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8/11/2019 Sharmapkimf9!12!2012 Licekbre

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International Mathematical Forum, Vol. 7, 2012, no. 11, 523 - 530

On the Direct Product of Intuitionistic

Fuzzy Subgroups

P. K. Sharma

P.G. Department of Mathematics

D.A.V. College, Jalandhar City, Punjab, India

[email protected]

Abstract

In this paper, we first discuss some properties of intuitionistic fuzzy subgroup of agroup. Then we obtained some equivalent characterizations of the direct product

of intuitionistic fuzzy subgroups by means of (,)-Cut sets.

Mathematics Subject Classification:03F55, 20K25, 03E72

Keywords: Intuitionistic Fuzzy Set (IFS) , Intuitionistic Fuzzy Subgroup (IFSG) ,

Intuitionistic Fuzzy Normal Subgroup (IFNSG), Coset , Direct Product ,

Intuitionistic fuzzy Conjugate subgroups, (,)-Cut Set

1. Introduction

After the introduction of the concept of fuzzy set by Zadeh [8] several researches

were conducted on the generalization of the notion of fuzzy set. The idea of

Intuitionistic fuzzy set was given by Atanassov [2]. In this paper we discuss some

results relating to the intuitionistic fuzzy subgroup of a group by means of their

(, ) cut sets. Then we obtained some equivalent characterizations of the direct

product of intuitionistic fuzzy subgroups.


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524 P. K. Sharma

2.

Preliminaries

Definition (2.1)[4] An IFS A = { < x, A(x), A(x) > : x G} of a group G is said

to be intuitionistic fuzzy subgroupof G ( In short IFSG) of G if

(i) A(xy) Min {A(x) , A(y) } (ii) A(x-1

) = A(x)

(iii) A(xy) Max { A(x) , A(y) } (iv) A(x-1

) = A(x) , for all x , y G

Proposition (2.2)[4] An IFS A = { < x, A(x), A(x) > : x G} of a group G is

intuitionistic fuzzy subgroup of G of G if and only if

A(xy-1) Min{

A(x),

A(y)}and

A(xy-1) Max{

A(x),

A(y)} holds x , y G

Definition (2.3)[5] Let A be Intuitionistic fuzzy set of a universe set X . Then

( , )-cut of A is a crisp subset C, (A) of the IFS A is given by

C, (A)= { x X : A(x) , A(x) } , where ,[0 ,1] with + 1 .

Theorem(2.4)[5]: If A is IFS of a group G. Then A is IFSG of G if and only if

C, (A) is a subgroup of group G for all , [0 , 1] with + 1

Definition (2.5)[5 ] Let G be a group and A be IFSG of group G . Let x G be a

fixed element. Then for every element g G , we define

(i) (xA)(g) = (xA

(g), xA

(g)) ,where xA

(g) = A(x

-1g) and

xA(g) =

A(x

-1g).

Then xA is called intuitionistic fuzzy left coset of G determined by A and x

(ii) Ax(g) = (Ax(g) , Ax(g)), where Ax(g) = A(gx -1

) and Ax(g) = A(gx -1

)

Then Ax is called the intuitionistic fuzzy right coset of G determined by A and x .

Definition (2.6)[5] An IFSG A of a group G is IFNSG of G if and only if

xA = Ax for all x G

Theorem (2.7)[5] Let A be intuitionistic fuzzy subgroup of a group G and x be

any fixed element of G . Then for all , [0, 1] with + 1 , we have

(i) x . C, (A) = C, (xA) (ii) C, (A).x = C, (Ax) .

Proposition(2.8)If A is IFS of a group G. Then A is IFSNG of G if and only if

C, (A) is a normal subgroup of group G , for all , [0,1] with + 1

Proof.Now A is IFNSG of G xA = Ax for all x G

C, (xA) = C, (Ax) , for all , [0, 1] with + 1

x . C, (A) = C, (A).x , for all x G

C, (A) is normal subgroup of G

Definition (2.9)Let A and B be two IFSG of group G. Then A and B are said to

be intuitionistic fuzzy conjugate subgroups of G if for some g G

A(x) = B(g-1

xg) and A(x) = A(g-1

xg) , for every x G .


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Direct product of intuitionistic fuzzy subgroups 525

Definition(2.10)[1] Let A be an intuitionistic fuzzy subset in a set S , the strongest

intuitionistic fuzzy relation on S , that is an intuitionistic fuzzy relation on A is V

given by V (x ,y) = Min { A(x) , A(y)} and V(x ,y) = Max {A(x) , A(y)} ,

for all x , y S .

3. Direct product of Intuitionistic Fuzzy Sets and their

properties

Definition (3.1)Let A ={< x,

A(x),

A(x) >: x

X} and B = {< y,

B(y) ,

B(y) >:

y Y} be any two IFSs of X and Y respectively . Then the Cartesian product of

A and B is denoted by A B and is defined as ;

A B = { : x X and yY}

Where AB(x, y) = Min{A(x), B(y)} and AB(x, y) = Max{A(x),B(y)} .

