A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings

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1 International Journal of Computational Science and Mathematics. ISSN Volume 3, Number 1 (2011), pp International Research Publication House A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings N. Palaniappan 1, P.S. Veerappan 2 and D. Ezhilmaran 2 1 Professor of Mathematics, Alagappa University, Karaikudi Tamilnadu, India. palaniappan.nallappan@gmail.com 2 Department of Mathematics, K.S.R. College of Technology, Tiruchengode , Tamilnadu, India. peeyesvee@yahoo.co.in, ezhil.devarasan@yahoo.com Abstract In this paper, we study some properties of intuitionistic fuzzy ideals of a Γ near ring and prove some results on these Mathematics Subject Classification: 16D25, 03E72, 03G25. Keywords and phrases: Γ near- ring, fuzzy set, intuitionistic fuzzy set, intuitionistic fuzzy ideal. Introduction The notion of a fuzzy set was introduced by L.A.Zadeh[10], and since then this concept have been applied to various algebraic structures. The idea of Intuitionistic Fuzzy Set was first published by K.T.Atanassov[1] as a generalization of the notion of fuzzy set. Γ- near-rings were defined by Bh.Satyanarayana [9] and G.L.Booth [2, 3] studied the ideal theory in Γ-near-rings. W. Liu[7] introduced fuzzy ideals and it has been studied by several authors. The notion of fuzzy ideals and its properties were applied to semi groups, BCK- algebras and semi rings. Y.B. Jun [5, 6] introduced the notion of fuzzy left (resp.right) ideals. In this paper, we introduce the notion of intuitionistic fuzzy ideals in Γ near- rings and study some of its properties. Preliminaries In this section we include some elementary aspects that are necessary for this paper.

2 62 N. Palaniappan et al Definition 2.1 A non empty set R with two binary operations + (addition) and. (multiplication) is called a near-ring if it satisfies the following axioms: i. (R, + ) is a group, ii. iii. (R,. ) is a semigroup, (x + y). z = x. z +y. z, for all x, y, z R. It is a right near-ring because it satisfies the right distributive law. Definition 2.2 A Γ-near-ring is a triple (M, +, Γ ) where i. (M, +) is a group, ii. is a nonempty set of binary operators on M such that for each α Γ, ( M, +, α ) is a near ring, iii. xα(yβz) = (xαy)βz for all x, y, z M and α, β Γ. Definition 2.3 A subset A of a Γ-near-ring M is called a left (resp. right) ideal of M if i. (A, +) is a normal divisor of (M, +), ii. uα(x + v) -uαv A ( resp. xαu A ) for all x A, α Γ and u,v M. Definition 2.4 A fuzzy set μ in a Γ-near-ring M is called a fuzzy left (resp.right) ideal of M if i. μ(x-y) min{ μ(x), μ(y) }, ii. μ(y + x-y ) μ(x), for all x, y M. iii. μ(uα(x + v) -uαv) μ(x) (resp. μ(xαu) μ(x) ) for all x, u, v M and α Γ. Definition 2.5 [1] Let X be a nonempty fixed set.an intuitionistic fuzzy set (IFS) A in X is an object having the form A = { < x, μ A (x), ν A (x) >/ x X }, where the functions μ A : X [ 0, 1] and ν A : X [ 0, 1] denote the degree of membership and degree of non membership of each element x X to the set A, respectively, and 0 μ A (x) + ν A (x) 1. Notation. For the sake of simplicity, we shall use the symbol A= < μ A, ν A > for the IFS A = {< x, μ A (x), ν A (x) > /x X }. Definition 2.6 [1]. Let X be a non-empty set and let Α = < μ Α, ν Α > and Β = < μ Β, ν Β > be IFSs in X. Then 1. Α Β iff μ Α μ Β and ν Α ν Β. 2. Α = Β iff Α Β and Β Α. 3. Α c = < ν Α, μ Α >. 4. Α I Β = ( μ Α μ Β, ν Α ν Β ). 5. Α U Β = ( μ Α μ Β, ν Α ν Β ). 6. Α = (μ Α,1 μ Α ), Α = (1 ν Α, ν Α ). Definition 2.7. Let µ and ν be two fuzzy sets in a Γ-near-ring For s, t [0, 1] the set U(µ,s) = { x µ(x) s } is called upper level of µ. The set L(ν,t)={x ν(x) t } is

