In this paper, we introduce the intuitionistic fuzzy bi ideals in G-near-rings and investigate some of their related properties

DHANDAPANI A

EZHILMARAN D

English

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Vol 5, Issue 1, 2017 ISSN - 2347-1573 ON INTUITIONISTIC FUZZY BI-IDEALS IN GAMMA NEAR RINGSThe fuzzy set theory developed by Zadeh and others has found many applications in the domain of mathematics. Gamma near-rings were defined by Satyanarayana and the ideal theory in gamma near rings was studied by Satyanarayana and Booth. In this paper, we introduce the intuitionistic fuzzy bi-ideals in G-near-rings and investigate some of their related properties.Keywords: G-near-rings, Intuitionistic fuzzy ideals, Intuitionistic fuzzy bi-ideals.1. INTRODUCTIONFollowing the introduction of fuzzy sets by Zadeh [20], the fuzzy set theory has been used for many applications in the domain of mathematics and elsewhere. The idea of “Intuitionistic Fuzzy Set (IFS)” was first published by Atanassov [1] as a generalization of the notion of fuzzy set. The concept of G-near-ring, a generalization of both the concepts nearing and G-ring, was introduced by Satyanarayana [16,17]. Later, several authors such as Booth and Satyanarayana [2,3,5-7,10-14,19] studied the ideal theory of G-near-rings. Later Jun et al. [8,9] considered the fuzzification of left (resp. right) ideals of G-near-rings. In this paper, we introduce the notion an intuitionistic fuzzy bi-ideal in a G-near-ring and some properties of such bi-ideals are investigated. The homomorphic property of intuitionistic fuzzy bi-ideals is established.2. PRELIMINARIESIn this section, we include some elementary aspects that are necessary for this paper.Definition 2.1 [16]: A nonempty 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. (R,.) is a semigroup,iii. (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 [17]: A G-near-ring is a triple (M, +, G) where,i. (M, +) is a group,ii. G is a nonempty set of binary operators on M such that for each a ∈ G, (M, +, a) is a near-ring,iii. x a (y b z) = (x a y)b z for all x, y, z ∈ M and a, b ∈ G.Definition 2.3 [17]: A subset A of a G-near-ring M is called a left (resp. right) ideal of M ifi. (A, +) is a normal divisor of (M, +),ii. u a (x + v) - uav ∈ A (resp. x a u ∈ A) for all x ∈ A, a ∈ G and u, v ∈ M.Definition 2.4 [18]: Let M be G-near-ring. A subgroup A of M is called a bi-ideal of M if (AGMGA)∩ (AGM) G*A⊆A. where the operation “*” is defined by,AG*B = {aγ(a’ + b) − aγa’)/a, a’ ∈ A, γ ∈ G, b ∈ B}.Definition 2.5 [17]: Let M be G-near-ring. A