<<

International Journal of Computer Applications (0975 – 8887) Volume 62– No.7, January 2013 Some Properties of Fuzzy Sets on Finite Type of

Kac-Moody Algebra G2

A. Uma Maheswari S.Krishnaveni Associate Professor Associate Professor Quaid-E-Millath Government College for Women M.O.P. Vaishnav College for Women (Autonomous), (Autonomous), Chennai – 600002 Chennai - 600034

~ ~ ABSTRACT Definition 3 [11]: The support of a fuzzy set A , S(A) is the The Kac-Moody algebras has been attracting the attention of a crisp set of all x  X such that ~ lot of Mathematicians because of its various connections and A (x)  0. applications to different branches of and Mathematical since the introduction of the subject in Definition 4 [11]: The (crisp) set of elements that belong to ~ 1968, developed simultaneously and independently by Kac the fuzzy set A , at least to the degree α is called the α - and Moody. On the other hand, Fuzzy sets first originated in a level set seminar paper by Lotfi A.Zadeh in 1965. The theory on fuzzy A {xX/μ ~ (x)  0}, sets has been applied not only to all branches of Mathematics α A but also acts as a tool for solving challenging problems in ~ A'α {xX/μ A (x)  0}, is called strong α - level set or science & technology and social problems. strong α  cut. Fuzzy approach on Kac-Moody algebras was initiated by ~ Definition 5 [11]: Let A be a fuzzy set on X. Then the set A.Uma Maheswari [4] in which fuzzy sets were defined on ~ {x  X/μ ~ (x)  1} is called the core of the fuzzy set A . the of Kac-Moody algebras. Some fuzzy A properties were studied for the finite type of Kac-Moody ~ ~ This set is denoted by core ( A ). A fuzzy set A is said to be algebras Al, Bl, Cl, Dl, E6, E7, E8, in [4], [7] & [9], for the normal if sup μ ~ x = 1. affine type of Kac-Moody algebras in [5] & [6] and x A   hyperbolic type HG2 in [8]. In this paper another fuzzy approach is attempted on the root system of finite type of Definition 6 [11]: The height of a fuzzy set is the largest Kac-Moody algebra G2. membership grade attained by any element in that set. Fuzzy sets are defined on the cartesian product of the root basis of G2 using an invariant, non degenerate, symmetric Definition 7 [11]: A fuzzy set is convex if bilinear form. Some of the basic properties like support, core, ~ ~ ~ A[x1  (1)x2 ]  min[A (x1),A (x2 )]  x1, x2  X normality, height, cardinality, relative cardinality, and Alternatively, a fuzzy set is convex if all α complement and convexity are studied; α – level sets and λ [0,1]. strong α – level sets are computed. α – cut decomposition, – level sets are convex. hamming distance and euclidean distance for the fuzzy set Definition 8 [1]: Let A be a fuzzy set on U and be a associated with G2 of finite type of Kac-Moody algebra are α also computed. number such that 0 <  1. Then by αA we mean a fuzzy set on U, denoted by αA which is such that ( )(x) = α Keywords A(x) for every x in U. This procedure of associating another Generlized , Kac-Moody algebra, root basis, fuzzy set with the given fuzzy set A is termed as restricted non- degenerate form, fuzzy set, core, normal, height, scalar multiplication. convexity, α -level set, strong α -level set, α – cut decomposition. Definition 9 [1]: Any fuzzy set A on U can be decomposed as A  sup{A /0 α 1}. we also write 1. INTRODUCTION  A  αA (or)A  αA . 1.1 Basic definitions of Fuzzy Sets  α α Definition 1 [11]: A classical (crisp) set is normally defined as a collection of elements or objects x  X that can be finite, Definition 10 [1]: Complement of a fuzzy set A, denoted by countable or overcountable. A, is defined as the fuzzy set on U, for which (A)(x) = 1 – Definition 2 [11]: If X is a collection of objects denoted A(x) for every x in U. generically by x, then a fuzzy set ~ A  {(x,μ ~ (x)) / x X }. μ ~ (x) is called the “membership A A Definition 11 [1]: Let A be a fuzzy set on U. Then by the ~ function” or “grade of membership” of x in A that maps X scalar cardinality of A we mean the number A(x) where to the membership space M.  the summation is over all the elements of U.

