On Intuitionistic Q-Fuzzy R-Subgroups of Near-Rings

International Mathematical Forum, 2, 2007, no. 59, 2899-2910 On Intuitionistic Q-Fuzzy R-Subgroups of Near-Rings Osman Kazancı, Sultan Yamak Serife Yılmaz Department of Mathematics, Faculty of Arts Sciences Karadeniz Technical University 61080, Trabzon, Turkey okazanci@ktu.edu.tr, syamak@ktu.edu.tr, syilmaz@ktu.edu.tr Abstract In this paper, we introduce the notion of intuitionistic Q-fuzzification of R-subgroups (subnear-rings) in a near-ring investigate some related properties. Characterization of intuitionistic Q-fuzzy R-subgroups (subnear-rings) is given. Mathematics Subject Classification: 03F055, 03E72, 16Y30 Keywords: Q-fuzzy set, intuitionistic Q-fuzzy set, intuitionistic Q-fuzzy R-subgroup (subnear-ring) 1 Introduction The theory of fuzzy sets which was introduced by Zadeh [18] is applied to many mathematical branches. Abou-Zaid [1], introduced the notion of a fuzzy subnear-ring studied fuzzy ideals of a near-ring. This concept discussed further by many researchers, among whom Cho, Davvaz, Dudek, Jun, Kim [6,7-8,9,10-11,12-14]. The notion of intuitionistic fuzzy sets was introduced by Atanassov [2-4] as a generalization of the notion of fuzzy sets. In [5], Biswas applied the concept of intuitionistic fuzzy set to the theory of groups studied intuitionistic fuzzy subgroups of a group. The notion of an intuitionistic fuzzy ideal of a near-ring is given by Jun in [11]. In [16], Yon et al. considered the intuitionistic fuzzification of a right(resp.left) R-subgroup of a near-ring. Also Cho at al., in [6] the notion of a normal intuitionistic fuzzy R-subgroup in a near-ring is introduced related properties are investigated. Recently, Davvaz at al. [8] considered the intuitionistic fuzzification of the concept of the H ϑ -submodules in a H ϑ - module Dudex at al. [9] considered the intuitionistic fuzzification

2900 O. Kazancı, S. Yamak S. Yılmaz of the concept of sub-hyperquasigroups in a hyperquasigroup. They investigated some properties of such hyperquasigroups. The notion of intuitionistic Q-fuzzy semiprimality in a semigroup is given by Kim[14]. Also Roh at al. [17] considered the intuitionistic Q-fuzzification of BCK/BCI-algebras. In this paper, we introduce the notion of intuitionistic Q-fuzzification of R-subgroups (subnear-rings) in a near-ring investigate some related properties. Characterization of intuitionistic Q-fuzzy R-subgroups (subnear-rings) is given. 2 Definition Preliminaries Definition 2.1 (15). A non empty-set R with two binary operations + is called a near-ring if it satisfies the following axioms. (i) (R, +) is a group, (ii) (R, ) is a semigroup, (iii) x (y + z) =x y + x z for all x, y, z R. Precisely speaking it is a left near-ring. Because it satisfies the left distributive law. We will use the word near-ring to mean left near-ring. We denote xy instead of x y. Note that x0 =0 x(-y) =-xy for all x, y R, but in general, 0x 0 for some x R. An R-subgroup of a near-ring R is a subset H of R such that (i) (H, +) is a subgroup of (R, +), (ii) RH H, (iii) HR H. If H satisfies (i) (ii), then it is called a left R-subgroup of R, if H satisfies (i) (iii) then it is called a right R-subgroup of R. A map f from a near-ring R into a near-ring S is called a homomorphism if f(x + y)= f(x)+f(y) f(xy)=f(x)f(y) for all x, y R. Let R be a near-ring. A fuzzy set μ in R is called a fuzzy subnear-ring in R if [1] (i) μ(x y) μ(x) μ(y), (ii) μ(xy) μ(x) μ(y) for all x, y R. A fuzzy set μ in R is called a fuzzy R-subgroup of R if (i) μ(x y) μ(x) μ(y),

