On Fuzzy Representations of Fuzzy G-Modules

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On Fuzzy Representations of Fuzzy G-Modules

THAMPY ABRAHAM

1

and SOURIAR SEBASTIAN

2

1

Department of Mathematics,

St. Peter's College, Kolenchery, Kerala - 682 311, India E-mail: thampyabraham2003@yahoo.co.in

2

Rev. Dr. A.O. Konnully Memorial Research Centre, Department of Mathematics,

St. Albert's College, Ernakulam, Kerala - 682 018, India.

ABSTRACTS

In this paper we study fuzzy representations of a fuzzy G- module  of a G-module M/N onto a fuzzy G-module  of a general linear space GL(V). This transformation is done on the basis of G-module representation theory. We also prove a fundamental theorem of G- module fuzzy representations.

Keywords: G-module, fuzzy G-module, fuzzy homomorphism, fuzzy

isomorphism, fuzzy representation.

INTRODUCTION

The notion of fuzzy subsets was introduced by L.A. Zadeh

7

in 1965. In 1971, Rosenfeld

4

defined the fuzzy subgroups an d ga ve some of its p rope rtie s.

Mathematicians like Abu Osman, Katsaras and Liu and Gu Wenxiang have studied fuzzy versions of various algebraic structures. G. Frobenius developed the theory of group representations at the end

of the 19th century. The theory of G- modules originated in the 20

th

century.

Representation theory was developed on

the basis of embedding a group G in to a

linear group GL (V). As a continuation of

the authors' work in

6to 11

, in this paper, we

prove a theorem related to the fuzzy

homomorphism an d a fund amen tal

the orem for G-module fu zzy

representations.

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1.1 Definition [1]

Let G be a finite group. A vector space M over a field K (a subfield of C) is called a G-module if for every g  G and m M, a product (called the right action of G on M) m.g M which satisfies the following axioms.

1. m.1

G

= m  m M (1

G

being the identify of G)

2. m. (g. h) = (m.g). h m  M, g, hG 3. (k

1

m

1

+ k

2

m

2

). G = k

1

(m

1

. g) + k

2

(m

2

. g), k

1

, k

2

 K, m

1

, m

2

M &

g  G.

In a similar manner the left action of G on M can be defined.

1.2. Definition [1]

Let M and M* be G-modules. A map ping :MM* is a G-module homomorphism if

1. (k

1

m

1

+ k

2

m

2

) = k

1

 (m

1

) + k

2

 (m

2

) 2. (gm) = g  (m),  k

1

, k

2

 K, m, m

1

,

m

2

 M & g  G.

1.3. Definition [1]

Let M be a G-module. A subspace N of M is a G-submodule if N is also a G- module under the action of G.

1.4. Definition [3]

Let G be a finite group and M be a G-module over K, which is a subfield of C.

Then a fuzzy G-module on M is a fuzzy subset µ of M such that.

i) µ (ax + by) > µ (x) µ (y), a, bK

& x, y M

ii) µ (gm) > µ (m),  g  G, m  M It may be recalled that by a fuzzy subset of M we mean a function µ:M [0,1].

1.5 Definition [1]

Let G be a group and M be a vector spa ce over a field K. A line ar representation of G with representation space M is a homomorphism of G into GL(M), where GL (M) is the group of units in Hom

k

(M, M) called the general linear group GL (M).

1.6 Definition [3]

Let G and G

1

be groups. Let µ be a fuzzy group on G and  be a fuzzy group on G

1

. Let f be a group homomorphism of G onto G1. Then f is called a weak fuzzy homomorphism of µ into  if f (µ) . The homomorphism f is a fuzzy homomorphism of µ onto  f (µ ) =. We say that µ is fuzzy homomorphic to  and we write µ .

Let f : GG

1

be an isomorphism.

The n f is ca lled a we ak fuzzy isomorphism f (µ)  and f is a fuzzy isomorphism if f (µ) = .

1.7 Definition [5]

Let G be a group and M be vector spa ce over K a nd T: GG

1

be a representation of G in M. Let µ be a fuzzy group on G and  be a fuzzy group on the range of T. Then the representation T is a fuzzy representation if T is a fuzzy homomorphism of µ onto .

1.8 Definition [2, 3]

Let f be a function defined from X to Y. The image of a fuzzy subset µ on X under 'f' is the fuzzy subset f (µ ) of Y defined by f ( µ) (y) =  {µ (x) | x  f

–1

(y)},

y  Y.

The pre-image of the fuzzy subset A

A

A

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 on Y under 'f' is the fuzzy subset on Y under ‘f‘ is the fuzzy subset f

–1

() of X defined by f

–1

() (x) =  { f (x) },  x  X.

1.9 Definition [3]

Let µ be a fuzzy subgroup of G and N be a normal subgroup of G.  on G/N is defined as  [x] =  {µ(z) |  [x] },  x  G, where [x] denotes the coset xN. Then  is a fuzzy subgroup of G/N.