26 International Journal of Computer Applications (0975 – 8887) Volume 62– No.7, January 2013

1.2 Basic Definitions on Kac-Moody Algebras This number is denoted by A or SC(A). n Definition 20 [3]: An matrix A  (ai,j)i,j1 is a Definition 12 [1]: Let A be a fuzzy set on U, we associate Generalized Cartan Matrix (abbreviated as GCM) if it L(A), a crisp subset on I = (0,1], called its level set, L(A) is satisfies the following conditions: defined as follows: aii = 2  i =1,2,….,n L(A) = {α  I | A(x) = α for some x  U} aij = 0  aji = 0 i, j = 1,2,…,n ~ Definition 13 [11]: For a finite fuzzy set A , the aij  0 i, j = 1,2,…,n. ~ ~ Let us denote the index set of A by N = {1,…,n}. A GCM cardinality A is defined as A =  ~ (x)  A A is said to decomposable if there exist two non-empty xX subsets I, J  N such that I  J = N and a = a = 0  i I ~ ij ji ~ A ~ and j J. If A is not decomposable, it is said to be A  is called the relative cardinality of A . indecomposable. X Definition 21 [2]: A GCM A is called symmetrizable if DA Definition 14 [1]: Let A be a non-empty fuzzy set on U and is symmetric for some diagonal matrix D = diag(q1,…,qn), supp(A) be finite. Its fuzzy cardinality, denoted by FC(A), is with q > 0 and q ’s are rational numbers. defined as the fuzzy set on N (the set of all the natural i i numbers) given by  α /SC(A α ), i.e. FC(A) =  α /n α n Definition 22 [2]: A realization of a matrix A  (aij )i,j1 is a where n α = SC(A α ) and  runs over all  in L(A) [here v N = {1,2,3,…}. triple ( H, Π , Π ) where l is the rank of A, H is a 2n - l dimensional complex , Π {α1,...,αn } and Definition 15 [1]: For a fuzzy set A on U, and a positive v v v real number α , we define the α -th power of A (denoted by Π {α1 ,...,α n } are linearly independent subsets of H* A α ), A α (x) = [A(x)] for all x in U. v and H respectively, satisfying α j (αi )  a ij for i, j = Definition 16 [1]: Dilation of A is denoted by Dil(A) and is 1,….,n. Π is called the root basis. Elements of  are given Dil(A) = [A(x)]0.5 for all x in U, that is, Dil(A) = A0.5 called simple roots. The root generated by Π is n Q  Zα . Definition 17 [1]: The Concentration is denoted by con(A)  i and is given by con (A)(x) = [A(x)]2 for x in U. i1

Definition 18 [1]: The Contrast Intensification of a fuzzy set Definition 23 [2]: The Kac-Moody algebra g(A) associated n A , denoted by Int(A), is defined as : with a GCM A  (aij )i,j1 is the generated by the Int(A)(x) = 2 [A(x)]2 for 0  A(x)  0.5 elements ei , fi , i 1, 2, ... n and H with the following 2 = 1  2[1 A(x)] for 0.5  A(x) 1 defining relations : Definition 19 [1]: Index of Fuzziness indicates a method of [h,h' ]  0, h,h'  H measuring the fuzziness associated with a fuzzy set. This v index is given by the following functions: [ei , f j ]   ij i

Hamm(A) =  A(x) A'(x) [h,e j ]   j (h)e j Euc(A) = [ (A(x) A'(x)) 2]1/2 [h, f j ]    j (h) f j , i, j  N where the summation signs are taken over all elements of 1aij supp(A) [assuming supp(A) to be finite]. Hamm is called (adei ) e j  0 hamming distance and Euc is called euclidean distance. Note (adf )1aij f  0 ,  i  j, i, j  N that Hamm(A) = the cardinality of supp(A), if A is a crisp set; i j and Hamm(A) = 0 if A(x) = 0.5 for all x in U. The Kac-Moody algebra g(A) has the root space Thus , 0 Hamm(A) n, where n is the cardinality of decomposition g(A)   gα (A) where   αQ supp(A). Another approach is also available which is based on the α - gα(A)  {x g(A)/[h, x]  α(h)x, for all h H} . An element cut with = 0.5. The index is based on this idea is given by α , α  0 in Q is called a root if gα  0 . Let f H(A)  n Q  Z α . has a partial ordering “  ” defined by  A(x)  A (x) 2 1/2   i 0.5 , f E(A)  (A(x)  A0.5(x))  i1 Where the summation is over all the elements in supp(A) is α  β if β  α  Q +, where α , β  Q finite. These can be normalized in case supp(A) is finite , so n that the normalized values lie between 0 and 1. We have the Definition 24 [2]: For any αQ and α  k α , define following:  i i i1  2f H(A)  2f E(A) f H A  ,f E A  support of α , written as supp α , by #(supp(A)) #(supp(A))0.5 supp α {iN / k  0}. Let Δ( Δ(A)) denote the set i

of all roots of g(A) and Δ the set of all positive roots of

g(A). We have Δ   Δ and Δ  Δ  Δ .