On intuitionistic Q-fuzzy R-subgroups of near-ring 2901 (ii) μ(rx) μ(x), (iii) μ(xr) μ(x) for all x, y, r R. Let X be a non-empty set. A mapping μ : X [0, 1] is called a fuzzy set in X. The complement of a fuzzy set μ in X, denoted by μ c, is the fuzzy set in X given by μ c (x) =1 μ(x) for all x X. In what follows, let Q R denote a set a near-ring, respectively, unless otherwise specified. A mapping μ : R Q [0, 1] is called a Q-fuzzy set in R. For any Q-fuzzy set μ in R any t [0, 1] we define two sets U(μ; t) ={x X μ(x, q) t, q Q} L(μ; t) ={x X μ(x, q) t, q Q} which are called an upper lower t-level cut of μ can be used to the characterization of μ. An intuitionistic Q-fuzzy set (IQFS for short) A is an object having the form A = {((x, q),μ A (x, q),λ A (x, q)) x X, q Q}, where the functions μ A : X [0, 1] λ A : X Q [0, 1] denote the degree of membership (namely, μ A (x, q)) the degree of nonmembership (namely, λ A (x, q)) of each element (x, q) X Q to the set A, respectively, 0 μ A (x, q)+λ A (x, q) 1 for all x X q Q. For the sake of simplicity, we shall use the symbol A =(μ A,λ A ) for the IQFS A = {(x, μ A (x, q),λ A (x, q)) x X, q Q}. Definition 2.2. A Q-fuzzy set μ is called a fuzzy R-subnear-ring of R over Q (shortly, Q-fuzzy R-subnear-ring of R) if (QF 1) μ(x y, q) μ A (x, q) μ A (y, q), (QF 2) μ(xy, q) μ A (x, q) μ A (y, q) for all x, y R q Q. Definition 2.3. A Q-fuzzy set μ is called a fuzzy R-subgroup of R over Q (shortly, Q-fuzzy R-subgroup of R) if μ satisfies (QF1) (QF R3) μ(rx, q) μ A (x, q), (QF R4) μ(xr, q) μ A (x, q) for all x, r R q Q.

2902 O. Kazancı, S. Yamak S. Yılmaz Definition 2.4 (16). Let θ be a mapping from X to Y. If A =(μ A,λ A ) B =(μ B,λ B ) are intuitionistic Q-fuzzy sets in X Y,respectively, then the inverse image of B under θ denoted by θ 1 (B), is an intuitionistic Q- fuzzy set in X defined by θ 1 (B) =(μ θ 1 (B), (λ θ 1 (B)), where μ θ 1 (B)(x, q) = μ B (θ(x),q) λ θ 1 (B)(x, q) =λ B (θ(x),q) for all x X, q Q, the image of A under f denoted by θ(a) =(μ θ(a),λ θ(a) ) where μ θ(a) (y, q) = λ θ(a) (y, q) = for all y Y,q Q. { x θ 1 (y) μ A (x, q) if θ 1 (y), 0, otherwise. { x θ 1 (y) λ A (x, q) if θ 1 (y), 1, otherwise. 3 Intuitionistic Q-fuzzy R-subgroups of nearrings In what follows, let Q R denote a set a near-ring, respectively, unless otherwise specified. Definition 3.1. An IQFS A =(μ A,λ A ) in R is called an intuitionistic Q-fuzzy subnear-ring of R if (IQF1) μ A (x y, q) μ A (x, q) μ A (y, q) λ A (x y, q) λ A (x, q) λ A (y, q), (IQF2) μ A (xy, q) μ A (x, q) μ A (y, q) λ A (xy, q) λ A (x, q) λ A (y, q) for all x, y R q Q. Definition 3.2. An IQFS A =(μ A,λ A ) in R is called an intuitionistic Q-fuzzy R-subgroup of R if A satisfies (IQF1) (IQF3) μ A (rx, q) μ A (x, q) λ A (rx, q) λ A (x, q), (IQF4) μ A (xr, q) μ A (x, q) λ A (xr, q) λ A (x, q) for all x, r R q Q. If A =(μ A,λ A ) satisfies (IQF1) (IQF3), then A is called an Q-fuzzy left R-subgroup of R, if A =(μ A,λ A ) satisfies (IQF1) (IQF4), then A is called an Q-fuzzy right R-subgroup of R. Example 1. Let R = {a, b, c, d} be a set with two binary operations as follows:

