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 = { xG, / ( 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 = { xG, 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 = {xG, 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 xG 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() = { aG / (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{gg-1(x , q) , gg-1(y , q) }, for all x , y in G and q Q. (ii) gg-1(x , q) = (g-1xg , q) = ((g-1xg)-1 , q) = (g-1x-1g , q) = gg-1(x-1 , q) , for all x , y in G and q Q. -1 Hence gg 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 gg-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 qQ. (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 gg is a subgroup of G. Now conditions are equivalent for each x , y in G. -1 gg-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 = gg-1(y,q). (v) xx = 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 gg-1 is also a Q-fuzzy subgroup of G Let and be two Q-fuzzy subgroups of G for all g G and qQ. (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) (gg-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. 35 (1971),512-517. Normal subgroups, Q-fuzzy Normalizer and Qfuzzy subgroups under homomorphism. [14] Samit kumar majumder and Sujit kumar sardar, On some properties of anti fuzzy subgroups, Interestingly, It has been observed that Q-fuzzy J.Mech.cont.and Math.Sci.,3(2),2008,337-343. concept adds an another dimension to the defined [15] Sherwood, H : Product of fuzzy subgroups, fuzzy normal subgroups. This concept can further Fuzzy sets and systems 11(1983) ,79-89. be extended for new results. [16] A.Solairaju and R.Nagarajan “ Q- fuzzy left Rsubgroups of near rings w.r.t T- norms”, References Antarctica journal of mathematics.vol 5,2008. [1] Biswas. R, Fuzzy subgroups and anti-fuzzy [17] A.Solairaju and R.Nagarajan , A New subgroups, Fuzzy sets and systems, 35(1990), Structure and Construction of Q-Fuzzy 121-124. Groups, Advances in Fuzzy mathematics, [2] Choudhury.F.P. and Chakraborty.A.B. and Vol 4 (1) (2009) pp.23-29. Khare.S.S., A note on fuzzy subgroups and [18] Vasntha Kandasamy.W.B., Smarandache Fuzzy homomorphism .Journal of Fuzzy Algebra, American Research mathematical analysis and applications 131, Press,2003. 537 -553 (1988 ). [19] Yunjie Zhand and Dong Yu , On fuzzy [3] Das. P.S, Fuzzy groups and level subgroups, Abelian subgroups, J.Math.Anal. Appl, 84(1981) 264-269. [20] Zadeh. L.A.Fuzzy sets, Information and [4] Dixit.V.N., Rajesh Kumar, Naseem Ajmal., Control,8(1965),338-353.
(i) (xy , q) = sup min { (x1 y1 , q), (x2 y2 , q)} x1y1= x x2y2 = y
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