27 International Journal of Computer Applications (0975 – 8887) Volume 62– No.7, January 2013

Proposition 25 [2]: A GCM n is symm- A  (a i,j )i,j  1 2. SOME FUZZY PROPERTIES ON THE etrizable if and only if there exists an invariant, bilinear, ROOT SYSTEMS OF FINITE KAC- symmetric, non-degenerate form on g(A). MOODY ALGEBRA G2 Now, for the finite type of Kac-Moody algebra G2, whose Definition 26 [2]: To every GCM A is associated a Dynkin is diagram S(A) defined as follows: S(A) has n vertices and vertices i and j are connected by max {|aij|,|aji|} number of lines if aij . aji  4 and there is an arrow pointing towards i if |aij| > 1. If aij. aji > 4, i and j are connected by a bold faced edge, equipped with the ordered pair (|aij| , |aji|) of . and the associated GCM with G2 is A =  2  1 and A =    3 2  Theorem 27 [2]: Let A be a real n x n matrix satisfying (m1), 1 0 2 1 D B where   and   . (m2) and (m3). D    B    0 3 1 2/3 (m1) A is indecomposable;   (m2) a ≤ 0 for i ≠ j; ij Let g(A) be the Kac – Moody Lie algebra associated with the (m3) a = 0 implies a = 0 ij ji GCM A . In the usual notation g(A) denote the Kac-Moody Then one and only one of the following three possibilities t algebra associated with G2. Let π be the set of simple roots holds for both A and A: ~ of g(A). Let ~ be the fuzzy set on (i) det A ≠ 0; there exists u > 0 such that A u > 0; Av ≥ 0 A  ((αi,α j), μA (αi,α j)) implies v > 0 or v = 0; ππ given by (1). (ii) co rank A=1; there exists u > 0 such that Au = 0; Av ≥ The following lemmas describe some properties of this fuzzy 0 implies Av = 0; set: (iii) there exists u > 0 such that Au < 0; Av ≥ 0, v ≥ 0 imply v = 0 . Lemma 1: Support of the fuzzy set Then A is of finite, affine or indefinite type iff (i), (ii) or (iii) ~ (respectively) is satisfied. A ={(α1,α1), (α2,α2), (α1,α2), (α2,α1)}  X.

Definition 28 [2]: A Kac- Moody algebra g(A) is said to be Proof: By the definition of support, ~ of finite, affine or indefinite type if the associated GCM A is Supp(A) { (α , α )X / μ (α , α )  0} of finite, affine or indefinite type respectively. i j i j  {(α1,α1), (α2,α2), (α1,α2), (α2,α1)}  X. Definition 29 [10]: A symmetric bilinear C- valued form < . , . > on a complex Lie algebra g(A) is said to be invariant Lemma 2: Height of the fuzzy set {(α ,α ), μ ~ (α ,α )} is 1. i j A i j if ( [ x, y ], z ) = ( x, [ y, z ] ) for all x, y, z  g(A). Proof: Height of the fuzzy set is 1 because the maximum Theorem 30 [10]: For any complex n x n matrix A the membership grade attained is 1. following statements are equivalent: (a). There exists an invariant, non-degenerate, symmetric, ~ Lemma 3: Core of the fuzzy set A is {(α1,α1)} bilinear C valued form  αi , α j  on g(A). (b). A is s ymmetrizable. Moreover, if these conditions are Proof: By the definition of core, satisfied, then ~ Core( A ) = ~ { (αi ,α j )X / μ A (αi ,α j ) 1} {(α1 ,α1 )} ( c )  . , .  | gα  gβ is ~ 0 if α,β  Δ {0} and α  β  0, Lemma 4: The fuzzy set A is normal.    non degenerately paired, if α,β  Δ {0} and α  β  0 Proof: Since Sup(α ,α )μ ~ (αi,αj)  1, the fuzzy set is In particular, the restriction of  . , .  to H is also non- i j A normal. degenerate. Note 31 [4]: Note that for the finite type of Kac-Moody Theorem 5: Let A be an indecomposable with the GCM algebra the rank of the GCM A = n.ie l = n.  2  1 . Let g(A) be the Kac – Moody algebra associated    3 2  In paper [4], the new concept of fuzzy sets on the root ~ systems of Kac- Moody algebras was introduced. The fuzzy with G2. Let A be the fuzzy set defined on ππ given by set on X = π π , where π = { α 1, α 2,…, n}, is defined as (1). Then for G2, the α - level sets and strong α - level sets follows: for α = 1,1/2 , 1/3,….,1/k… are given by,