On intuitionistic Q-fuzzy R-subgroups of near-ring 2903 + a b c d a a b c d b b a d c c c d b a d d c a b a b c d a a a a a b a a a a c a a a a d a a b b Then (R, +, ) is a near ring. We define an IFQS A =(μ A,λ A ) in R as follows: for every q Q, μ A (a, q) =1,μ(b, q) = 1,μ(c, q) =0=μ(d, q), 3 λ A (a, q) =0,λ A (b, q) = 1 3,λ A(c, q) =1=λ A (d, q). By routine calculation, we can check that A =(μ A,λ A ) is both an intuitionistic Q-fuzzy subnear-ring an intuitionistic Q-fuzzy R-subgroup of R. Example 2. Let R = {a, b, c, d} be a set with two binary operations as follows: + a b c d a b c d a a b c d a a a a a b b a d c b a a a a c c d b a c a a a a d d c a b d a b c d Then (R, +, ) is a near ring. Let Q = {1, 2, 3, 4} let A =(μ A,λ A )bean intuitionistic Q-fuzzy set in R defined by μ A (a, 0) = μ A (a, 1) = μ A (a, 2) = μ A (a, 3) = μ A (a, 4) = 1, μ A (b, 0) = μ A (b, 1) = μ A (b, 2) = μ A (b, 3) = μ A (b, 4) = 2 3, μ A (c, 0) = μ A (c, 1) = μ A (c, 2) = μ A (c, 3) = μ A (c, 4) = 1 3, μ A (d, 0) = μ A (d, 1) = μ A (d, 2) = μ A (d, 3) = μ A (d, 4) = 1 3, λ A (a, 0) = λ A (a, 1) = λ A (a, 2) = λ A (a, 3) = λ A (a, 4) = 0, λ A (b, 0) = λ A (b, 1) = λ A (b, 2) = λ A (b, 3) = λ A (b, 4) = 1 4, λ A (c, 0) = λ A (c, 1) = λ A (c, 2) = λ A (c, 3) = λ A (d, 4) = 1 2, λ A (d, 0) = λ A (d, 1) = λ A (d, 2) = λ A (d, 3) = λ A (d, 4) = 1 2.

2904 O. Kazancı, S. Yamak S. Yılmaz We can check that A =(μ A,λ A ) is both intuitionistic Q-fuzzy subnear-ring intuitionistic Q-fuzzy R-subgroup of R. Lemma 3.3. If an IQFS A =(μ A,λ A ) in R satisfies the condition (IQF1), then (i) μ A (0,q) μ A (x, q) λ A (0,q) λ A (x, q), (ii) μ A ( x, q) =μ A (x, q) λ A ( x, q) =λ A (x, q) for all x R q Q. Proposition 3.1. If an IQFS A =(μ A,λ A ) in R satisfies the condition (IQF1), then (i) μ A (x y, q) μ A (0,q) implies μ A (x, q) =μ A (y, q), (ii) λ A (x y, q) λ A (0,q) implies λ A (x, q) =λ A (y, q) for all x, y R q Q. Theorem 3.4.If{A i } is a family of intuitionistic Q-fuzzy R-subgroups (subnear-rings) of R. Then A i is an intuitionistic Q- fuzzy R-subgroup (subnear-ring) of R, where A i = {(x, μ Ai (x, q), λ Ai (x, q)) x X, q Q}. Proof. Let x, y R, q Q. Then ( μ Ai )(x y, q) = μ Ai (x y, q) (μ Ai (x, q) μ Ai (y, q)) =( μ Ai (x, q)) ( μ Ai (y, q)) = ( μ Ai )(x, q) ( μ Ai )(y, q). ( λ Ai )(x y, q) = λ Ai (x y, q) (λ Ai (x, q) λ Ai (y, q)) =( λ Ai (x, q)) ( λ Ai (y, q)) = ( λ Ai )(x, q) ( λ Ai )(y, q). Let x, r R, q Q. Then ( μ Ai )(xr, q) = μ Ai (xr, q) μ Ai (x, q) =( μ Ai )(x, q). ( λ Ai )(xr, q) = λ Ai (xr, q) λ Ai (x, q) =( λ Ai )(x, q). Similarly, we get ( μ Ai )(rx, q) ( μ Ai )(x, q) ( λ Ai )(rx, q) ( λ Ai )(x, q). Hence A i is an intuitionistic Q-fuzzy R-subgroup of R. We can prove intuitionistic Q-fuzzy subnear-ring similarly. So we omit the proof.

