Ijcot v3i11p115

International Journal of Computer & Organization Trends –Volume 3 Issue 11 – Dec 2013

On Q-Fuzzy Normal Subgroups T.Priya1, T.Ramachandran2, K.T.Nagalakshmi3 Department of Mathematics, PSNA College of Engineering and Technology, Dindigul-624 622, Tamilnadu , India.

2 Department of Mathematics, M.V.Muthiah Government Arts College for Women, Dindigul-624001,Tamilnadu , India.

3 Department of Mathematics, K L N College of Information and Technology, pottapalayam- 630611, Sivagangai District,Tamilnadu , India.

1

Abstract In this paper, we introduce the concept of Q-fuzzy normal group and Q-fuzzy left ( right) cosets and discussed some of its properties.

Keywords Fuzzy group, Q-fuzzy group, Q-fuzzy normal group, Q-fuzzy normalizer, Q-fuzzy left ( right) cosets.

AMS Subject Classification (2000): 20N25, 03E72, 03F055 , 06F35, 03G25.

2.3 Definition [6,18] Let G be a group. A fuzzy subgroup  of G is said to be normal if for all x, y  G ,  (xyx-1) =  (y) or  (xy) =  (yx).

2.4 Definition[3] Let  be a fuzzy subgroup of a group G. For any t [0,1], we define the level subset of  is the set, t = { xG, /  ( x ) ≥ t }.

2.5 Definition [17] Let Q and G be any two sets. A mapping  : G x Q →[0, 1] is called a Q-fuzzy set in G.

1.Introduction The concept of fuzzy sets was initiated by Zadeh[ 20]. Then it has become a 2.6 Definition [10] vigorous area of research in engineering, medical A Q-fuzzy set  is called a Q-fuzzy science, social science, graph theory etc. Rosenfeld subgroup of a group G, if for x, y  G, q  Q, [13] gave the idea of fuzzy subgroups.. A.Solairaju and R.Nagarajan [16,17 ] introduce and define a (i)  (xy, q) ≥ min {  (x,q), (y,q)} new algebraic structure of Q-fuzzy groups. In this paper we define a new algebraic structure of Q(ii)  (x-1, q ) =  (x, q) fuzzy normal subgroups and Q-fuzzy left (right) cosets and study some of their properties.

3.Q-Fuzzy Normal Subgroups

2. Preliminaries In this section we site the fundamental definitions that will be used in the sequel.

2.1 Definition [20] Let X be any non empty set. A fuzzy subset  of X is a function  : X → [0,1].

2.2 Definition [1,2] Let G be a group. A fuzzy subset  of G is called a fuzzy subgroup if for all x, y  G, (i)  (xy) ≥ min { (x), (y)} (ii)  (x-1 ) =  (x)

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3.1 Definition Let G be a group. A Q- fuzzy subgroup  of G is said to be normal if for all x,y  G and q  Q, (xyx-1,q) = (y, q) or  (xy, q) ≥ ( yx, q) .

3.2 Definition Let  be a Q-fuzzy subgroup of a group G. For any t [0,1], we define the level subset of  is the set, t = { xG, q  Q /  ( x , q ) ≥ t }.

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International Journal of Computer & Organization Trends –Volume 3 Issue 11 – Dec 2013 Theorem 3.1 Theorem 3.2 Let G be a group and  be a Q-fuzzy subset Let G be a group and  be a Q-fuzzy subset of G. Then  is a Q – fuzzy subgroup of G iff the of G. Then  is a Q- fuzzy normal subgroup of G level subsets, t , t[0,1], are subgroup of G. iff the level subsets t, t[0,1], are normal subgroup of G. Proof Let  be a Q-fuzzy subgroup of G and the level Proof subset t = {xG, q  Q /  (x,q) ≥ t, t[0,1] }.

Let  be a Q- fuzzy normal subgroup of G Let x , y t . Then  (x , q) ≥ t &  (y , q) ≥ t. and the level subsets t , t[0,1], is a subgroup of G. Now  (xy-1, q) ≥ min {  (x , q) ,  (y-1 , q) } = min {  (x , q) ,  (y , q) }

Let xG and a t , then  ( a, q) ≥ t.

Now,  (xax-1,q) =  ( a, q) ≥ t, since  is a Q-fuzzy normal subgroup of G.

≥ min { t , t }

That is ,  (xax-1,q) ≥ t.

Therefore,  (xy-1,q ) ≥ t

Therefore, xax-1  t.

This implies xy-1  t .

Hence t is a normal subgroup of G. Theorem 3.3

Thus t is a subgroup of G.

Conversely, let us assume that t be a subgroup of Then G.

ii.  is a Q-fuzzy normal  N() = G.

Let x , y  t . Then  (x , q) ≥ t and  (y , q) ≥ t.

Also,

 (xy-1, q )

iii.  is a Q-fuzzy normal subgroup of the group N().