|  αi ,α j  | i. A1  {(α1,α1)} ~ μ A (αi ,α j )  (1) max |  αp ,αq  | p,q  1,2,...,n ii. A1/2  {(α1,α1), (α1,α2), (α2,α1)} where  . , .  denotes bilinear, invariant, non-degenerate iii. A1/3  {(α1,α1), (α2,α2), (α1,α2), (α2,α1)},| A1/3 |  4 symmetric form on X defined by  αi , α j  = bij, where iv. A1/4  A1/3, A1  A1/2  A1/3  A1/4  ...  A1/k  ....X, ~ B = (bij) and A = DB. Then A = ~ forms | A1/k | 4, k  3,4,... ((αi,αj), μA (αi,αj)) a fuzzy set on π π . v. A'1/2  {(α1, α1)}.

vi. A'1/3  {(α1,α1), (α1,α2), (α2,α1)}.

28 International Journal of Computer Applications (0975 – 8887) Volume 62– No.7, January 2013

~ vii A'1/4 {(α1,α1),(α2 ,α2 ),(α1,α2 ),(α2 ,α1)}  X, | A'1/4 |  4 Hence the fuzzy set A corresponding to G2 of finite type of Kac-Moody algebra is convex. viii. A'  A' , ~ 1/5 1/4 Theorem 7: Let A be the fuzzy set defined on ππ , A'1/2  A'1/3  A'1/4  A'1/5  ...  A'1/k  ...  X, where  denotes the root basis for the finite Kac-Moody algebra given by equation (1). Then the α-cut decomposition | A'1/ k |  4, k  4,5,... Proof: (i) To compute 1- level Set: for the fuzzy set for the finite family G2 is 1A1 1/2A1/2  1/3A . A { (α ,α )X / μ ~ (α ,α )  1 } {(α ,α )} 1/3 1 i j A i j 1 1 (ii) To compute 1/2 –level Set: Proof: From the Theorem No [5] for the finite family G2, A { (α ,α )X / μ ~ (α ,α )  1/2 } 1/2 i j A i j  {(α1,α1),(α1,α2 ),(α2 ,α1)} We have A1  A1/2  A1/3  A1/4  ... A1/k  ... X. By (iii) To compute 1/3 -level set: the definition of α-cut decomposition, A {(α ,α )X / μ ~ (α ,α )  1/3} 1/3 i j A i j The α-cut decomposition for the fuzzy set for the finite  {(α ,α ),(α ,α ),(α ,α ),(α ,α )}  X, | A | 4 1 1 2 2 1 2 2 1 1/3 family G2 : 1A1 1/2A1/2  1/3A1/3. (iv) To compute 1/4 – level set: ~ A1/4 {(αi ,α j)X / μA (αi ,α j)  1/4} A1/3 Lemma 8: Let A be an indecomposable with the GCM From the above result,  2  1 . Let g(A) be the Kac – Moody Lie algebra    3 2  A1  A1/2  A1/3  A1/4  ...  A1/k  ...X, associated with G2. Let be the fuzzy set defined on π π | A | 4, for k  3,4,... 1/k given by equation (1). Then has the following properties: (v) To compute strong 1/2 - level set: (a) The cardinality | | = 7 / 3. A' {(α ,α )X / μ ~ (α ,α )  1/2}  {(α1,α1)} 1/2 i j A i j (b) Relative cardinality || || = 7 / 12. (vi) To compute strong 1/3 –level set: (c) The membership function of the complement of a A' {(α ,α )X / μ ~ (α ,α )  1/3} 1/3 i j A i j normalized fuzzy set corresponding to the finite type  {(α ,α ), (α ,α ), (α ,α )} 1 1 1 2 2 1 G ,  (α ,α )X are listed below : (vii) To compute strong 1/4 – level set: 2 i j ~ i.μ ~ (α ,α )  0 i  j 1. A'1/4  {(αi ,α j)X / μA (αi ,α j) 1/4} C A i j  {(α ,α ),(α ,α ),(α ,α ),(α ,α )}  X, 1 1 2 2 1 2 2 1 ii. μ ~ (αi ,αi1) 1/2, i 1. C A | A'1/4 |  4. iii. μ ~ (αi ,α j)  2/3 for i  j  2. (viii) To compute strong 1/5 - level set: C A ~ 1 1 1 1 ~ A'1/5 {(αi ,α j)X / μA (αi ,α j) 1/5} A'1/4 iv. FC(A) 1 .  . 2 3 3 4 From the above results, Proof: From equation (1), the fuzzy set corresponding to A'1/2  A'1/3  A'1/4  .A'1/5.  ...  A'1/k  ...  X. the finite type of Kac-Moody algebra associated with G2 Since A'  A' , | A' | 4 for k  4,5,... contains 1 element in X having membership grade 1, 2 1/3 1/k 1/k elements in X having membership grade 1/2 and 1 element in ~ X having membership grade 1/3. Lemma 6: A fuzzy set A corresponding to the finite type of By the definition of cardinality, Kac-Moody algebra G2, defined by (1) is convex. ~ (a) | A | μ ~ (x)  7/3. Proof: Consider the finite type of family G2. The table  A xX showing all possible membership grades for the elements of X ~ ~ | A | 7 and the conditions for checking convexity is listed below: (b) || A ||  . Table 1. Possible membership grades attained by the elements | X | 12 ~ of X (c) Since the fuzzy set A corresponding to the classical μ ~ (x ) μ ~ (x ) μ ~ [x  (1 λ)x ] min[(μ ~ (x1), μ ~ (x 2)] A 1 A 2 A 1 2 A A algebra G2, is normal,  (αi ,α j)X , the membership ~ function of the complement of a normalized fuzzy set A are 1 1 1 1 listed below :

