Anti Q-Fuzzy Right R -Subgroup of Near-Rings with Respect to S-Norms

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1 International Journal of Fuzzy Mathematics and Systems. ISSN Volume 2, Number 2 (2012), pp Research India Publications Anti Q-Fuzzy Right R -Subgroup of Near-Rings with Respect to S-Norms 1 G.Subbiah and 2 R.Balakrishnan 1 Associate Professor, Department of Mathematics, Sri K.G.S Arts College, Srivaikuntam Tamilnadu, India 2 Associate Professor, PG & Research Department of Mathematics V.O.C College Thoothukudi Tamilnadu India subbiahkgs@gmail.com, balakri@yahoo.com Abstract In this paper, we introduce the notion of anti Q- fuzzification of right R- subgroups in a near-ring and investigate some related properties. Characterization of anti Q- fuzzy right R-subgroups and onto homomorphic image of anti Q- fuzzy right R- subgroup with the inf property with respect to a s-norm are given. Index Terms: Q- fuzzy set, Imaginable, inf property, anti Q- fuzzy right R- subgroups, s-norm. Mathematics Subject Classification: 03F055, 03E72. Introduction The theory of fuzzy sets which was introduced by Zadeh [6] is applied to many mathematical branches. Abou-zoid [1], introduced the notion of a fuzzy sub near-ring and studied fuzzy ideals of near-ring. This concept discussed by many researchers among cho, Davvaz, Dudek, Jun, Kim [2],[3],[4]. In [5], considered the intuitionistic fuzzification of a right (resp left ) R- subgroup in a near-ring. Also cho.at.al in [4] the notion of normal intuitionistic fuzzy R- subgroup in a near-ring is introduced and related properties are investigated. The notion of intuitionistic Q- fuzzy semi primality in a semi group is given by Kim [3]. A.Solairaju and R.Nagarajan introduced the concept of Structures of Q- fuzzy groups [7]. In this paper, We introduce the notion of anti Q- fuzzification of right R- subgroups in a near ring and investigate some related properties. Characterization of anti Q- fuzzy right- subgroups with respect to S-norm are given.

2 172 G. Subbiah and R. Balakrishnan Preliminaries Definition 2.1: A non empty set with two binary operations + and. is called a near-ring if it satisfies the following axioms ( R,+ ) is a group. ( R,. ) is a semi group. x. (y+z) = x.y + x. z for all x,y,z ε R. Precisely speaking it is a left near-ring. Because it satisfies the left distributive law As R subgroup of a near- ring R is a subset H of R such that ( H, + ) is a subgroup of ( R, + ). RH H HR H. If H satisfies (i) and (ii) then it is called left R- subgroup of R and if H satisfies (i) and (iii) then it is called a right R- subgroup of R. A map f : R S is called homomorphism if f(x+y) = f(x) + f (y) for all x,y in R. Definition 2.2: Let Q and G be a set and a group respectively. A mapping μ: G Q [0,1] is called a Q fuzzy set. Definition 2.3: Let R be a near ring. A fuzzy set μ in R is called anti fuzzy sub near ring in R if (i) μ(x+y) Max { μ(x), μ(y) } (ii) μ(xy) Max { μ(x), μ(y) } for all x,y in R. Definition 2.4: A Q -fuzzy set μ is called a anti Q-fuzzy right R- subgroup of R over Q if μ satisfies (AQFS1) μ(x+y,q ) S { μ(x,q ), μ(y,q) } (AQFS2) μ(xr, q ) μ(x q). (AQFS3) A(0,q ) = 1 Definition 2.5: By a s- norm S, we mean a function S: [0,1] [0,1] [0,1] satisfying the following conditions (S1) S(x,0) = x (S2) S(x,y) S(x,z) if y z (S3) S(x,y) = S(y,x) (S4) S(x, S(y,z) ) = S(S(x,y),z), for all x,y,z ε [0,1]. Definition 2.6: Define S n (x 1,x 2,, x n ) = S (x i, S n-1 (x 1,x 2, x i-1, x i+1,x n )) for all 1 i n, n 2, S 1 = S. Also define S (x 1,x 2, ) = lim S n (x 1,x 2,,x n ) as n. Definition 2.7: By the union of Q-fuzzy subsets A 1 and A 2 in a set X with respect to s-norm S we mean the Q-fuzzy subset A = A 1 U A 2 in the set X such that for any x ε X A(x,q) = (A 1 UA 2 ) (x,q) = S (A 1 (x,q), A 2 (x,q)). By the union of a collection of Q-fuzzy subsets { A 1,A 2, } in a set X with respect to a s- norm S we mean the Q-fuzzy subset UA i such that for any x ε A, (UA i )(x,q) = S (A 1 (x,q), A 2 (x,q),}.