On intuitionistic Q-fuzzy R-subgroups of near-ring 2905 Lemma 3.5. If A = (μ A,λ A ) is an intuitionistic Q-fuzzy R-subgroup (resp. subnear-ring) of R, then so is A =(μ A,μ c A ). Proof. It is sufficient to show that μ c A satisfies the conditions (IQF1), (IQF3) (IFQ4) of Definition 3.1 3.2. For any x, y R, q Q, we have μ c A (x y, q) = 1 μ A(x y, q) 1 (μ A (x, q) μ A (y, q)) = (1 μ A (x, q)) (1 μ A (y, q)) = μ c A (x, q) μc A (y, q) Hence the condition (IQF1) of Definition 3.1 is valid. For any x, r R, q Q, we have μ c A (rx, q) =1 μ A(rx, q) 1 (μ A (x, q)) = μ c A (x, q). Hence the condition (IQF3) of Definition 3.2 is valid. μ c A (xr, q) =1 μ A(xr, q) 1 (μ A (x, q)) = μ c A (x, q). Hence the condition (IQF4) of Definition 3.2 is valid. Therefore A is an intuitionistic Q-fuzzy R-subgroup of R. We can prove intuitionistic Q-fuzzy subnear-ring similarly. So we omit the proof. Lemma 3.6. If A = (μ A,λ A ) is an intuitionistic Q-fuzzy R-subgroup (subnear-ring) of R, then so is A =(λ c A,λ A ). Proof. The proof is similar to the proof of Lemma 3.5. Combining the above two Lemmas it is not difficult to verify that the following Theorem is valid. Theorem 3.7. A = (μ A,λ A ) is an intuitionistic Q-fuzzy R-subgroup (subnear-ring) of R if only if A A are intuitionistic Q-fuzzy R- subgroups (subnear-rings) of R. Corollary 3.8. A = (μ A,λ A ) is an intuitionistic Q-fuzzy R-subgroup (subnear-ring) of R if only if μ A λ c A are Q-fuzzy R-subgroups (subnearrings) of R. Theorem 3.9. If an IQFS A =(μ A,λ A ) is an intuitionistic Q-fuzzy R- subgroup (subnear-ring) of R. Then the sets R μa = {x M μ A (x, q) =μ A (0,q)} R λa = {x M λ A (x, q) =λ A (0,q)} are R-subgroups (subnear-rings) of R for all q Q.