Proposition (3.2)If A and B be two IFS of X and Y respectively. Then

C, ( A B ) = C, ( A) C, ( B) , for all , [ 0,1] with 0 + 1.

Proof. Let ( x, y ) C, ( A B ) be any element

AB(x, y) and AB(x, y)

Min {A(x), B(y)} and Max { A(x) , B(y) }

A(x) , B(y) and A(x) , B(y)

A(x) , A(x) and B(y) , B(y)

x C, ( A) and y C, ( B)

( x, y ) C, ( A) C, ( B)


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526 P. K. Sharma

Hence C, ( A B ) = C, ( A) C, ( B)

Theorem (3.3)Let A and B be IFSG of group G1and G2 respectively. Then

A B is also IFSG of group G1G2.

Proof. Since A and B be IFSG of group G1and G2 respectively.

Therefore C, ( A) and C, ( B) are subgroup of group G1and G2 respectively

for all , [ 0,1] with 0 + 1. [ by theorem (2.4)]

C, ( A) C, ( B) is subgroup of group G1G2

C, ( A B ) is subgroup of group G1G2

A B is IFSG of group G1G2 [ by theorem (2.4) ]

Theorem (3.4) Let A and B be IFNSG of group G1and G2 respectively. Then

A B is also IFNSG of group G1G2.

Proof. SinceA and B be IFNSG of group G1and G2 respectively. Then by

Proposition (2.8) C, ( A) , C, ( B) are normal subgroup of G1, G2 respectively

C, ( A) C, ( B) is normal subgroup of group G1G2

C, ( A B ) is normal subgroup of group G1G2

A B is IFNSG of group G1G2 [ by Proposition (2.8) ]

Remark(3.5) If A and B are IFS of group G1and G2 respectively. If A B is

also IFSG of group G1G2, then it is not necessarily that both A and B should be

IFSG of G1G2.


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Example(3.6):Let G1= { e1, a }, where a2= e1and let G2= { e2, x , y , xy },

where x2= y

2= e2 and xy = yx . Then

G1G2= { (e1, e2) , (e1, x ), (e1, y) , (e1, xy), (a , e2), (a , x), (a , y) , (a , xy) }

Let A = {< e1, 0.7, 0.2 > , < a , 0.6, 0.3 >} and B = {< e 2, 0.9, 0.1 > , < x ,1, 0 >,

< y , 0.8, 0.2 >, < xy , 0.7, 0.2 > } be IFS of G1and G2 respectively. Then

A B = { < (e1, e2) , 0.7, 0.2 > , < (e1, x ), 0.7 , 0.2 > , < (e1, y), 0.7, 0.2 > ,

< (e1, xy), 0.7, 0.2 > , < (a , e2) ,0.6, 0.3 > , < (a , x), 0.6, 0.3 > ,< (a , y), 0.6, 0.3>,

, < (a , xy) , 0.6 , 0.3 > }

Here A B is also IFSG of group G1G2, where as A is IFSG of G1but B is not

IFSG of G2 as C0.8, 0.2( B) = { x , y } is not subgroup of G2.

Proposition(3.7) :Let A and B be IFSs of the groups G1and G2, respectively.

Suppose that e1and e2 are the identity element of G1and G2, respectively. If A x B

is an IFSG of G1xG2, then at least one of the following two statements must holds.

(i) B(e2) A(x) and B(e2) A(x), for all x in G1 ,

(ii) A(e1) B(y) and A(e1) B(y), for all y in G2.

Proof :Let A x B is an IFSG subgroup of G1xG2 .

If possible, let the statements (i) and (ii) does not holds.

Then we can find x in G1and y in G2such that A(x) >B(e2), A(x) A(e1), B(y

) Min{A(e1), B(e2

)} = AxB (e1, e2

).

and, AxB ( x, y ) = Max{A(x), B(y)}


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528 P. K. Sharma

Proof.Let x , y G1be any element . Then (x, e2) and (y, e2

) are in G1x G2.

Since A(x) B(e2) and A(x) B( e2

) holds for all x in G1, we get,

A( xy-1

) = Min{A(xy-1

) , B(e2e2)} = AxB ( (xy

-1), (e2e2

) )

= AxB [ (x, e2)(y

-1, e2

) ]

Min{ AxB(x , e2) , AxB(y-1

, e2)}

= Min{Min{A(x), B(e2)}, Min{A(y-1

), B(e2)}}

= Min{ A(x) , A(y-1

) }

Min{ A(x) , A(y)}.

Thus A( xy-1

) Min{ A(x) , A(y)} , also

A( xy-1

) = Max{ A(xy-1

), B(e2e2) } = AxB ( (xy

-1), (e2e2

) )

= AxB [ (x, e2)(y

-1, e2

) ]

Max{ AxB(x, e2) , AxB(y

-1, e2

) }

= Max{Max{A(x),B(e2)}, Max{A(y-1

),B(e2)}}

= Max{A(x), A(y-1

) }

Max{A(x), A(y)}.