3 Anti Q-Fuzzy Right R -Subgroup of Near-Rings 173 Definition 2.8: By the direct product of fuzzy sets { A 1,A 2,} with respect to s- norm S we mean the Q-fuzzy subset A = A i such that A(x 1,x 2,.x n } q = ( A i ) {(x 1,x 2,x n ) q } = S n (A 1 (x 1,, q), A 2 (x 2, q). A n (x n, q)} Proposition 2.1: For a S-norm, then the following statement holds S(x,y) max{x,y}, for all x,y ε [0,1] Properties of Q- Fuzzy Left R- Subgroups Proposition 3.1: Let S be a s- norm. Then every imaginable anti Q- fuzzy right R- subgroup μ of a near ring R is anti Q- fuzzy right R-subgroup of R. Proof: Assume μ is imaginable Q- fuzzy right R- subgroup of R, then we have μ (x+y, q) S { μ(x,q), μ(y,q) }and μ (xr, q) μ (x,q) for all x,y in R. Since μ is imaginable, we have max { μ(x,q), μ(y,q) } = S { max { μ(x,q), μ(y,q), max { μ(x,q), μ(y,q) } } S ( μ(x,q), μ(y,q) ) max { μ(x,q), μ(y,q) } And so S( μ(x,q), μ(y,q) ) = max { μ(x,q), μ(y,q) }. It follows that μ(x+y, q ) S( μ(x,q), μ(y,q) )= max { μ(x,q), μ(y,q) } for all x,y ε R. Hence μ is anti Q- fuzzy right R- subgroup of R Proposition 3.2: If μ is anti Q- fuzzy right R- subgroups of a near ring R and Ө is an endomorphism of R, then μ[ө] is anti Q- fuzzy right R- subgroup of R. Prof: For any x,y ε R, we have μ[ө] ( x+y, q ) = μ ( Ө(x+y, q ) ) = μ ( Ө(x,q ), Ө(y,q) ) S { μ( Ө(x,q)), μ( Ө(y,q) ) } = S { μ[ө] (x,q), μ[ө] (y,q ) } μ[ө] (xr, q ) = μ( Ө (xr, q ) μ ( Ө(x,q) ) μ [Ө] (x,q) Hence μ[ө] is anti Q- fuzzy right R- subgroup of R Proposition 3.3: An onto homomorphism s of anti Q- fuzzy right R- subgroup of near ring R is anti Q- fuzzy right R- subgroup. Proof: Let f : R R 1 be an onto homomorphism of near rings and let λ be anti Q- fuzzy right R- subgroup of R 1 and μ be the pre image of λ under f, then we have μ(x+y, q) = λ ( f(x+y, q )) = λ ( f(x,q), f(y,q) ) S ( λ (f(x,q)), λ(f(y,q)) )

4 174 G. Subbiah and R. Balakrishnan μ(xr,q) S (μ(x,q), μ(y,q) ) = λ(f (xr,q)) λ (f(x,q) ) = μ(x,q). Proposition 3.4: An onto homomorphic image of anti Q- fuzzy right R- subgroup with the inf property is anti Q- fuzzy right R- subgroup. Prof: Let f: R R 1 be an onto homomorphism of near rings and let μ be a inf property of anti Q-fuzzy right R- subgroups of R Let x 1, y 1 ε R 1, and x 0 ε f -1 (x 1 ), y 0 ε f -1 (y 1 ) be such that μ(x 0, q ) = inf μ(h,q), μ(y 0,q) = inf μ(h,q) (h,q)εf -1 (x 1 ) (h,q)εf -1 (y 1 ) Respectively, then we can deduce that μ f ( x 1 +y 1, q ) = inf μ(z,q) (z,q) ε f -1 (x 1 +y 1,q) max { μ(x 0,q), μ(y 0,q) = max inf μ(h,q), inf μ(h,q) (h,q)εf -1 (x 1,q) (h,q)εf -1 (y 1,q) = max { μ f (x 1,q), μ f (y 1,q) } μ f (xr,q) = inf μ(z,q) (z,q)εf -1 (x 1 r 1, q) μ(y 0, q) = inf μ(h,q) (h,q)εf -1 (y 1,q) = μ f (y 1,q). Hence μ f is anti Q- fuzzy right R- subgroup of R 1 Proposition 3.5: Let S be a continuous s-norm and let f be a homomorphism on a near ring R. If μ is anti Q- fuzzy right R- subgroup of R, then μ f is anti Q- fuzzy right R- subgroup of f(s) Proof: Let A 1 = f -1 (y 1,q), A 2 = f -1 (y 2,q) and A 12 = f -1 (y 1 +y 2, q) where y 1,y 2 ε f(s), qε Q consider the set A 1 -+A 2 = { x ε S / (x,q) = (a 1,q) + (a 2,q) } for some (a 1,q) εa 1 and (a 2,q) ε A 2 If (x,q) ε A 1 +A 2, then (x,q) = (x 1,q) + (x 2,q) for some (x 1,q) ε A 1 and (x 2,q) ε A 2 so that we have f (x,q) = f(x 1,q) - f(x 2,q) = y 1 - y 2 (ie) (x,q) ε f -1 ((y 1,q) + (y 2,q)) = f -1 (y 1 +y 2, q) = A 12. Thus A 1 +A 2 c A 12.