3 A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings 63 called lower level of ν. Definition 2.8. Let Α be an IFS in a Γ ring Μ. For each pair < t, s > [0, 1] with t + s 1, the set Α < t, s > = { x X / μ Α (x) t and ν Α (x) s } is called a <t, s> level subset of Α. Definition 2.9. Let Α = < μ Α, ν Α > be an intuitionistic fuzzy set in Μ and let t [0, 1]. Then the sets U(μ Α ; t) = {x Μ : μ Α (x) t} and L(ν Α ; t) = {x Μ : ν Α (x) t} are called upper level set and lower level set of Α respectively. Intuitionistic fuzzy ideals In what follows, let Μ denote a Γ near-ring unless otherwise specified. Definition 3.1. An IFS A = < μ Α, ν Α > in Μ is called an intuitionistic fuzzy left ( resp. right ) ideal of a Γ near-ring M if 1. μ Α (x y) { μ Α (x) μ Α (y) }, 2. μ Α (y + x y) μ Α (x), 3. μ Α (uα(x+v)-uαv) μ Α (x) ( resp. μ Α (xαu) μ Α (x) ), 4. ν Α (x y) { ν Α (x) ν Α (y) }, 5. ν Α (y+x y) ν Α (x), 6. ν Α (uα(x+v)-uαv) ν Α (x) ( resp. ν Α (xαu) ν Α (x) ), for all x, y, u, v Μ and α Γ. Example 3.2. Let R be the set of all integers then R is a ring. Take M = Γ = R. Let a, b Μ, α Γ, suppose aαb is the product of a, α, b R. Then M is a Γ- near-ring. Define an IFS Α = < μ Α,ν Α > in R as follows. μ Α (0) = 1 and μ Α (±1) = μ Α (±2) = μ Α (±3) =... = t and ν Α (0) = 0 and ν Α (±1) = ν Α (±2) = ν Α (±3) =... = s, where t [0, 1], s [0, 1] and t + s 1. By routine calculations, clearly A is an intuitionistic fuzzy ideal of a Γ- near-ring R. Theorem 3.3. Α is an ideal of a Γ near-ring M if and only if à = < μ Ã, ν à > where 1 x A 0 x A μ à (x) = ν à (x) = 0 Otherwise 1 Otherwise is an intuitionistic fuzzy left (resp.right) ideal of Μ. Proof ( ) : Let Α be a left (resp.right) ideal of Μ. Let x, y, u, v Μ and α Γ. If x, y Α, then x y Α, y + x y Α and (uα(x +v) -uαv) Α. Therefore

4 64 N. Palaniappan et al μ à (x y) = 1 { μ à (x) μ à (y) },μ à (y + x y) = 1 μ à (x) and μ à (uα(x +v) -uαv) = 1 = μ à (x) (resp. μ à (xαu) = μ à (x) = 1), ν à (x y) = 0 {ν à (x) ν à (y) },ν à (y + x y) = 0 ν à (x) and ν à (uα(x +v) -uαv) = 0 = ν à (x) (resp. ν à (xαu) = ν à (x) = 0). If x Α or y Α then μ à (x) = 0 or μ à (y) = 0. Thus we have μ à (x y) { μ à (x) μ à (y) },μ à (y + x y) μ à (x) and μ à (uα(x +v) -uαv) μ à (x) (resp. μ à (xαu) μ à (x)), ν à (x y) {ν à (x) ν à (y) },ν à (y + x y) ν à (x) and ν à (uα(x +v) -uαv) ν à (x) (resp. ν à (xαu) ν à (x)). Hence à is an intuitionistic fuzzy left (resp.right) ideal of Μ. ( ) : Let à be an intuitionistic fuzzy left (resp.right) ideal of Μ. Let x, y Μ and α Γ. If x, y, u, v Α, then μ à (x y) { μ à (x ) μ à ( y)} = 1 ν à (x y) {ν à (x ) ν à ( y)} = 0 So x y Α. μ à (y + x y) μ à (x ) = 1 ν à (y + x y) ν à (x ) = 0 So (y + x y) Α. Also μ à (uα(x +v) -uαv) μ à (x) = 1 ( resp. μ à (xαu) = μ à (x) = 1) ν à (uα(x +v) -uαv) ν à (x) = 0 ( resp. ν à (xαu) = ν à (x) = 0) So (uα(x +v) -uαv) Α. Hence Α is a left (resp.right) ideal of Μ. Theorem 3.4. Let Α be an intuitionistic fuzzy left (resp.right) ideal of Μ and t [0,1], then I. U(μ Α ; t) is either empty or an ideal of Μ. II. L(ν Α ; t) is either empty or an ideal of Μ. Proof. (i) Let x, y U(μ Α ; t). Then μ Α (x y) { μ Α (x) μ Α (y) } t, Hence x y L(ν A ; t). μ Α (y + x y) μ Α (x) t Hence (y + x y) U(μ Α ; t).