Therefore, A( xy-1

) Max{A(x), A(y).} Hence A is an IFSG of G1.

Corollary (3.9)Let A and B be IFS of the groups G1 and G2respectively suchthat B(y ) B(e1

) and B(y

) B( e1

) holds for all yG2, e1being the identity

element of G1.If A x B is an IFSG of G1x G2, then B is IFSG of group G2.

From Theorem (3.4) and Proposition (3.7) and (3.8) , we have the following:

Corollary(3.10)Let A and B be IFS of the groups G1and G2respectively. If Ax B

is an IFSG of G1x G2, then either A is IFSG of G1 or B is IFSG of group G2 .

Theorem (3.11)Let A , C be IFSGs of the groups G1and B , D be IFSG of the

group G2 respectively such that A and C are intuitionistic fuzzy conjugate

subgroups of G1 and B , D are intuitionistic fuzzy conjugate subgroups of G2 .

Then the IFSG A x B of G1x G2 is conjugate to the IFSG C x D of G1x G2.

Proof. SinceA and C are intuitionistic fuzzy conjugate subgroups of G1

s g1G1such that A(x) = C(g1-1

xg1) and A(x) = C(g1-1

xg1), x G1

Also B , D are intuitionistic fuzzy conjugate subgroups of G2sg2G2s.t.

B(y) = D(g2-1

yg2) and B(y) = D(g2-1

yg2) , y G2

Now AxB (x , y) = Min{ A( x ) , B ( y)} = Min { C(g1-1

xg1) , D(g2-1

yg2)}

= CxD ((g1-1

xg1) , (g2-1

yg2) ) = CxD ((g1-1

, g2-1

)( x , y) (g1, g2))

Similarily, AxB (x , y) = Max{A( x ) , B ( y)}= Max{C(g1-1

xg1) , D(g2-1

yg2)}

= CxD ((g1-1

xg1), (g2-1

yg2))= CxD ((g1-1

, g2-1

)( x , y) (g1, g2))


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Hence the IFSG A x B of G1x G2 is conjugate to the IFSG C x D of G1x G2.

Theorem(3.12) Let A be an IFS of a group of G and V be the strongest

intuitionistic fuzzy relation on G . Then A is an IFSG of G if and only if V is an

IFSG of GG.

Proof. Firstly,let A be IFSG of group G.

Let x = (x1, x2) , y = (y1, y2) be any two element of GG , we have

V(xy-1

) = V[(x1, x2) (y1, y2)-1

] = V[(x1, x2) (y1-1

, y2-1

)] = V[(x1y1-1

, x2y2-1

)]

Min{A(x1y1-1

) , A(x2y2-1

)}

= Min{{ Min{A(x1) , A(y1)} , Min{A(x2) , A(y2)}}

= Min{{ Min{A(x1) , A(x2)} , Min{A(y1) , A(y2)}}

= Min { V(x1, x2) , V(y1, y2) }

= Min { V(x) , V(y) }

Similarly , we have

V(xy-1

) = V[(x1, x2) (y1, y2)-1

] = V[(x1, x2) (y1-1

, y2-1

)] = V[(x1y1-1

, x2y2-1

)]

Max{A(x1y1-1

) , A(x2y2-1

)}

= Max{{ Max{A(x1) , A(y1)} , Max{A(x2) , A(y2)}}

= Max{{ Max{A(x1) , A(x2)} , Max{A(y1) , A(y2)}}

= Max { V(x1, x2) , V(y1, y2) }

= Max { V(x) ,

V(y) }

Thus V is IFSG of GG

References

[1] N. Anitha and K. Arjunan , Notes on intuitionistic fuzzy ideals of Hemiring,

Applied Mathematical Science, Vol. 5 , 2011 , no. 68, 3393-3402

[2] K.T Atanassov , Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986) ,no.1, 87-96

[3] G. Deschrijver, E. E. Kerre, On the Cartesian product of intuitionistic fuzzy

sets, Notes on Intuitionistic Fuzzy Sets, 7 (2001) , pp. 14-22 .

[4] N Palaniappan, S Naganathan and K Arjunan, A study on Intuitionistic

L-Fuzzy Subgroups, Applied Mathematical Sciences, vol. 3 , 2009, no. 53 ,

2619-2624.


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530 P. K. Sharma

[5] P.K.Sharma, (,)-Cut of Intuitionistic fuzzy groups, International

Mathematics Forum , Vol.6 , 2011 , no. 53 , 2605-2614.

[6] Asok K. Ray , On the product of fuzzy subgroups, Fuzzy Sets and Systems

105(1999), 181-183.

[ 7] Shaoquan Sun , Some properties of the direct product of fuzzy algebras,

Fuzzy System and Mathematics, vol. 18 , 2004 , issue z1.

[8] L.A Zadeh , Fuzzy sets, Information and Control 8 , (1965), 338-353.

Received: September 2011

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