1/2 1 (2 - λ)/ 2 1/2 ~ i. μC A (αi ,α j ) 11  0 for i  j 1 1 1/2 (1+λ) /2 1/2 ii. For,i  1,μ ~ (α ,α )  11/2  1/2 C A i i iii. For i  j  2, μ ~ (α ,α ) 1 (1/3)  2/3. 1/2 1/2 1/2 1/2 C A i j iv. By the definition of 1/3 1/3 1/3 1/3 ~  FC(A)   where n = SC(A ), 1/3 1 (3-2 λ)/3 1/3 n 1 1 1 1 1 1/3 (2λ+1)/3 1/3 1 .  . 2 3 3 4 1/3 1/2 (3- λ)/6 1/3 1/2 1/3 (λ+2)/6 1/3 Lemma 9: Let A be an indecomposable with the GCM . Let g(A) be the Kac – Moody Lie algebra From the above table, we see that for every x1 , x2 X and  λ  [ 0 , 1],  ~[x  (1)x ]  min[ ~ (x ), ~ (x )]. A 1 2 A 1 A 2

29 International Journal of Computer Applications (0975 – 8887) Volume 62– No.7, January 2013

~ given by equation (1).The index is given by the following associated with G2. Let A be the fuzzy set defined on π π given by equation (1). Then the Dilation and Concentration functions which are based on the α-cut with α = 0.5: are given by ~ 1 ~ 1 (i) fH (A)  (ii) fE (A)  ~ 3 9 (i) Dil(A) {1,.7,.5} ~ The normalized values of the index functions are given by (ii) ConA{1,.25,.09}. * ~ 1 * ~ 1 Proof: (i) By the definition of Dilation, (iii) fH (A)  (iv) fE (A)  ~ 0.5 ~ 6 9 Dil(A) = [A(x)] for all x in A , Proof: By the definition, = { 1, .7, .5} ~ ~ ~ 1 (i) f (A)   A(x)  A (x)  (ii) By the definition of Concentration, H 0.5 3 ~ ~ 2 ~ con(A)(x)  [A] x A ~ ~ ~ 2 1/2 1 (ii) f E(A)  (A(x)  A0.5(x))   ={ 1, .25, .09 }. 9 Lemma 10: Let A be an indecomposable with the GCM By the definition of the normalized values of the index  2  1 functions are given by   . Let g(A) be the Kac – Moody Lie algebra ~    ~ 2f H(A) 1  3 2  (iii) f H A ~  #(supp(A)) 6 associated with G2. Let be the fuzzy set defined on ~  ~ 2f E(A) 1 given by equation (1). Then the contrast intensification of a (iv) f E A  ~ ~ 0.5 9 fuzzy set A is given by #(supp(A)) ~ ~ .5,.18, if 0  A(x)  .5 3. CONCLUSION Int A(x)   ~  We can further compute the level cuts for various families of 1, if .5  A(x) 1  affine, hyperbolic and non hyperbolic of Kac-Moody algebras; Other structural properties on the fuzzy nature of Proof: By the definition of contrast intensification of a fuzzy these algebras can also be studied. set , 4. REFERENCES ~ 2 [1] Ganesh.M., 2009. Introduction to fuzzy sets and fuzzy Int( A )(x) = 2 [ (x)] for 0  (x)  0.5 logic. New Delhi: PHI Learning Private Limited. = 2(.5)2 and 2(.3)2 = { .5, .18} [2] Kac.V.G., 1990. Infinite Dimensional Lie Algebra, 3rd ed., Cambridge : Cambridge University Press. Int( )(x) = 1  2[1  (x)]2 for 0.5 A(x) 1 = 1 2[1 1 ]2 = 1. [3] Moody,R.V., 1968 . A new class of Lie algebras. J. Hence, Algebra. 