5 A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings 65 Let x Μ, α Γ and u, v U(μ Α ; t). Then μ Α (uα(x +v) -uαv) μ Α (x) t and so (uα(x +v) -uαv) U(μ Α ; t). Hence U(μ Α ; t) is an ideal of Μ. III.Let x, y L(ν A ; t). Then ν A (x - y) {ν A (x) ν A (y)} t. Hence x y L(ν A ; t). ν A (y + x y) ν A (x) t. Hence (y + x y) L(ν A ; t). Let x Μ, α Γ and u, v L(ν Α ; t). Then ν Α (uα(x +v) -uαv) ν Α (x) t and so (uα(x +v) -uαv) L(ν Α ; t). Hence L(ν Α ; t) is an ideal of Μ. Theorem 3.5. Let Ι be the left (resp.right) ideal of Μ. If the intuitionistic fuzzy set A= < μ Α,ν Α > in Μ is defined by p if x I u if x I μ Α (x) = and ν Α (x) = s Otherwise v Otherwise for all x Μ and α Γ, where 0 s < p, 0 v < u and p + u 1, s + v 1, then Α is an intuitionistic fuzzy left (resp.right) ideal of Μ and U(μ Α ; p) = Ι = L(ν Α ; u). Proof. Let x, y Μ and α Γ. If at least one of x and y does not belong to I, then μ Α (x y) s = {μ Α (x) μ Α (y)}, ν Α (x y) v = {ν Α (x) ν Α (y)}. If x, y Ι, then x y I and so μ Α (x y) = p = {μ Α (x) μ Α (y)} and ν Α (x y) = v = {ν Α (x) ν Α (y)}. μ Α (y + x y) s = μ Α (x), ν Α (y + x y) v = ν Α (x). If x, y Ι, then (y + x y) I and so μ Α (y + x y) = p = μ Α (x) and ν Α (y + x y) = v = ν Α (x). If u, v I, x Μ and α Γ, then (uα(x +v) -uαv) I, μ Α (uα(x +v) -uαv) = p = μ Α (x) and ν Α (uα(x +v) -uαv) = u = ν Α (x). (resp. μ Α (xαu) = p = μ Α (x) and ν Α (xαx) = u = ν Α (x) ) If y I, then μ Α (uα(x +v) -uαv) = s = μ Α (x), ν Α (uα(x +v) -uαv) = v = ν Α (x). (resp., μ Α (xαu) = s = μ Α (x) and ν Α (xαu) = v = ν Α (x) ) Therefore Α is an intuitionistic fuzzy left (resp.right) ideal. Definition3.6. Let f be a mapping from a Γ near-ring Μ onto a Γ near-ring N. Let Α