≥ t, since xy-1  t = min {t , t} = min {  (x , q) ,  (y , q) }

That is,

 (xy-1, q ) ≥ min {  (x , q) ,  (y , q) }

Hence  is a Q-fuzzy subgroup of G. 3.3 Definition Let G be a group and  be a Q-fuzzy subgroup of G. Let N() = { aG /  (axa-1, q) =  (x ,q) , for all x G , q Q }. Then N() is called the Q-fuzzy Normalizer of .

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Let  be a Q-fuzzy subgroup of a group G. i. N() is a subgroup of G.

Proof i. Let a , b  N() then  (axa-1 , q) =  (x , q) , for all x  G.  (bxb-1 , q) =  (x , q) , for all x  G. Now  (abx(ab)-1 , q) =  (abxb-1a-1 , q) =  (bxb-1 , q ) =  (x , q) Thus we get,  (abx(ab)-1 , q ) =  (x , q).  ab N() Therefore, N() is a subgroup of G.

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International Journal of Computer & Organization Trends –Volume 3 Issue 11 – Dec 2013 ii. Clearly N()  G ,  is a Q-fuzzy normal subgroup of G. Let a G, then  (axa-1 , q) =  (x , q).

Then a  N()  G  N().

Hence N() = G. Conversely, let N() = G. Clearly  (axa-1 , q) =  (x , q) , for all x  G and a  G. Hence  is a Q – fuzzy normal subgroup of G. iii.

From (2) ,  is a Q-fuzzy normal subgroup of a group N().

=  ((g-1xg)(g-1 yg) , q) ≥ min{  (g-1xg, q) ,  (g-1 yg , q)} ≥ min{gg-1(x , q) , gg-1(y , q) }, for all x , y in G and q Q. (ii) gg-1(x , q) = (g-1xg , q) =  ((g-1xg)-1 , q) =  (g-1x-1g , q) = gg-1(x-1 , q) , for all x , y in G and q Q. -1 Hence gg is a Q-fuzzy subgroup of G.

3.4 Definition Let  be a Q-fuzzy subset of G. Let xf : G x Q → G x Q [ fx : G x Q → G x Q ] be a function defined by xf(a , q) = (xa , q) [fx(a , q) = Theorem 3.6 : If  is a Q-fuzzy normal subgroup of (ax , q) ]. A Q-fuzzy left ( right) coset x (x ) is G, then gg-1 is also a Q-fuzzy normal subgroup of defined to be xf() (fx()).

It is easily seen that (x)(y , q) =  (x-1 y , q) and G , for all g  G and qQ. (x)(y , q) = (yx-1 , q) , for every (y , q) in G x Q. Proof : Let  be a Q-fuzzy normal subgroup of G. Theorem 3.4 : -1 Let  be a Q-fuzzy subset of G. Then the following then gg is a subgroup of G. Now conditions are equivalent for each x , y in G. -1 gg-1(xyx-1,q) = ( g-1(xy x-1) g, q) (i)  ( xyx , q ) ≥  (y , q) = ( xy x-1, q) (ii)  ( xyx-1 , q ) =  (y , q) = ( y, q) (iii)  (xy , q) =  (yx ,q) = ( g y g-1, q) (iv) x = x -1 = gg-1(y,q). (v) xx =  Theorem 3.7 : Proof : Straight forward.

Let  and  be two Q-fuzzy subgroups of G. Then  is a Q-fuzzy subgroup Theorem 3.5 : of G. If  is a Q-fuzzy subgroup of G, Proof : then gg-1 is also a Q-fuzzy subgroup of G Let  and  be two Q-fuzzy subgroups of G for all g  G and qQ. (i) () (xy-1, q) = min ( (xy-1, q) , (xy-1, q)) ≥ min { min{(x , q) ,  (y ,q)}, min{(x , q) ,  (y ,q)}}

Proof : Let  be a Q-fuzzy subgroup of G. Then (i) (gg-1) (xy , q) =  (g-1(xy)g , q)

≥ min{ min{(x , q) , (x ,q)}, min{(y, q),(y ,q)}} = min { () (x , q) , () (y , q)}

= (g-1(xgg-1 y)g , q)

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International Journal of Computer & Organization Trends –Volume 3 Issue 11 – Dec 2013 Thus () (xy-1, q) ≥ min{()(x, q),()(y,q)}

(ii) () (x , q) = {  (x , q) ,  (x , q)} = {  (x-1 , q) ,  (x-1 , q)} = { () (x-1 , q )}.

Hence  is a Q-fuzzy subgroup of G.

homomorphism. (i) If  is a Q-fuzzy normal subgroup of H , then f-1() is a Q-fuzzy normal subgroup of G. (ii) If f is an epimorphism and  is a Q-fuzzy normal subgroup of G, then f() is a Q-fuzzy normal subgroup of H.

