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J. Comp. & Math. Sci. Vol.3 (4), 458-463 (2012)

Semi-simple Fuzzy G- Modules SOURIAR SEBASTIAN* and PRATHISH ABRAHAM Department of Mathematics, St. Albert’s College, Ernakulam, Kochi-682018, Kerala, INDIA. Department of Mathematics, Union Christian College, Aluva, Ernakulam-683102, Kerala, INDIA. (Received on: July 5, 2012) ABSTRACT The concept of fuzzy G- modules and its properties including reducibility and injectivity are already defined. In this paper we extend this idea to define semi-simplicity of fuzzy G- modules. The existence of a semi-simple fuzzy G-module for every finite dimensional G-module is proved and the relationships of semisimplicity with other properties of fuzzy G- modules are also discussed. Keywords: Fuzzy-G-modules, Direct sum of Fuzzy G-modules, Complete reducibility, Fuzzy Injectivity, Semi-simple fuzzy G- modules.

1. INTRODUCTION The introduction of fuzzy sets by Zadeh led way to the fuzzification of Algebraic structures. Fuzzy groups and groupoids are defined by Rosen ield . Fuzzification of G-modules, their complete reducibility and injectivity are discussed by Shery10. In this paper we define semisimplicity of fuzzy G-Modules using direct sum of fuzzy G-modules. We prove the existence of semi-simple fuzzy G-modules on every finite dimensional G-module. We

also obtain the relationship between complete reducibility and semi-simplicity of fuzzy G- modules and relate fuzzy injectivity with fuzzy semi-simplicity. 2. PRELIMINARIES Given a finite group G, a vector space M over a field K is said to be a Gmodule if for every g ∈ G and m ∈ M there exists a product ‘gm’ called action of G on M satisfying (i) 1 m m, m M (ii) gh m g hm , g, h G, m M

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Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (4), 458-463 (2012)

(iii) g k 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 Ďˆ 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 Âľ on M (i.e. a function Âľ: M 0,1") such that (i) Âľ ax by ' min(Âľ x , Âľ y ) , a, b K and x, y M and (ii) Âľ gm ' Âľ m , 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 Âľ is a fuzzy G-module on M , ∀i , then Âľ defined by Âľ x = min (Âľ x , Âľ x , ‌ , Âľ x ), x

x x . . . x M and x M is fuzzy G-module on M called the direct sum of fuzzy G-modules ¾ , i 1,2, ‌ , n. A fuzzy G-module ¾ 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 ¾ on M , i 1,2 , such that ¾ ¾ ¾ . If ¾ and - are fuzzy G-modules on Gmodules M and M* then ¾ is - injective if i)

M is M* injective and

ii) - (m) ≤ Âľ(Ďˆ(m)), for every Ďˆ ∈ Hom(M*,M). The standard fuzzy compliment [3] of a fuzzy set Âľ on X is defined as Âľ. x 1 / Âľ x . The standard fuzzy intersection [3] and standard fuzzy union [3] of two fuzzy sets Âľ and Âľ on X are defined by Âľ 0 Âľ x min1Âľ x , Âľ x 2 and Âľ 3 Âľ x max1Âľ x , Âľ x 2. 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 Âľ on M is said to be semi-simple if M is semi-simple and Âľ Âľ where Âľ is a fuzzy G-module on M , i.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

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

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= .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 Âľ Âľ and Âľ Âľ are two semi-simple fuzzy G-modules on M, then Âľ 0 Âľ 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 ¾ 0 ¾ x min1¾ x , ¾ x 2 , 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 ? 0 ? EF 5 GHIIJ 4 / K;7HL= ;6 8

= B 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 ι , ι , ‌, ι 2 is a basis for M. Let M span 1ι 2. Then M is semi-simple with M M . Define ¾ on M by ¾ c ι c ι . . . c ι 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 n 1

Define Âľ on M by

Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

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Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (4), 458-463 (2012)

? A 1, EG A 0 1 , EG U S 0 E 1 Then ? ? 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 8

and ? ? where ? VF are fuzzy Gmodules on the irreducible G-submodules 8 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 { ? , ? .} = 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 ¾ on a finite dimensional G-module M is semi-simple if ¾ is linear as a function from M to [0, 1] and, ¾ 0 1 for all fuzzy G-modules ¾ on G-submodules M of M. Proof: Since ? is completely reducible, M is completely reducible and hence is semisimple. Let 8 be the G-submodule of M spanned by the basis vector W of a basis 1 W , W , ‌W 2of M.. Then 8 8 , and . . . , , . . . , min , , ‌ , , (1) 0,

As Âľ is completely reducible, for the decomposition 8 X ;G 8 where, X

8 , ¾ is decomposed into ? ? ? where ? 567 ? are fuzzy G-modules on 8 567 X . Hence ? A min 1 ? A , ? V A V 2 CD=>= A V A A Y A . Similarly for every decomposition 8 X ;G 8 we can find fuzzy G-modules ? 567 ? so that ? A min 1 ? A , ? V A V 2 CD=>= A V X , E 1,2,3, ‌ , 6, (2) Each of these n equations gives the inequalities ? A Z ? A , 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)

Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (4), 458-463 (2012)

? A , ‌ , ? A 2 (3) Equation (2) gives that ? A ? A 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 ? ?

where ? VF are fuzzy G-modules on 8 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 Âľ is ^ injective, we obtain M is M injective and ^ m Z Âľ(Ďˆ m ) for every Ďˆ Hom M , 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 Ďˆ is any homomorphism in Hom Mŕ­§ , 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 Ďˆ toHom M , M . Then (5) and (6) gives ^ m Z Âľ φ m Âľ Ďˆ m for every Ďˆ Hom M , M . This proves that Âľ is ^ injective, for every i. On assuming the converse, by proposition 4.3, M is M injective. Let Ďˆ Hom M , 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 Ďˆ= 0 is an extension of φ from M to M. If N M , then Ďˆ c m c m . . . c m φ c m is an extension of φ from M to M.. If N = M , k ] 6, then Ďˆ c m c m . . . c m φ c m c m . . . c m gives the required extension. This proves that every Gsubmodule M is M injective M

Since Âľ is ^ injective, ^ m Z Âľ Ďˆ m , i. Hence ^ m Z min`Âľ(Ďˆ m )a i 1,2,3, ‌ , n.

4.4 Proposition

Hence Âľ 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 Âľ on M, Âľ is ^ injective if and only if Âľ is ^ injective for every i Proof: Since ^ is semi-simple fuzzy Gmodule on M , M M , and ^ ^ , where ^ is a fuzzy G-module on

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m min m , m , ‌ , m , i 1,2,3, ‌ , n.

m

!

Âľ Ďˆ mଵ Ďˆ mଶ Ďˆ mŕ­Ź Âľ Ďˆ m

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Armand Borel, Semi-simple Groups and Riemannian Symmetric Spaces, Hindustan Book Agency(1998). Charles W Curtis and Irving Reiner, Representation Theory of Finite

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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,

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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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