6 66 N. Palaniappan et al be an intuitionistic fuzzy ideal of Μ. Now Α is said to be f invariant if f(x) = f(y) implies μ Α (x) = μ Α (y) and ν Α (x) = ν Α (y). Definition 3.7 [2]. A function f : Μ Ν, where Μ and Ν are Γ near-rings, is said to be a Γ homomorphism if f(a +b) = f(a) + f(b), f(aαb) = f(a)αf(b), for all a, b Μ and α Γ. Definition 3.8 Let f : X Y be a mapping of a Γ near-ring and Α be an intuitionistic fuzzy set of Y. Then the map f -1 (A) is the pre-image of Α under f, if μ f (Α) (x) = μ Α (f(x)) and ν f (Α) (x) = ν Α (f(x)), for all x X. Definition 3.9. Let f be a mapping from a set X to the set Y. If A = <μ A, ν A > and B = < μ B, ν B > are intuitionistic fuzzy subsets in X and Y respectively, then the image of Α under f is the intuitionistic fuzzy set f(α) = < μ f(a), ν f(a) > defined by -1 μ f (A) μ 1 A( x) if f ( y) ϕ, (x) = x f ( y) 0 Otherwise, -1 ν f (A) ν 1 A( x) if f ( y) ϕ, (x) = x f ( y) 1 Otherwise, for all y Y. (b) the pre image of Α under f is the intuitionistic fuzzy set f (B) = < μ f (B), ν f (B) > defined by -1 μ f (B) μ 1 B( y) if f ( x) ϕ, (x) = y f ( x) 0 Otherwise, -1 ν f (B) ν 1 A( y) if f ( x) ϕ, (x) = y f ( x) 1 Otherwise, -1-1 for all x X, where μ f (B) (x) = μ B (f(x)) and ν f (B) (x) = ν B (f(x)). Theorem Let Μ and Ν be two Γ near-rings and θ : Μ Ν be a Γ epimorphism and let Β = < μ Β, ν Β > be an intuitionistic fuzzy set of N. If θ (Β) = < μ θ (Β), ν θ (Β) > is an intuitionistic fuzzy left (resp.right) ideal of Μ, then Β is an intuitionistic fuzzy left (resp.right) ideal of Ν. Proof. Let x, y, u, v Ν and α Γ, then there exists a, b, c, d Μ such that θ(a) = x, θ(b) = y, θ(c) = u, θ(d) = v. It follows that μ Β (x y) = μ Β (θ(a) θ(b)) = μ Β (θ(a b)) = μ θ (Β) (a b) { μ θ (Β) (a) μ θ (Β) (b)} = {μ Β (θ (a)) μ Β (θ(b))}

7 A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings 67 = {μ Β (x) μ Β (y)} ν Β (x y) = ν Β (θ(a) θ(b)) = ν Β (θ(a b)) = ν θ (Β) (a b) {ν θ (Β) (a) ν θ (Β) (b)} = {ν Β (θ(a)) ν Β (θ(b))} = {ν Β (x) ν Β (y)}. μ Β (y + x y) = μ Β (θ(b) + θ(a) θ(b)) = μ Β (θ(b + a b)) = μ θ (Β) (b + a b) μ θ (Β) (a) = μ Β (θ(a)) = μ Β (x) ν Β (y + x y) = ν Β (θ(b) + θ(a) θ(b)) = ν Β (θ(b + a b)) = ν θ (Β) (b + a b) ν θ (Β) (a) = ν Β (θ(a)) = ν Β (x). Also μ Β (uα(x +v) -uαv) = μ Β (θ(c)α(θ(a)+θ(d)) θ(c)αθ(d)) = μ Β (θ(cα(a+d)-cαd)) = μ θ (Β) (cα(a+d)-cαd) μ θ (Β) (a) = μ Β (θ(a)) = μ Β (x) ν Β (uα(x +v) -uαv) = ν Β (θ(c)α(θ(a)+θ(d)) θ(c)αθ(d)) = ν Β (θ(cα(a+d)-cαd)) = ν θ (Β) (cα(a+d)-cαd) ν θ (Β) (a) = ν Β (θ(a)) = μ Β (x) Similarly, μ Β (xαu) μ Β (x) and ν Β (xαu) ν Β (x). Hence B is an intuitionistic fuzzy left (resp. right) ideal of Ν. Theorem An intuitionistic fuzzy set Α= < μ Α, ν Α > in a Γ near-ring M is an intuitionistic fuzzy left (resp. right) ideal if and only if Α < t, s> ={ x Μ μ Α (x) t, ν Α (x) s} is a left (resp.right) ideal of M for μ Α (0) t, ν Α (0) s. Proof.( ) Suppose that Α = < μ Α, ν Α > is an intuitionistic fuzzy left (resp.right) ideal of M and let μ Α (0) t, ν Α (0) s. Let x, y, u, v Α < t, s > and α Γ. Then μ Α (x) t, ν Α (x) s and μ Α (y) t, ν Α (y) s. Hence μ Α (x y) {μ Α (x) μ Α (y) } t, ν Α (x y) {ν Α (x) ν Α (y) } s. μ Α (y + x y) μ Α (x) t, ν Α (y + x y) ν Α (x) s. μ Α (uα(x +v) -uαv) μ Α (x) t and ν Α (uα(x +v) -uαv) ν Α (x) s (resp. μ Α (xαu) μ Α (x) t and ν Α (xαu) ν Α (x) s). Therefore x y Α < t,s>, (y + x y) Α < t,s> and (uα(x +v) -uαv) Α < t,s> for all x, y A < t,s> and α Γ. So Α < t,s> is a left (resp. right) ideal of M. ( ) Suppose that Α < t,s> is a left (resp. right) ideal of M for μ Α (0) t and ν Α (0) s. Let x, y M be such that μ Α (x) = t 1, ν Α (x) = s 1, μ Α (y) = t 2 and ν Α (y) = s 2.