subgroup Q of M is called a quasi-ideal of Mif (QGM) ∩ (MGQ) ∩ (MG)*Q ⊆ Q.Definition 2.6 [9]: Let M and N be G-near-rings. A mapping f: M → N is said to be a homomorphism if f(a a b) = f(a) a f(b) for all a, b ∈ M and a ∈ G.Definition 2.7 [9]: A fuzzy set m in a G-near-ring M is called a fuzzy left (resp. right) ideal of M if,i. m(x−y) ≥ min{m(x), m(y)},ii. m(y + x−y) ≥ m(x), for all x, y ∈ M,iii. m(u a (x + v) − u a v) ≥ m(x) (resp. m(x a u) ≥ m (x)) for all x, u, v ∈ M and a ∈ G.Definition 2.9 [1]: Let X be a nonempty fixed set. An IFS A in X is an object having the form A = {< x, mA(x), nA(x) >/x ∈ X}, where the functions mA: X → [0, 1] and nA: X → [0, 1] denote the degree of membership and degree of nonmembership of each element x ∈ X to the set A, respectively, and 0 ≤ mA(x) + nA(x) ≤ 1.Notation: For the sake of simplicity, we shall use the symbol A= <mA, nA > for the IFS,A = {<x, mA(x), nA(x) >/x ∈ X}.Definition 2.10 [1]: Let X be a nonempty set and let A = <mA, nA> mn and B = <mB, nB> be IFSs in X. Then:1. A ⊂ B if mA ≤ mB and nA ≥ nB2. A = B if A ⊂ B and B ⊂ A3. Ac = <nA, mA>4. A ∩ B = (mA∧mB, nA∨nB)5. A ∪ B = (mA∨mB, nA∧nB)6. □A = (mA, 1−mQ), ◊A = (1−nA, nA)Definition 2.11 [14]: Let A be an IFS in a G-near-ring M. For each pair <τ, σ> ϵ [0, 1] ∈with t + s ≤ 1, the set A<t, s> = {x ∈X/mA(x) ≥ t and nnA(x) ≤  s} is called a <t, s>  level subset of A.Definition 2.12 [14]: Let A = <mA,nA>mn be an IFS in M and let t ∈ [0, 1]. Then the sets U(mA;t) = {x∈M: mA(x) ≥ t} and Ln(nA; t) = {x∈M: nA(x) ≤ t} are called upper level set and lower level set of A, respectively.3. INTUITIONISTIC FUZZY BI-IDEALS OF Γ-NEAR-RINGSIn what follows, M will denote a G-near-ring unless otherwise specified.Definition 3.1: An intuitionistic fuzzy ideal A = <µA, nA> of M is called an intuitionistic fuzzy bi-ideal of M if,Research ArticleReceived: 25 November 2016, Revised and Accepted: 22 December 2016ABSTRACT1Department of Mathematics, School of Advanced Sciences, VIT University, Vellore - 632 014, Tamil Nadu, India. 2Department of Mathematics, Salalah College of Technology, Salalah, Sultanate of Oman. Email: ezhil.devarasan@yahoo.com, swettha@gmail.comEZHILMARAN D1*, DHANDAPANI A2Innovare Journal of Eng. & Tech, Vol 5, Issue 1, 2017, 1-4 Ezhilmaran and Dhandapani 2i. mA(x−y) ≥ {mA(x) ∧ mA(y)},ii. mA(y + x−y) ≥ mA(x),iii. mA((x α y β z) ∧ (x a (y + z) – x a z)) ≥{mA(x) ∧ mA(z)}for all x, y, z ∈ M, α, β ∈ G.iv. nA(x−y) ≤ {nA(x) ∨ nA(y)},v. nA(y + x−y) ≤ nA(x),vi. nA((x α y β z) ∨ (x a (y + z) – x a z)) ≤ n{nA(x) ∨ nA(z)} for all x, y, z ∈ M, α, β ∈ G.Example 3.2.: Let