10, pp. 211-230. [4] Uma Maheswari, A, & Krishnaveni, S, 2012, “Fuzziness ~ on the Root System of Finite Type of Kac-Moody ~ .5,.18, if 0  A(x)  .5 Int A(x)   ~  Algebras”, Proceedings of the International Conference 1, if .5  A(x)  1  on Mathematical Modeling And Applied Soft Computing , Vol 1 ISBN 978-81-92375 Lemma 11: Let A be an indecomposable with the GCM [5] Uma Maheswari, A., 2012, “Fuzzy Decomposition on . Let g(A) be the Kac – Moody Lie algebra the affine Type of Kac-Moody Algebras ”, IJPAM Vol 80. No 1 2012 -75 – 91. associated with G2. Let be the fuzzy set defined on [6] Uma Maheswari, A., 2012, “Fuzzy Decomposition on (1) (1) (1) given by equation (1). Then the Hamming distance and the affine Type of Kac-Moody Algebras F4 , E7 E8 (1) Euclidean distances are given by & G2 ”, IJSER Vol 3-Issue 7. ~ ~ Hamm(A)  2/3&Euc(A)  (10/9)1/2 [7] Uma Maheswari, A., & Gayathri, V, 2012, “Fuzzy Properties On Finite type Of Kac-Moody algebras “, Proof: By the definition of Hamming distance and Euclidean Lecture notes on Springer-Verlag CCIS 305 pp-352-363. [8] Uma Maheswari, A., 2012, “Fuzziness on the hyperbolic distance, ~ ~ ~ algebras HG2”, Mathematical Sciences International Hamm(A)  | A(x)  A'(x) | Research Journals- ISSN- 2278- 8697.

 1 1   1 1   1 2  2 [9] Uma Maheswari, A., & Krishnaveni, S., 2012,  1 0             “Fuzziness on the Root System of Finite Type of Kac- 2 2 2 2 3 3 3       ~ ~ ~ Moody Algebras E6, E7 & E8 ”, IEEE Proceedings of the Euc(A)  [(A(x)  A'(x))2 ]1/2 International Workshop on Mathematical Modeling ICECCS 2012, Kochin. 1/2  2  1/2 2  1 10 [10] Wan Zhe-Xian.,1991. Introduction to Kac–Moody  1        . 3 9 Algebra. Singapore : World Scientific Publishing       Co.Pvt.Ltd. Lemma 12: Let A be an indecomposable with the GCM . Let g(A) be the Kac – Moody Lie algebra [11] Zimmermann. H.J, 2001. Fuzzy Set Theory and its applications, 4th ed., London: Kluwer Academic Publisher. associated with G2. Let be the fuzzy set defined on

30