8 68 N. Palaniappan et al Then x and y. We may assume that t 2 t 1 and s 2 s 1 without loss of generality. It follows that so that x, y. Since is an ideal of M, we have x y and (uα(x +v) -uαv) for all α Γ. μ Α (x y) t 1 t 2 = {μ Α Α(x) μ Α (y)}, ν Α (x y) s 1 s 2 = {ν Α Α(x) ν Α (y)}. μ Α (y + x y) t 1 t 2 = μ Α (x), ν Α (y + x y) s 1 s 2 = ν Α (x). μ Α (uα(x +v) -uαv) t 1 t 2 = μ Α (x) and ν Α (uα(x +v) -uαv) s 1 s 2 = ν Α (x). (y + x y) Therefore A is an intuitionistic fuzzy left (resp.right) ideal of M. Theorem If the IFS Α = < μ Α, ν Α > is an intuitionistic fuzzy left (resp.right) ideal of a Γ near-ring Μ, then the sets Μμ Α ={x Μ /μ Α (x) = μ Α (0)} and Μν Α ={x Μ /ν Α (x) = ν Α (0)} are left (resp.right) ideals. Proof. Let x, y, u, v Μμ Α and α Γ. Then μ Α (x) = μ Α (0), μ Α Α(y) = μ Α (0). Since Α is an intuitionistic fuzzy left (resp.right) ideal of a Γ near-rinμ Α (y)} = μ Α (0). M, we get μ Α (x y) {μ Α (x) But μ Α (0) μ Α (x y). So x y Μμ Α. μ Α (y + x y) μ Α (x) = μ Α (0). But μ Α (0) μ Α (y + x y). So (y + x y) Μμ Α. μ Α (uα(x +v) -uαv) μ Α (x) = μ Α (0) ( resp. μ Α (xαu) μ Α (x) = μ Α (0)). Hence (uα(x +v) -uαv) Μμ Α. Therefore Μμ Α is a left (resp.right) ideal of M. Similarly, let x, y, u, v Μν Α and α Γ. Then ν Α (x) = ν Α (0), ν Α (y) = ν Α (0). Since Α is an intuitionistic fuzzy left (resp.right) ideal of a Γ near-rinν Α (y)} = ν Α (0). Μ, ν Α (x y) {ν Α (x) But ν Α (0) ν Α (x y). So x y Μν Α. ν Α (y + x y) ν Α (x) = ν Α (0). But ν Α (0) ν Α (y + x y). So (y + x y) Μν Α. ν Α (uα(x +v) -uαv) ν Α (x) = ν Α (0) ( resp. ν Α (xαu) ν Α (x) = v A (0) ). Hence (uα(x +v) -uαv) Μν Α. Therefore Μν Α is a left (resp.right) ideal of Μ. Definition A Γ near-ring M is said to be regular if for each a M there exists

