Semi-simple Fuzzy G- Modules
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(2) 459. Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (4), 458-463 (2012). (iii) gk m k m k gm k gm , g G, m , m M, k , k K A subspace of M, which itself is a G-module with the same action is called Gsubmodule of M. It can be seen that the intersection of G-submodules is again a Gsubmodule. A non-zero G- module M is irreducible if the only G- submodules of M are M and {0}.Otherwise it is reducible. A non-zero G module M is completely reducible if for every G-submodule N of M there exists a G-submodule N of M such that M=N N . It is well known that Gsubmodules of completely reducible Gmodules are completely reducible. For Gmodules M and M*, M is M* injective if, for every submodule N of M*, any homomorphism φ from N to M can be extended as a homomorphism ψ from M* to M. A G- module M is semi simple if there exists a family of irreducible G sub modules M i such that M M . It is evident that completely reducible G-modules are semi simple. A fuzzy G-module over a G-module M is a fuzzy set µ on M (i.e. a function µ: M 0,1") such that (i) µax by ' min(µx, µy) , a, b K and x, y M and (ii) µgm ' µm, m M and g G. The standard fuzzy intersection of finite number of fuzzy G- modules is again a fuzzy G- module, while standard union and compliment need be so. If M M is a G-module and µ is a fuzzy G-module on M , ∀i , then µ defined by µ x = min (µ x , µ x , … , µ x ), x . x x . . . x M and x M is fuzzy G-module on M called the direct sum of fuzzy G-modules µ , i 1,2, … , n. A fuzzy G-module µ on M is completely reducible if (i) M is completely reducible, (ii) M has at least one proper G-submodule and (iii) Corresponding to any proper decomposition M M of M, there exists fuzzy G-modules µ on M , i 1,2 , such that µ µ µ . If µ and - are fuzzy G-modules on Gmodules M and M* then µ is - injective if i). M is M* injective and. ii) - (m) ≤ µ(ψ(m)), for every ψ ∈ Hom(M*,M). The standard fuzzy compliment [3] of a fuzzy set µ on X is defined as µ.x 1 / µx. The standard fuzzy intersection [3] and standard fuzzy union [3] of two fuzzy sets µ and µ on X are defined by µ 0 µ x min1µ x, µ x2 and µ 3 µ x max1µ x, µ x2. Throughout this paper we are applying standard fuzzy operations for union, intersection and complementation. 3. SEMI–SIMPLE FUZZY G-MODULES 3.1 Definition A fuzzy G-module µ on M is said to be semi-simple if M is semi-simple and µ µ where µ is a fuzzy G-module on M , i.. Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497).
(3) Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (4), 458-463 (2012). 3.2 Example Let 4 11, /12 567 8 9(√2) ;<=> 9. Then M is a semi-simple Gmodule with M = 9(√2) 9 √29. Let ? from M to [0, 1] be defined as ?(5 @√2) = 1, if a = 0, b = 0 = .8, if a ≠ 0, b = 0 = .2, if b ≠ 0 Define ? from Q to [0, 1] by ? A = 1, if x = 0 = .8, if x ≠ 0 and ? from √29 to [0, 1] by ? A = 1, if x = 0. 460. = .8, if x ≠ 0 Then ? 567 ? are fuzzy Gmodules on Q and √29 respectively such that ? ? ? . Hence ? is a semi-simple fuzzy G-module over M. 3.3 Proposition Let M be a semi-simple G-module with decomposition M M. . If. µ µ and µ µ are two semi-simple fuzzy G-modules on M, then µ 0 µ is also a semi-simple fuzzy G-module on M, where 0 denotes standard fuzzy intersection.. Proof: The standard fuzzy intersection of fuzzy G- modules is a fuzzy G-module defined by µ 0 µ x min1µ x, µ x2 , x x x . . . x M =min1min1 ? A , ? A , … ? A 2, min 1? A , ? A , … ? A 22 min1min ? A , ? A ", min ? A , ? A " , … min ? A , ? A "2 = min1 ? 0 ? A , ? 0 ? A , … … ? 0 ? A 2 = min1B A , B A , … , B A 2, CD=>= B
(4) ?
(5) 0 ?
(6) EF 5 GHIIJ 4 / K;7HL= ;6 8
(7) =
(8) B
(9) A Hence ? 0 ? is semi simple M 3.4 Proposition Any finite dimensional G- module with dimension at least 2 has a semi-simple fuzzy G-module. Proof: Assume that M is a G-module with dimension n ' 2, and 1 α , α , …, α 2 is a basis for M. Let M span 1α 2. Then M is semi-simple with M M . Define µ on M by µc α c α . . . c α 1, if c c . . . c 0. 1 , if c S 0, c . . . c 0 2 1 , if c S 0, c . . . c 0 3 . . .. 1 , if c S 0 n1. Define µ on M by. Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497).
(10) 461. Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (4), 458-463 (2012). ?
(11) A 1, EG A 0 1 , EG U S 0. E1 Then ?
(12) ?
(13) and hence the result. M 4. SEMI-SIMPLICITY AND OTHER PROPERTIES The semi-simplicity of a fuzzy Gmodule is related to properties like complete reducibility and fuzzy injectivity of fuzzy G-modules. These relationships are derived in the following propositions 4.1 Proposition For any finite dimensional Gmodule M, semi-simple fuzzy G-modules on M are completely reducible. Proof: Let ? be a semi-simple fuzzy Gmodule on M. Assume that 8
(14) 8
(15) and ?
(16) ?