R be the set of all integers then R is a ring.Take M = G = R. Let a, b ∈ M,  a ∈ G, suppose aab is the product of a, a, b ∈ R.Then M is a G -near-ring.Define an IFS A = <µA, nA>mn in R as follows.mA(0) = 1 and mA(±1) = mA(±2) = ±mA(±3) = .... = t andnA(0) = 0 and ±nA(±1) = nA(±2) = nA(±3) = .... = s,Where, t ∈ [0, 1), s ∈ (0, 1] and t + s ≤ 1.By routine calculations,Clearly, A is an intuitionistic fuzzy bi-ideal of a G-near-ring M.Lemma 3.3.: If B is a bi-ideal of M then for any 0 < t, s < 1, there exists an intuitionistic fuzzy bi-ideal C = < mC, nC > of M such that C<t, s> = B.Proof: Let C → [0, 1] be a function defined by{      t       if  x     0       if  x ,∈= ∉m B,(x)B B  nB x BB,( ).= ∉∈{      1       if  y     s       if  yFor all x ∈ M and the pair s, t ∈ [0, 1]. Then C<t, s> = B is an intuitionistic fuzzy bi-ideal of M with t + s ≤ 1.Now suppose that B is a bi-ideal of M. For all x, y ∈ B, such that,x−y ∈ B, We have,µc(x−y) ≥ t = {µc(x) ∧ µc(y)},nc(x−y) ≤ s = {nc(x) ∨ nc(y)},µc(y + x−y) ≥ t = µc(x),nc(y + x−y) ≤ s = nc(x),Also, for all x, y, z ∈ B and α, β ∈ G such that xαyβz ∈ B, we have,µc((x α y β z) ∧ (x a (y + z) – x a z))) ≥ t = {µc(x) ∧ µc(z)},nc((x α y β z) ∨ (x a (y + z) –x a z))) ≤ s = {nc(x) ∨ nc(z)}.Thus C<t, s> is an intuitionistic fuzzy bi-ideal of M.Lemma 3.4.: Let B be a nonempty subset of M. Then B is a bi-ideal of M if and only if the IFS B̄ = <cB, χ̄B > is an intuitionistic fuzzy ideal of M.Proof: Let x, y ∈ B. From the hypothesis, x−y ∈ B.i. If x, y ∈ B, then χB(x) = 1, χ̄B(x) = 0, χB(y) = 1 and χ̄B(y) = 0. In this case,χB(x−y) = 1 ≥ {χB(x) ∧ χB(y)}.χ̄B(x−y) = 0 ≤ {χ̄B(x) ∨ χ̄B(y)}.ii. If x ∈ B, y ∉ B, then χB(x) = 1, χ̄B(x) = 0, χB(y) = 0, and χ̄B(y) = 1. Thus,χB(x−y) = 0 ≥ {χB(x) ∧ χB(y)}χ̄B(x−y) = 1 ≤ {χ̄B(x) ∨ χ̄B(y)}.iii. If x ∉ B, y ∈ B, then χB(x) = 0, χ¯B(x) = 1 and χB(y) = 1 and χ¯B(y) = 0. Thus,χB(x−y) = 0 ≥ {χB(x) ∧ χB(y)}.χ̄B(x−y) = 1 ≤ {χ̄B(x) ∨ χ̄B(y)}.iv. If x ∉ B, y ∉ B, then χB(x) = 0, χ̄B(x) = 1, χB(y) = 0 and χ̄B(y) = 1. Thus,χB(x−y) ≥ 0 = {χB(x) ∧ χB(y)}.χ̄B(x−y) ≤ 1 = {χ̄B(x) ∨ χ̄B((y)}.Thus (i) of Definition 3.1 holds good.Let x, y ∈ B. From the hypothesis, y + x−y ∈ B.i. If x, y ∈ B, then χB(x) = 1, χ̄B(x) = 0, χB(y) = 1 and χ̄B(y) = 0. In this case,χB(y + x–y) = 1 ≥ χB(x).χ̄B(y + x–y) = 0 ≤ χ̄B(x).ii. If x ∈ B, y ∉ B, th χ̄Ben χB(x) = 1, χ̄B(x) = 0, χB(y) = 0 and χ̄B(y) = 1. Thus,χB(y + x–y) = 0 ≥ χB(x).χ̄B(y + x–y) = 1 ≤ χ̄B(x).iii. If x ∉ B, y ∈ B, then χB(x) = 0, χ̄B(x) = 1, χB(y) = 1 and χ̄B(y) = 0. Thus,χB(y + x−y) = 0 ≥ χB(x).