Proof : (i) Let f: G x Q → H x Q is a group Remark : If i , i is a Q-fuzzy subgroup of G, Q-homomorphism and let  be a Q-fuzzy then   i is a Q-fuzzy subgroup of G. i Normal subgroup of H. Theorem 3.8 : Now for all x , y  G , we have The intersection of any two Q-fuzzy f-1()(xyx-1, q) =  ( f (xyx-1 , q)) normal subgroups of G is also a Q-fuzzy normal subgroup of G. =  ( f(x)f(y)f(x)-1 , q) Proof : Let  and  be two Q-fuzzy normal subgroups of G. According to theorem 3.7,  is a Q-fuzzy subgroup of G.

=  ( f(y) , q) = f-1()(y , q) Hence f-1() is a Q-fuzzy normal subgroup of G. (ii) Let  be a Q-fuzzy normal subgroup of G. Then f() is a Q- fuzzy subgroup of H.

Now for all x , y in G , we have () (xyx-1 , q) = min ( (xyx-1 , q) , (xyx-1 , q) ) = min (  (y , q) , (y , q) )

Now, for all u , v in H, we have f()(uvu-1, q) = sup  (y , q) = sup  (xyx-1, q) f(y)=uvu-1

f(x)=u; f(y) = v

= sup  (y ,q) = f()(v , q)

=( ) ( y , q)

f(y) = v

Hence  is a Q-fuzzy normal subgroup of G.

(since f is an epimorphism) Hence f() is a Q-fuzzy normal subgroup of H.

Remark : If i , i are Q-fuzzy normal subgroup of G, then   i is a Q-fuzzy normal subgroup of G. Definition 3.6 : i Let  and  be two Q-fuzzy subsets of G. The product of  and  is defined to be the Definition 3.5 : The mapping f: G x Q → H x Q is said to be Q-fuzzy subset  of G is given by (x , q) = sup min ( (y , q) , (z , q) ) , x  G. a group Q-homomorphism if yz = x

(i) f: G → H is a group homomorphism (iii)f(xy , q) = (f(x)f(y) , q) , for all x , y  G and q  Q. Theorem 3.9 : Let f: G x Q → H x Q is a group Q-

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Theorem 3.10: If  &  are Q-fuzzy normal subgroups of G , then  is a Q-fuzzy normal subgroup of G .

Proof : Let  &  be two Q-fuzzy normal subgroups of G.

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International Journal of Computer & Organization Trends –Volume 3 Issue 11 – Dec 2013 Level subgroups and union of fuzzy subgroups, Fuzzy Sets and Systems, 37, (1990),359-371.

[5] Malik D.S. and Mordeson J.N. Fuzzy subgroups of abelian groups, Chinese Journal of Math., vol.19 (2) , 1991. ≥ sup min{min{(x1,q),(y1,q)}, [6] Mashour A.S, Ghanim M.H. and Sidky F.I. , x1y1 = x min{(x2,q),(y2,q) } } Normal Fuzzy Subgroups,Univ.u Novom Sadu x2y2 = y Zb.Rad.Prirod.-Mat.Fak.Ser.Mat. 20(2) ,1990, ≥ min{supmin{(x1,q),(y1 , q)},sup min{(x2 , q) , 53-59. x1y1 = x (y2 , q)}} [7] Mehmet sait EROGLU, The homomorphic x2 y2 = y image of a fuzzy subgroup is always a Fuzzy subgroup, Fuzzy sets and Systems, 33(1989)  (xy , q) ≥ min {  (x , q) ,  (y , q) } 255 – 256.

[8] Mohamed Asaad, Groups and Fuzzy subgroups, Fuzzy sets and systems 39(1991) 323-328. (ii)  (x-1 , q) = sup min { (z-1 , q) , (y-1, q) } (yz)-1= x-1 [9] Mustafa Akgul, Some properties of fuzzy groups, Journal of mathematical analysis and applications 133, 93-100 (1988). = sup min { (z , q) , (y, q) } x = yz [10] Muthuraj.R., Sithar Selvam.P.M., Muthuraman.M.S., Anti Q-fuzzy group and its = sup min { (y , q) , (z , q) } lower Level subgroups, International journal x = yz of Computer Applications 3(3),2010,16-20.

[11] Prabir Bhattacharya, Fuzzy Subgroups: Some =  (x , q).

Characterizations,J.Math. Anal. Appl.128 (1987),241 – 252.

[12] Rajesh kumar, Homomorphism and fuzzy Hence  is a normal Q-fuzzy subgroup of G . (fuzzy normal) subgroups, Fuzzy sets and Systems, 44(191) 165 – 168.

4. Conclusion [13] Rosenfeld.A., fuzzy groups, J. math. Anal.

In this article we have discussed Q-fuzzy Appl.

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(i)  (xy , q) = sup min { (x1 y1 , q), (x2 y2 , q)} x1y1= x x2y2 = y

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