(17) where ?
(18) VF are fuzzy Gmodules on the irreducible G-submodules 8
(19) of M. Let N be any G-submodule of M. Then N is spanned by the elements1 W , W , …W 2 of a basis 1 W , W , …W , W , … , W 2 of M. Let XV be the sub module spanned by the remaining basis vectors. Then 8 X X and for any A A A . . . A 8, we have min , , … , = min min , , … , min , , … . = min {
(20) ?
(21) ,
(22) ?
(23) .} = B B , where B 567 B are fuzzy G modules on N and X . This shows that ? is completely reducible. M. 4.2 Proposition A completely reducible fuzzy Gmodule µ on a finite dimensional G-module M is semi-simple if µ is linear as a function from M to [0, 1] and, µ 0 1 for all fuzzy G-modules µ on G-submodules M of M. Proof: Since ? is completely reducible, M is completely reducible and hence is semisimple. Let 8
(24) be the G-submodule of M spanned by the basis vector W
(25) of a basis 1 W , W , …W 2of M.. Then 8
(26) 8
(27) , and . . . , , . . . , min , , … , , (1) 0, . As µ is completely reducible, for the decomposition 8 X ;G 8 where, X .
(28) 8
(29) , µ is decomposed into ? ? ? where ? 567 ? are fuzzy G-modules on 8 567 X . Hence ?A min 1 ? A , ? VA V2 CD=>= A V A A Y A . Similarly for every decomposition 8
(30) X
(31) ;G 8 we can find fuzzy G-modules ?
(32) 567 ?
(33) so that ?A min 1 ?
(34) A
(35) , ?
(36) VA
(37) V2 CD=>= A
(38) V X
(39) , E 1,2,3, … , 6, (2) Each of these n equations gives the inequalities ?A Z ?
(40) A
(41) , E 1,2,3, … , 6. The inequalities together implies ?A Z min 1 ? A ,. Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497).
(42) Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (4), 458-463 (2012). ? A , … , ? A 2 (3) Equation (2) gives that ?A
(43) ?
(44) A
(45) which together with (1) proves ?A ' min 1 ? A , ? A , … , ? A 2. (4). Inequalities (3) and (4) together give ?A min 1 ? A , ? A , … , ? A 2, thereby making ?
(46) ?
(47) where ?
(48) VF are fuzzy G-modules on 8
(49) VF. This proves that M is semi-simple. M 4.3 Proposition If M is a semi-simple G-module, then M is M injective for every G-module M. M . On assuming µ is ^ injective, we obtain M is M injective and ^m Z µ(ψm) for every ψ HomM , M.. (5). Since M is M injective and M is a Gsubmodule of M, M is M injective, i=1,2,…n and ^m ^ m i 1,2, … , n.. (6). Let ψ is any homomorphism in HomM୧ , M. As M is M injective, every homomorphism from M to M can be extended as a homomorphism from M to M. Let φ is an extension of ψ toHom M , M. Then (5) and (6) gives ^ m Z µφm µψm for every ψ HomM , M. This proves that µ is ^ injective, for every i. On assuming the converse, by proposition 4.3, M is M injective. Let ψ HomM , M and m M. Then m m, and. Proof: Semi-simplicity of M gives M = M . Let N be any G-submodule of M and φ be a homomorphism from N to M. If N = {0}, then φ = 0 and ψ= 0 is an extension of φ from M to M. If N M , then ψc m c m . . . c m φc m is an extension of φ from M to M.. If N = M, k ] 6, then ψc m c m . . . c m φc m c m . . . c m gives the required extension. This proves that every Gsubmodule M is M injective M. Since µ is ^ injective, ^ m Z µψm , i. Hence ^m Z min`µ(ψm )a i 1,2,3, … , n.. 4.4 Proposition. Hence µ is ^ injective. M. If G is a finite group and ^ is a semisimple fuzzy G-module on a M*, then for any fuzzy G- module µ on M, µ is ^ injective if and only if µ is ^ injective for every i Proof: Since ^ is semi-simple fuzzy Gmodule on M , M M, and ^ . ^, where ^ is a fuzzy G-module on. 462. m min m , m , … , m , i 1,2,3, … , n.. m . !. µψmଵ ψmଶ ψm୬ µψm. REFERENCES 1.. 2.. Armand Borel, Semi-simple Groups and Riemannian Symmetric Spaces, Hindustan Book Agency(1998). Charles W Curtis and Irving Reiner, Representation Theory of Finite. Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497).
(50) 463. 3.. 4.. 5.. 6.. Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (4), 458-463 (2012). Groups and Associative Algebras, Wiley Eastern (1962). George J Klir and Bo Yuan, Fuzzy sets and Fuzzy Logic: Theory and Applications, Prentice Hall, India (1995). Hiram Paley and Paul M. Weichsel, A First Cousre in Abstract Algebra, Halt, Renehart, Winstion Inc. (1996) Joachim Lambek, Lectures on Rings and Modules, Blaisdell Publishing Company (1966). Musli C, Representation of Finite Groups, Hindustan Book Agency,. 7.. 8.. 9. 10.. India (1993). Musli C, Introduction to Rings and Modules, Narosa Publishing House, India (1992). Phillipe Gille and Tamas Szamwely, Central simple Algebras and Galois Cohomology, Cambridge University Press (2006). Rosenfield A, Fuzzy Groups, J Math Anal. Appl. (1971). Shery Fernandez, A Study of Fuzzy GModules, Ph.D Thesis, MG University, Kerala (2004).. Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497).
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