5 Anti Q-Fuzzy Right R -Subgroup of Near-Rings 175 It follows that μ f (y 1 +y 2, q) = inf { μ(x,q)/ (x,q) ε f -1 ((y 1,q)+(y 2,q))} = inf{ μ(x,q) / (x,q) ε A 12 } inf { μ(x,q)/ (x,q) ε A 1 -A 2 } inf { μ((x 1,q)- (x 2,q) ) / (x 1,q) ε A 1 and (x 2,q) ε A 2 } inf { T(μ(x 1,q), μ(x 2,q))/ (x 1,q) ε A 1 and (x 2,q) ε A 2 } Since S is continuous. For every ε > 0, we see that if inf { μ(x 1,q) / (x 1,q) ε A 1 } + (x 1 *, q) } δ and inf { μ(x 2,q) / (x 2,q) ε A 2 } + (x 2 *,q) } δ S{inf{μ(x 1,q) / (x 1,q) ε A 1 }, inf { μ(x 2, q) / (x 2,q) ε A 2 } + S ((x 1 *,q), (x 2 *,q) ε Choose (a 1,q) ε A 1 and (a 2,q) ε A 2 such that inf { μ(x 1,q) / (x 1.q) ε A 1 } + μ(a 1,q) δ and inf { μ(x 2,q) / (x 2,q) ε A 2 } + μ(a 2,q) δ. Then we have S{inf{ μ(x 1,q) / (x 1,q) ε A 1 }, inf { μ(x 2,q) / (x 2,q) ε A 2 } + S(μ(a 1,q), μ(a 2,q) ε consequently, we have μ f (y 1 +y 2, q ) inf { S(μ(x 1,q), μ(x 2,q)) / (x 1,q) ε A 1,(x 2,q) ε A 2 } S (inf{μ(x 1,q) / (x 1,q) ε A 1 }, inf{μ(x 2,q) / (x 2,q)εA 2 } S (μ f (y 1,q), μ f (y 2,q) } Similarly we can show μ f (xr,q) μ f (y,q). Hence μ f is anti Q- fuzzy right R- subgroup of f(r). Proposition 3.6: If R is a near ring with identity and a s-norm S for all x ε [0,1] satisfies the condition S(x,x)=x. Then condition (AQFS2) in Definition (2.4) for any r ε R is equivalent to the condition A(xr,q) = A(x,q). (AQFS2 1 ) Proof: Let condition (AQFS1 ) and (AQFS2 ) be fulfilled and 1 be the identity element in the near ring R. Then A(x,q) = A(xr + (1- xr),q ) S (A(xr,q), A(1-xr,q)) S (A(xr, S(A(x,q), A(-xr,q))) S(A(x,q), S(A(x,q), A(-xr))). Taking into consideration conditions (S2) and (S5) for the s-norm S and again applying ( AQFS2), we obtain S(A(x,q), S(A(,q), A(- xr,q))) = S(S(A(x,q), A(x,q),A(-xr,q)) = S (A(x,q), A(-xr,q)) S (A(x,q), A(xr,q)). Thus we have A(x,q) = S(A(x,q), A(x,q)) S (A(x,q), A(xr,q)).From here, using condition (S4), we conclude that A(x,q) A(xr,q) (1) From (1) and condition (AQFS2) we obtain (AQFS2 1 ). Proposition 3.7: Let A : B [0,1] be the characteristic function of a subset B is contained in R and R be an R-subgroup. Then A is anti Q-fuzzy right R-subgroup of R with respect to a s-norm S if and only if B is a subgroup of R.