2906 O. Kazancı, S. Yamak S. Yılmaz Proof. Let x, y R μa q Q. Then μ A (x, q) = μ A (0,q), μ A (y, q) = μ A (0,q). Since A =(μ A,λ A ) is an intuitionistic Q-fuzzy R-subgroup of R, we get μ A (x y, q) μ A (x, q) μ A (y, q) =μ A (0,q). By using Proposition 3.1 we get μ A (x y, q) =μ A (0,q). Hence x y R μa. For every r R, x R μa, we have μ A (rx, q) μ A (x, q) =μ A (0,q). By using Proposition 3.1 we get μ A (rx, q) =μ A (0,q). Hence rx R μa. Similarly xr R μa. Therefore R μa is a R-subgroup of R. We can prove subnear-ring similarly. So we omit the proof. Similarly R λa is a R-subgroup (subnear-ring) of R. Theorem 3.10. If A =(μ A,λ A ) is an intuitionistic Q-fuzzy R-subgroup (subnear-ring) of R, then the sets U(μ A ; t) L(λ A ; t) are R-subgroups (subnearrings) of R for all q Q, t Im(μ A ) Im(λ A ). Proof. Let t Im(μ A ) Im(λ A ) [0, 1] let x, y U(μ A ; t) q Q. Then μ A (x, q) t, μ A (y, q) t. Since A =(μ A,λ A ) is an intuitionistic Q- fuzzy R-subgroups of R, we have μ A (x y, q) μ A (x, q) μ A (y, q) t. Hence x y U(μ A ; t). Let x U(μ A ; t) q Q. Then μ A (x, q) t. Since A = (μ A,λ A ) is an an intuitionistic Q-fuzzy R-subgroup of R, we have μ A (rx, q) μ A (x, q) t for all r R, which implies that rx U(μ A ; t). Similarly xr R μa. Therefore U(μ A ; t) is a R-subgroup of R. We can prove subnear-ring similarly. So we omit the proof. Similarly L(λ A ; t) is a R-subgroup (subnearring) of R. Theorem 3.11.IfA =(μ A,λ A ) is an intuitionistic Q-fuzzy set in R such that all non-empty level sets U(μ A ; t) L(λ A ; t) are R-subgroups (subnearrings) of R, then A =(μ A,λ A ) is an intuitionistic Q-fuzzy R-subgroup (subnearring) of R. Proof. Assume that all non-empty level sets U(μ A ; t) L(λ A ; t) are R- subgroups of R. If t 0 = μ A (x, q) μ A (y, q) t 1 = λ A (x, q) λ A (y, q) for x, y R, q Q, then x, y U(μ A ; t 0 ) x, y L(λ A ; t 1 ). So x y U(μ A ; t 0 ) x y L(λ A ; t 1 ). Hence μ A (x y, q) t 0 = μ A (x, q) μ A (y, q) λ A (x y, q) t 1 = λ A (x, q) λ A (y, q) which implies that the condition (IQF1) is valid. Now t 2 = μ A (x, q) t 3 = λ A (x, q) for some x R, q Q, then x U(μ A ; t 2 ) x L(λ A ; t 3 ). Since U(μ A ; t 2 ), L(λ A ; t 3 ) are R-subgroups of R, we get rx U(μ A ; t 2 ) rx L(λ A ; t 3 ) for all r R. Therefore μ A (rx) t 2 = μ A (x, q) λ A (rx) t 3 = λ A (x, q) which verify the condition (IQF2). Similarly the condition (IQF3) is valid. Therefore A =(μ A,λ A )isan intuitionistic Q-fuzzy R-subgroup of R. We can prove subnear-ring similarly. So we omit the proof.

On intuitionistic Q-fuzzy R-subgroups of near-ring 2907 Corollary 3.12. Let I be a R-subgroup (subnear-ring) of R. If Q-fuzzy sets μ A, λ A in R are defined by { p if x I, μ A (x, q) = s otherwise. { u if x I, λ A (x, q) = v otherwise. for all x R, q Q where 0 s<p, 0 u<v p + u 1, s + v 1, then A =(μ A,λ A ) is an intuitionistic Q-fuzzy R-subgroup (subnear-ring) of R U(μ; p) =I = L(λ; u). Proof. Let x, y R, q Q. If at least one of x y does not belong to I, then μ A (x y, q) s = μ A (x, q) μ A (y, q), λ A (x y, q) v = λ A (x, q) λ A (y, q). If x, y I, then x y I so μ A (x y, q) =p = μ A (x, q) μ A (y, q), λ A (x y, q) =v = λ A (x, q) λ A (y, q). If x I, then rx, xr I for all r R so μ A (rx) =μ A (xr) =p = μ A (x, q), λ A (rx) =λ A (xr) =u = λ A (x, q). If x/ I, then μ A (rx) =μ A (xr) = s = μ A (x, q) λ A (rx) =λ A (xr) =v = λ A (x, q) for all r R, q Q. Therefore A =(μ A,λ A ) is an intuitionistic Q-fuzzy R-subgroup of R. We can prove subnear-ring similarly. So we omit the proof. Obviously U(μ; p) = I = L(λ; u). Corollary 3.13. Let χ I be the characteristic function of a R-subgroup (subnear-ring) I of a near-ring R. Then Ĩ =(χ I,χ c I) is an intuitionistic Q- fuzzy R-subgroup (subnear-ring) of R. Theorem 3.14. Let R S be two near-rings θ : R S a homomorphism. If B =(μ B,λ B ) is an intuitionistic Q-fuzzy R-subgroup (subnearring) of S, then the preimage θ 1 (B) =(μ θ 1 (B),λ θ 1 (B)) of B under θ is an intuitionistic Q-fuzzy R-subgroup (subnear-ring) of R. Proof. Assume that B =(μ B,λ B ) is an intuitionistic Q-fuzzy R-subgroup of S let x, y, r R, q Q. Then μ θ 1 (B)(x y, q) = μ B (θ(x y),q)=μ B (θ(x) θ(y),q) μ B (θ(x),q) μ B (θ(y),q)=μ θ 1 (B)(x, q) μ θ 1 (B)(y, q). λ θ 1 (B)(x y, q) = λ B (θ(x y),q)=λ B (θ(x) θ(y),q) λ B (θ(x),q) λ B (θ(y),q)=λ θ 1 (B)(x, q) λ θ 1 (B)(y, q). μ θ 1 (B)(rx, q) = μ B (θ(rx),q)=μ B (θ(r)θ(x),q) μ B (θ(x),q)=μ θ 1 (B)(x, q).