9 A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings 69 an x M and α, β Γ such that a = aαxβa. Definition Let Α = < μ Α, ν Α > and Β = < μ Β, ν Β > be two intuitionistic fuzzy subsets of a Γ-near-ring Μ. The product ΑΓΒ is defined by μ ΑΓΒ ( μa( u ) μb( v) ) if x ( uγ( v w) uγw), u, v, w, γ (x) = x= ( uγ( v+ w) uγw ) = + Μ Γ 0 otherwise, A( ) B( ) ν ΑΓΒ ν u v ( ( ) ),,,, (x) = x ( u ( v w) u w( ν ) if x = uγ v + w uγw u v w Μ γ Γ = γ + γ ) 1 otherwise. Theorem If Α = < μ Α, ν Α > and Β = < μ Β, ν Β > are two intuitionistic fuzzy left (resp. right) ideals of Μ, then Α Β is an intuitionistic fuzzy left (resp. right) ideal of Μ. If Α is an intuitionistic fuzzy right ideal and Β is an intuitionistic fuzzy left ideal, then ΑΓΒ Α Β. Proof. Suppose Α and Β are intuitionistic fuzzy ideals of Μ and let x, y, z, z' Μ and α Γ. Then μ Α Β (x y) = μ Α (x y) μ Β (x y) [μ Α (x) μ Α (y)] [ μ Β (x) μ Β (y)] = [μ Α (x) μ Β (x)] [ μ Α (y) μ Β (y)] = μ Α Β (x) μ Α Β (y), ν Α Β (x y) = ν Α (x y) ν Β (x y) [ν Α (x) ν Α (y)] [ v Β (x) ν Β ( y)] = [ν Α (x) ν Β (x)] [ ν Α (y) ν Β (y)] = ν Α Β (x) ν Α Β (y). μ Α Β (y + x y) = μ Α (y + x y) μ Β (y + x y) [μ Α (x)] [μ Β (x)] = μ Α Β (x) μ Α Β (y), ν Α Β (y + x y) = ν Α (y + x y) ν Β (y + x y) [ν Α (x)] [v Β (x)] = ν Α Β (x) ν Α Β (y). Since A and B are intuitionistic fuzzy ideals of M, we have μ Α (xα(y+z)-xαz) μ Α (x), ν Α (xα(y+z)-xαz) ν Α (x) and μ Β (yαx) μ Β (x), ν Β (yαx) ν Β (x). Now μ Α Β (xα(y+z)-xαz) = μ Α (xα(y+z)-xαz) μ Β (xα(y+z)-xαz) μ Α (x) μ Β (x) = μ Α Β (x)(resp. μ Α Β (yαx) μ Α Β (x)), ν Α Β (xα(y+z)-xαz) = ν Α (xα(y+z)-xαz) ν Β (xα(y+z)-xαz) ν Α (x) ν Β (x) = ν Α Β (x)(resp. ν Α Β (yαx) ν Α Β (x)). Hence Α Β is an intuitionistic fuzzy left ideal of M.

10 70 N. Palaniappan et al To prove the second part if μ ΑΓΒ (x) = 0 andν ΑΓΒ (x) = 1, there is nothing to show. From the definition of ΑΓΒ, μ Α (x) = μ Α (yα(z+z') yαz') μ Α (z), ν Α (x) = ν Α (yα(z+z') yαz') ν Α (z). Since Α is an intuitionistic fuzzy right ideal and Β is an intuitionistic fuzzy left ideal, we have μ Α (x) = μ Α (zαy) μ Α (z), ν Α (x) = ν Α (zαy) ν Α (z), μ Β (x) = μ Β (zαy) μ Β (z), ν Β (x) = ν Β (zαy) ν Β (z). Hence by Definition 3.5, μ ΑΓΒ (x) = μ Α (x) μ Β (x) = μ Α Β (x) and ν ΑΓΒ (x) = x= α y z x= α y z {μ Α (y) μ Β (z)} {ν Α (y) ν Β (z)} ν Α (x) ν Β (x) = ν Α Β (x) which means that ΑΓΒ Α Β. Theorem A Γ near-ring M is regular if and only if for each intuitionistic fuzzy right ideal A and each intuitionistic fuzzy left ideal B of M, AΓB = A B. Proof. ( ) Suppose R is regular. AΓB A B. Thus it is sufficient to show that A B AΓB. Let a M and α, β Γ. Then, by hypothesis, there exists an x M such that a = aαxβa. Thus μ A (a) = μ A (aαxβa) μ A (aαx) μ A (a), ν A (a) = ν A (aαxβa) ν A (aαx) ν A (a). So μ A (aαx) = μ A (a) and ν A (aαx) = ν A (a). On the other hand, μ AΓB (a) = a= aαxβa [μ A (aαx) μ B (a)] [μ A (a) μ B (a)] = μ A B (a) and ν AΓB (a) = a= aαxβa [ν A (aαx) ν B (a)] [ν A (a) ν B (a)] = ν A B (a). Thus A B AΓB. Hence AΓB = A B. References [1] Atanassov. K., 1986, Intuitionistic fuzzy sets, Fuzzy sets and systems, 20(1), pp [2] Booth. G.L., 1988, A note on Γ -near-rings Stud. Sci. Math. Hung. 23, pp

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