
In this paper, we introduce the notion of an action $Y_X$as a generalization of the notion of a module,and the notion of a norm $vt: Y_Xto F$, where $F$ is a field and $vartriangle(xy)vartriangle(y') =$ $ vartriangle(y)vartriangle(xy')$ as well as the notion of fuzzy norm, where $vt: Y_Xto [0, 1]subseteq {bf R}$, with $bf R$ the set of all real numbers. A great many standard mappings on algebraic systems can be modeled on norms as shown in the examples and it is seen that $mathrm{Ker}vt ={yvt(y)=0}$ has many useful properties. Some are explored, especially in the discussion of fuzzy norms as they relate to the complements of subactions $N_X$ of $Y_X$.
141
147
Jeong Soon
Han
Jeong Soon
Han
Department of Applied Mathematics,
Hanyang University ,
Ahnsan, 426791, Korea
Department of Applied Mathematics,
Korea
han@hanyang.ac.kr
Hee Sik
Kim
Hee Sik
Kim
Department of Mathematics,
Hanyang University ,
Seoul, 133791, Korea
Department of Mathematics,
Hanyang
Korea
heekim@hanyang.ac.kr
J
Neggers
J
Neggers
Department of Mathematics,
University of Alabama,
Tuscaloosa, AL 354870350, U. S. A
Department of Mathematics,
University
United States
jneggers@as.ua.edu
(Fuzzy) norm
(Sub) action
Kernel
[bibitem{BS}## T. Bag and S. K. Samata, textit{A comparative study of fuzzy norms on a linear space}, Fuzzy Sets and Systems##textbf{159} (2008), 670684.##bibitem{K}##A. K. Katsaras, textit{Fuzzy topological vector space} II, Fuzzy Sets and Systems textbf{12} (1984), 143154.##bibitem{O}##O. T. O'meara, textrm{Introduction to quadratic forms}, SpringerVerlag, Berlin, 1963.##bibitem{RS}##J. R. Raftery and T. Sturm, textit{On completions of pseudonormed $BCK$algebras and pseudometric universal algebras}, Math. Japonica textbf{33} (1988), 919929.##bibitem{ZS}##O. Zariski and P. Samuel, textrm{Commutative algebra}, D. Van Nostrand, Toronto, textbf{I, II} (1958).##]
Fuzzy Subgroups of Rank Two Abelian pGroup
Fuzzy Subgroups of Rank Two Abelian pGroup
2
2
In this paper we enumerate fuzzy subgroups, up to a natural equivalence, of some finite abelian pgroups of rank two where p is any prime number. After obtaining the number of maximal chains of subgroups, we count fuzzy subgroups using inductive arguments. The number of such fuzzy subgroups forms a polynomial in p with pleasing combinatorial coefficients. By exploiting the order, we label the subgroups of maximal chains in a special way which enables us to count the number of fuzzy subgroups.
1
In this paper we enumerate fuzzy subgroups, up to a natural equivalence, of some finite abelian pgroups of rank two where p is any prime number. After obtaining the number of maximal chains of subgroups, we count fuzzy subgroups using inductive arguments. The number of such fuzzy subgroups forms a polynomial in p with pleasing combinatorial coefficients. By exploiting the order, we label the subgroups of maximal chains in a special way which enables us to count the number of fuzzy subgroups.
149
153
S
Ngcibi
S
Ngcibi
Department of Mathematics (P&A),
University of Fort Hare,
Alice, 5700, South Africa
Department of Mathematics (P&A),
South Africa
sngcibi@ufh.ac.za
V
Murali
V
Murali
Department of Mathematics (P&A),
Rhodes University,
Grahamstown, 6140, South Africa
Department of Mathematics (P&A),
South Africa
v.murali@ru.ac.za
B. B
Makamba
B. B
Makamba
B. B. Makamba,
Department of Mathematics (P&A),
University of Fort Hare,
Alice, 5700, South Africa
B. B. Makamba,
Department of Mathematics
South Africa
bmakamba@ufh.ac.za
Equivalence
Fuzzy subgroup
pgroups
Keychain
[bibitem{mur:01} V.Murali, and B.B.Makamba, On an Equivalence of Fuzzy Subgroups I , Fuzzy Sets and Systems, 123 (2001) 163168.##bibitem{mur:03} V.Murali and B.B. Makamba, On an Equivalence of Fuzzy Subgroups II, Fuzzy sets and Systems, 136 (2003), 93104.##bibitem{mur:04} V.Murali and B.B. Makamba, Counting the number of fuzzy subgroups of an abelian group of order $p^n q^m$, Fuzzy sets and Systems, 144 (2004), 459470.##bibitem{ros:71} A.Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl., 35 (1971) 512517.##bibitem{zad:65} L.A.Zadeh, Fuzzy sets, Inform. and control, 8 (1965) 338353.##]
Persiantranslation Vol. 7, No.2, June 2010
2
2
1

157
167