1.10. A fundamental theorem of fuzzy representations (see [11])

Let G be a group and M be a vector space over a field K. If T : G GL (M) is a fuzzy representation of G then  : G/NGL (M) is a fuzzy representation of G/N where N is a normal subgroup of G

1.11. Definition

Let M and M* be G-modules Let T be a G- module homomorphism of M into M*. Let µ be a fuzzy G-module of M and  be a fuzzy G-module of M*. Then T is called a G-module fuzzy homomorphism of µ onto  if T (µ) =, written as µ  . If T (µ)

, T is called a weak fuzzy G-module homomorphism.

If T:MM* is a G-modu le isomorphism and T (µ) =  then T is a fuzzy G-module isomorphism of µ onto . If T (µ)  then T is called a weak fuzzy G- module isomorphism of µ into .

1.12 Example

Let G = { 1, -1 }, M = R

4

over R and M* = R.

Let { e

1

, e

2

, e

3

, e

4

} be the standard basis for M. Define µ : M [0,1] by µ (k

1

e

1

+ k

2

e

2

+ k

3

e

3

+ k

4

e

4

)

= 1 if k

i

= 0, i

= 1/2 if k

1

0, k

2

= k

3

= k

4

= 0

= 1/3 if k

2

 0, k

3

= k

4

= 0

= 1/4 if k

3

 0, k

4

= 0

= 1/5 if k

4

 0

Then µ is a fuzzy G-module of M. Define

on R by (0) = 1,  (x) = ½ if x  0. It can be verified that  is a fuzzy G-module on R. Define f : R

4

 R by

f (x) = 

4i=1

x

i

, x = (x

1

x

2

, x

3

, x

4

) For a, b  R and x, y M,

f(ax + by) = f { a (x

1

x

2

, x

3

, x

4

) + b ( y

1

, y

2

,y

3

, y

4

)}

= f {(a x

1

+ by

1

, ax

2

+ by

2

, ax

3

+ by

3

, ax

4

+ by

4

)}

= 

4i=1

(ax

i

+ by

i

) = a

4i

x

i

+ b

4i

y

i

= a f (x) + b f (y) For g  G & x  M,

f(gx) = f {g (x

1

x

2

, x

3

, x

4

) } = 

4i = 1

g x

i

= g 

4i

x

i

= g { f (x) }

f is a G -module homomorphism and f (µ) (0) =  {µ(x) | x  f

–1

(0) } = 1

For w  0, f (µ) (w) =  {µ (x) | x  f

–1

(w) }

= 1/2

 f (µ) = .  f is a G-module fuzzy homomorphism of µ onto v.

1.13 Definition

Let G be a finite group and M and V be G-modules over K.Then T: M GL (V) is called a G-module representation if T is a G-module homomorphism of M into GL (V).

1.14 Definition

Let G be a finite group and let M

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and V be G-modules over K. Let T be a G- module homomorphism of M into GL (V).

Let µ be a fuzzy G-module of M and  be a fuzzy G-module of T (M). Then T is called a G-module fuzzy representation if T is a G-module fuzzy homomorphism of µ onto

.

1.15 Example

Let G = { 1, -1}. Let M = C and V be any G-module over R.

Define T : MGL (V) by T(m) = Tm where Tm : VV is defined by Tm () = mv for m

 M and   V. We can show that T is a G- module homomorphism.

For k

1

, k

2

, R and m

1

, m

2

M, T (k

1

m

1

+ k

2

m

2

) = T

k1m1

+ k

2m2

T

k1m1 +k2m2

() = (k

1

m

1

+ k

2

m

2

) () = k

1

(m

1

) + k

2

(m

2

)

= k

1

Tm

1

() + k

2

Tm

2

()

= (k

1

Tm

1

+ k

2

Tm

2

) ()

 T k

1

m

1

+ k

2

m

2

= k

1

T

m1

+ k

2

T

m2

 T (k

1

m

1

+ k

2

m

2

) = k

1

T (m

1

) + k

2

T (m

2

) For g G and m M, T (gm) = Tgm Tgm () = (gm)  = g (m) = g Tm () = (gTm) ()

 Tgm = g Tm . ie, T (gm) = g T (m). T is a G-module homomorphism

Define µ on M by (x + iy)

= 1 if x = y = 0

= 1/2 if x  0, y = 0

= 1/3 if y  0

It can be verified that µ is a fuzzy G-module of M

Define  on T (m) by  (Tm) = (Tx + iy)

= 1 if x = y = 0

= 1/2 if x  0, y = 0

= 1/3 if y  0

For k

1

, k

2

 R and T

m1

, T

m2

 T (M),  ( k

1

T

m1

+ k

2

T

m2

) = (Tk

1

m

1

+ k

2m2

)

Let m

1

= x

1

+ iy

1

and m

2

= x

2

+ iy

2

.Then k

1

m

1

+ k

2

m

2

= (k

1

x

1

+ k

2

x

2

) + i (k

1

y

1

+ k

2

y

2

)

If x

1

= x

2

= y

1

= y

2

= 0, (T

k1m1 + k2m2

) = 1 If x

1

 0 or x

2

 0, y

1

= 0, y

2

= 0,  (T

k1m1 + k2m2

) = 1/2

If y

1

 0 or y

2

 0,  (T

k1m1 + k2m2

) = 1/3 Then v is a fuzzy G-module on T(M) T (µ) (T

x+iy

) =  {µ (x + iy) | x + iy 

T

–1

(T

x + iy

)