6 176 G. Subbiah and R. Balakrishnan Proof: Let A be anti Q-fuzzy R- sub group of R with respect to S. Then, according to (AQFS2), A(xr,q) A(x,q) = 1. Hence xr ε B. Finally, according to condition (AQFS3), A(0,q) = 1. Therefore, 0ε B. Thus, B is a subgroup of R. Conversely, Let B be a sub group of R. Then for any x,y ε R, A(x+y, q) S(A(x,q), A(y,q)}. Indeed, for any x,y ε B, A(x+y,q) = 1 1 = S(1,1) = S (A(x,q), A(y,q)} For any x ε B, and y is not in B, S(A(x, q), A(y, q)) = S(1.0) = 0 A(x+y, q) For any x is not in B, and y ε B, S(A(x,q), A(y,q)) = S(0,1) = 0 A(x+y, q) Finally, for any x,y does not belong to B, S(A(x, q), A(y, q)) = S(0,0) = 0 A(x+y, q) Further for all x ε R, and rε R, we have A(xr, q) A(x, q). Indeed, for all x ε B we have xr ε B, hence A(xr, q) = 1 A(x, q), and for all x does not belong to B we have A(x,q) = 0 A(x, q). Finally, since 0 ε B, we have A(0,q) = 1. Therefore, A is anti Q-fuzzy right R- subgroup of R with respect to S. Proposition 3.8: The Union of any collection of anti Q-fuzzy R- sub group of R is anti Q-fuzzy R-subgroup of R. Proof: For all x,y ε R, and any r ε R, we have UA i (x+y) q = S (A 1 (x+y) q, A 2 (x+y) q, ) S (S (A 1 (x,q), A 1 (y,q)), S (A 2 (x,q), A 2 (y,q)), ) = S (S (A 1 (x,q), A 2 (x,q),, ), S (A 1 (y,q), A 2 (y,q), )) = S ((UA i )(x,q), (UA i )(y,q)); (UA i )(xr,q) = S A 1 (xr,q),a 2 (xr,q),, ) S (A 1 (x,q), A 2 (x,q),) = (UA i ) (x,q); (UA i ) (0,q) = S (A 1 (0,q), A 2 (0,q),) = S (1,1,) = 1. Proposition 3.9: Let { R 1,R 2,,R n } be a collection of R-subgroups and R = A i be its direct product.let {A 1,A 2,,A n } be anti Q-fuzzy right subgroups of R subgroups { R 1,R 2,, R n } with respect to a s-norm S. Then A = A i is anti Q-fuzzy right R- subgroup of R with respect to the s-norm S. Proof: Let x,y ε R, x = (x 1,x 2, x n ), and y = (y 1,y 2, y n ). Also let r ε R. Then A(x+y, q) = A (x 1 +y 1, x 2 +y 2,, x n +y n ) q = S n (A 1 (x 1 +y 1 ) q, A 2 (x 2 +y 2 ) q,, A n (x n +y n ) q ) S n ( S (A 1 (x 1, q),a 1 (y 1, q)),s(a 2 (x 2, q), A 2 (y 2, q)),s(a n (x n, q), A n (y n, q))) = S (S n (A 1 (x 1, q), A 2 (x 2, q), A n (x n, q)), S n (A 1 (y 1, q), A 2 (y 2, q), A(y n, q))) = S (A(x,q), A(y,q)), A(xr,q) = A(x 1r,x 2r,,x nr ) q = S n (A 1 (x 1r ), A 2 (x 2r ),, A n (x nr ) q ) S n (A 1 (x 1, q) A 2 (x 2, q), A n (x n, q)), A(0, q) = A (0 1,0 2,, 0 n ) q = S n ((A 1 (0 1, q), A 2 (0 2,,q), A n (0 n, q)) = S n (1,1,,1) = 1. Therefore, A is anti Q-fuzzy R- subgroup of R with

7 Anti Q-Fuzzy Right R -Subgroup of Near-Rings 177 respect to S. Conclusion Y.U. Cho, Y.B.Jun, investigated the concept On intuitionistic fuzzy R- subgroup of near rings. Osman kazanci, Sultanyamark and Serifeyilmaz introduced the intutionistic Q- fuzzy R-subgroups of near rings. In this paper we investigate the notion of anti Q- fuzzy right R- subgroup of near ring w.r.t S-norm and characterization of them. References [1] S. Abou-Zoid, On Fuzzy sub near rings and ideals, Fuzzy sets. Syst. 44 (1991), [2] Y.U. Cho, Y.B.Jun, On Intuitionistic fuzzy R- subgroup of near rings, J. Appl. Math. And computing, 18 (1-2) (2005), [3] K.H.Kim, Y.B. Jun, On Fuzzy R- subgroups of near rings, J. fuzzy math 8 (3) (2000) [4] K.H.Kim, Y.B.Jun, Normal Fuzzy R- subgroups of near rings, J. fuzzy sets. Syst.121(2001) [5] Osman Kazanci, Sultan Yamark and Serife Yimaz On intuitionistic Q- fuzzy R- subgroups of near rings, International mathematical forum, 2, 2007, 59, ( ) [6] A.Solairaju and R.Nagarajan, Q- fuzzy left R- subgroup of near rings w.r.t T- norms, Antarctica Journal of Mathematics,5, no.2(2008), [7] A.Solairaju and R.Nagarajan, A New structure and construction of Q- fuzzy groups, Advances in Fuzzy Mathematics, 4, No.1(2009), [8] L.A.Zadeh, Fuzzy set, inform. control 8 (1965)

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