2908 O. Kazancı, S. Yamak S. Yılmaz λ θ 1 (B)(rx, q) = λ B (θ(rx),q)=λ B (θ(r)θ(x),q) λ B (θ(x),q)=λ θ 1 (B)(x, q). Similarly μ θ 1 (B)(xr, q) μ θ 1 (B)(x, q) λ θ 1 (B)(xr, q) λ θ 1 (B)(x, q). Therefore θ 1 (B) =(μ θ 1 (B),λ θ 1 (B)) is an intuitionistic Q-fuzzy R-subgroup of R. We can prove subnear-ring similarly. So we omit the proof. If we strengthen the condition of θ, then we can construct the converse of Theorem 3.14 as follows. Theorem 3.15. Let θ : R S be an epimorphism let B =(μ B,λ B ) is an intuitionistic Q-fuzzy set in S. If θ 1 (B) =(μ θ 1 (B),λ θ 1 (B)) is an intuitionistic Q-fuzzy R-subgroup (subnear-ring) of R, then B =(μ B,λ B ) is an intuitionistic Q-fuzzy R-subgroup (subnear-ring) of S. Proof. Let x, y, s S, q Q. Then there exist a, b, r R such that θ(a) =x, θ(b) = y, θ(r) = s. It follows that μ B (x y, q) = μ B (θ(a) θ(b),q)=μ B (θ(a b),q) = μ θ 1 (B)(a b, q) μ θ 1 (B)(a, q) μ θ 1 (B)(b, q) = μ B (θ(a),q) μ B (θ(b),q)=μ B (x, q) μ B (y, q). λ B (x y, q) = λ B (θ(a) θ(b),q)=λ B (θ(a b),q) = λ θ 1 (B)(a b, q) λ θ 1 (B)(a, q) λ θ 1 (B)(b, q) = λ B (θ(a),q) λ B (θ(b),q)=λ B (x, q) λ B (y, q). μ B (rx, q) = μ B (θ(c)θ(a),q)=μ B (θ(ca),q)=μ θ 1 (B)(ca, q) μ θ 1 (B)(a, q) =μ B (θ(a),q)=μ B (x, q). λ B (sx, q) = λ B (θ(r)θ(a),q)=λ B (θ(ra),q)=λ θ 1 (B)(ra, q) λ θ 1 (B)(a, q) =λ B (θ(a),q)=λ B (x, q). Similarly μ B (xs, q) μ B (x, q) λ B (sx, q) λ B (x, q). Therefore B = (μ B,λ B ) is an intuitionistic Q-fuzzy R-subgroup of S. We can prove subnearring similarly. So we omit the proof. ACKNOWLEDGEMENTS. The authors are highly grateful to the referees for their valuable comments suggestions for improving the paper.

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2910 O. Kazancı, S. Yamak S. Yılmaz [16] E.H. Roh, K.H.Kim, J.G. Lee, Intuitionistic Q-Fuzzy Subalgebras of BCK/BCI-Algebras, International Mathematical Forum, 1 (24) (2006) 1167-1174. [17] Y.H Yon, Y.B. Jun K.H. Kim, Intuitionistic fuzzy R-subgroups of near-rings, Soochow J. Math, 27 (3) (2001) 243-253. [18] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353. Received: April 25, 2007