T (µ) (T

m

) =  {  (z) z  T

-1

(T

m

) }

=  { µ (z) | T (z) = Tm}

=  { µ (m) |T (m) = Tm}

=  { µ (x + iy) | T (x+iy) = T

x + iy

}

= 1 when x = 0, y = 0

= 1/2 when x  0, y = 0

= 1/3 when y  0

= (Tm)

 T (µ)=.  T is a G-module fuzzy representation of µ onto 

1.16 Proposition [1]

If M is a G-module and N is a G- submodule of M, then M/N is a G-module 1.17 Definition

Let G be a finite group. Let M be a G-module over K and N be a G-submodule of M. Let µ be a fuzzy G-module on M.

Then the fuzzy subset  on M/N, defined by [x] =  {µ (z) | z  [x], x  M}, where [x] = x + N, is a fuzzy G-module on M/N.

The fuzzy G-module  is called the

quotient fuzzy G-module or factor fuzzy

G-module of the fuzzy G-module µ of M

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relative to the G-submodule N.

Proof

For a, b k and [x], [y]  M/N,

{a [x] + b [y] =  { [ax + Gy]}

> [x] ^  [y]

> [x]

{g [x]} =  [gx] >  [x]

 is a fuzzy G-module on M/N ? 1.18 Theorem

Let G be a finite group and M & M*

be G-modules. Let µ be a fuzzy G-module of M and be a fuzzy G-module of T (M).

Let T be a G-module fuzzy homomorphism of µ onto . Then  : M/NM* is a G- module fuzzy homomorphism of  onto  where  is a fuzzy G-module on M/N and N is a G-submodule of M.

Proof

Given that T is a G-module fuzzy homomorphism of µ onto . T (µ) = . Now we have to prove that : M/N M* is a G- module fuzzy homomorphism of  onto

. i.e., to show that  is a G-module homomorphism of M/N into M* and  ()

= . Define  : M/NM* by [x] = T (x), x

M where [x] = x + N. Then  is a G- module homomorphism if  { a [x] + b [y] = a  [x] + b  [y] and  { g [x]} = g  [x],

 [x], [y]  M/N , a, b  K & g  G.

{a [x] + b [y] } =  { [ax + by] }

= T (ax + by)

= a T (x) + b T (y)

= a  [x] + b  [y]

{g [x] } =  [gx] = T (gx)=g T (x) = g  [x]

is a G-module homomorphism.

Now to show that  () = 

() () =  { [x] | [x] 

–1

(y),y  (M/N)}

=  { [x] | [x] = y, y  T (M) }

=  {µ { (z) |z  [x] T (x) = yT (M)}

=  {µ(z) z  [x], T (x) = y T (M), x  M}

= T (µ ) (y)

() = T (µ) = 

 is a G-module fuzzy homomorphism of  onto 

1.19 Theorem

Let G be a finite group, M and V be G-modules over K. Let T : M GL (V) be a G-module fuzzy representation of µ onto

 where µ is a fuzzy G-module of M and  is a fuzzy G-module of T (M). Then : M/

NGL (V) is a G-module fu zzy representation of  onto  where  is a fuzzy G-module of M/N.

Proof

Given that T is G-module fuzzy representation of µ onto  where µ is a fuzzy G-module of M and  is a fuzzy G- module of T (M). We have to show that

 : M/NGL (V) is a G-module fuzzy representation of  onto  where  is a fuzzy G-module of M/N, N being a G-submodule of M.We know that GL(V) is a G-module.

Hence by theorem 3.15, is a G-module homomorphism of µ onto . Now to show that  () = .

() (Tx) =  { [x] | [x]

–1

(Tx)}

=  { [x] |  [x] = Tx}

=  { {µ (z)|z  [x]},  [x] = T [x], x  M}

=  {µ (z) | z  [x],  [x] = Tx, x  M}

= T (µ ) (Tx)

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() = T (µ ) = 

 is a G-mo dule f uzzy representation of  onto 

1.20 Example

Let G = {1, -1}, M = C and M* = R.

Then M and M* are G-modules over R.

Define f : MM* by f (x + iy) = x + y. Then f is a G-module homomorphism. Define µ on M by

µ (x + iy) = 1 if x = y = 0

= 1/2 if x  0, y = 0 = 1/3 if y  0

Define  on T (M) by  (0) = 1 and  (z) =

½ when z  0. Then µ and  are fuzzy G- modules on M and T (M) respectively.

Consider any G-submodule R of M and let M/R = {u + R | u  M}. Then M/R is a G- module. Define : M/RR by {u + R} = f(u),uM.Then  is a G-modu le homomorphism of M/R into R.

Let  : M/R[0,1] be defined by  [u]

= {u + R }, u  M

= v { µ (z) | z  [u], u  M}

() (0) =  { [u] | [u]

–1

(0) } = 1 For r  0, ( ) ( r) =  { [u] | [u] 

–1

( r) } = 1/2.

 () = .  is a G-module fuzzy representation of  onto .

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