**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

^{6}

^{to 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.

**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.

**called a G-module if for every g G and**

### 1. m.1

_{G}

### = m m M (1

_{G}

### being the identify of G)

### 2. m. (g. h) = (m.g). h m M, g, hG 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 :MM* is a G-module** **homomorphism if**

**map ping :MM* 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.

**N of M is a G-submodule if N is also a 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.

**Then a fuzzy G-module on M is a fuzzy**

### i) µ (ax + by) > µ (x) µ (y), a, bK

### & 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

**spa ce over a field K. A line ar**

**representation of G with representation**

_{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 .

**G onto G1. Then f is called a weak fuzzy**

**homomorphism of µ into if f (µ) . The**

### Let f : GG

^{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) = .**

**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: GG

^{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 .**

**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

**The pre-image of the fuzzy subset**

### A

### A

### 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/NGL (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)

**G-module fuzzy homomorphism**

**, T is called a weak fuzzy G-module** **homomorphism.**

**, T is called a weak fuzzy G-module**

**homomorphism.**

### If T:MM* 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 .**

**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) =

^{4}

_{i=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}

### )}

### =

^{4}

_{i=1}

### (ax

_{i}

### + by

_{i}

### ) = a

^{4}

_{i}

### x

_{i}

### + b

^{4}

_{i}

### 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}

### ) } =

^{4}

_{i = 1}

### g x

_{i}

### = g

^{4}

_{i}

### 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).

**(V) is called a G-module representation**

**1.14** **Definition**

### Let G be a finite group and let M

### 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

**a G-module fuzzy representation if T is**

### .

**1.15** **Example**

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

### Define T : MGL (V) by T(m) = Tm where Tm : VV 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**

**quotient fuzzy G-module or factor fuzzy**

**G-module of the fuzzy G-module µ of M**

**G-module of the fuzzy G-module µ of M**

### 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/NM* 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/NM* 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) = yT (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/

### NGL (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/NGL (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)

### () = 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 : MM* 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/RR by {u + R} = f(u),uM.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}