χ̄B(y + x−y) = 1 ≤ χ̄B(x).iii. If x ∉ B, y ∉ B, then χB(x) = 0, χ̄B(x) = 1, χB(y) = 0 and χ̄B(y) = 1.Thus,χB(y + x−y) ≥ 0 = χB(x).χ̄B(y + x−y) ≤ 1 = χ̄B(x).Thus (ii) of Definition 3.1 holds good.Let x, y, z ∈ B and α, β ∈ G. From the hypothesis, x α y β z, x a (y + z) – x a z ∈ B.i. If x, z ∈ B, then χB(x) = 1, χ̄B(x) = 0, χB(z) = 1 and χ̄B(z) = 0. Thus,χB(µ((x α y β z) ∧ (x a (y + z) – x a z))) = 1 ≥ {χB(x) ∧ χB(z)}.χ̄B(µ((x α y β z) ∨ (x a (y + z) – x a z))) = 0 ≤ {χ̄B(x) ∨ χ̄B(z)}.ii. If x ∈ B, z ∉ B, then χB(x) = 1, χ̄B(x) = 0, χB(z) = 0 and χ̄B(z) = 1. Thus,χB(µ((x α y β z) ∧ (x a (y + z) – x a z))) = 0 ≥ {χB(x) ∧ χB(z)}.χ̄B(µ((x α y β z) ∨ (x a (y + z) – x a z))) = 1 ≤ {χ̄B(x) ∨ χ̄B(z)}.iii. If x ∉ B, z ∈ B, then χB(x) = 0, χ̄B(x) = 1, χB(z) = 1 and χ̄B(z) = 0. Thus,χB(µ((x α y β z) ∧ (x a (y + z) – x a z))) = 0 ≥ {χB(x) ∧ χB(z)}.χ̄B(µ((x α y β z) ∨ (x a (y + z) – x a z))) = 1 ≤ {χ̄B(x) ∨ χ̄B(z)}.iv. If x ∉ B, z ∉ B, then χB(x) = 0, χ̄B(x) = 1, χB(z) = 0 and χ̄B(z) = 1. Thus,χB(µ((x α y β z) ∧ (x a (y + z) – x a z))) ≥ 0 = {χB(x) ∧ χB(z)}.χ̄B(µ((x α y β z) ∨ (x a (y + z) – x a z))) ≤ 1 = {χ̄B(x) ∨ χ̄B(z)}.Thus (iii) of Definition 3.1 holds good.Conversely, suppose that IFS B̄ = <cB, χ̄B> is an intuitionistic fuzzy ideal of M. Then by Lemma 3.3, χB is two-valued, Hence B is a bi-ideal of M.This completes the proof.Theorem 3.5.: If {Ai}i∈∧ is a family of intuitionistic fuzzy bi-ideals of M then ∩Ai is an intuitionistic fuzzy bi-ideals of M, where ∩Ai = {ΛµAi, VνAi},ΛµAi (x) = inf {µAi (x)| i ∈ Λ, x ∈ M} and VνAi(x) = sup {νAi (x) | i ∈ V, x ∈ M}.Proof: Let x, y ∈ M. Then we have,ΛµAi(x − y) = inf {{µAi(x) ∧ µAi(y)}|i ∈ Λ, x, y ∈ M}  = {{inf (µAi(x)) ∧ inf (µAi(y))}|i ∈ Λ, x, y ∈ M}  = {{inf (µAi(x)|i ∈ Λ, x ∈ M)} ∧ {inf (µAi(y)|i ∈ Λ, y ∈ M)}}  = {ΛµAi(x) ∧ ΛµAi(y)}.Let x, y ∈ M. Then we have,ΛµAi(y + x−y) = inf {µAi(x)|i ∈ Λ, x, y ∈ M}        = ΛµAi(x).Let x, y, z ∈ M and α, β ∈ G.ΛµAi((x α y β z) ∧ (x a (y + z) – x a z)) = inf{{µAi(x) ∧ µAi(z)}|i ∈ Λ, x, z ∈ M} = {{inf (µAi(x)) ∧ inf (µAi(z))}|i ∈ Λ, x, z ∈ M} = {{inf (µAi(x)|i ∈ Λ, x ∈ M)} ∧ {inf (µAi|i ∈ Λ, z ∈ M)}} = {ΛµAi(x) ∧ ΛµAi(z)}.Let x, y ∈ M. Then we have,VνAi(x − y) = sup {{νAi(x) ∨ νAi(y)}|i ∈ V, x, y ∈ M} = {{sup (νAi(x)) ∨ sup (νAi(y))}|i ∈ V, x, y ∈ M} = {{sup (νAi(x)|i ∈ V, x ∈ M)} ∨ {sup (νAi(y)|i ∈ V, y ∈ M)}} = {VνAi(x) ∨ VνAi(y)}.Let x, y ∈ M. Then we have,VνAi(y + x−y) = sup{νAi(x)|i ∈ V, x, y ∈ M}       = VνAi(x).Innovare Journal of Eng. & Tech, Vol 5, Issue 1, 2017, 1-4 Ezhilmaran and Dhandapani 3Let x, y, z ∈ M and α, β ∈ G.VνAi((x α y β z) ∨ (x a (y + z) – x a z)) = sup{{νAi(x) ∨ νAi(z)}|i ∈ V, x, z ∈ M}  = {{sup(νAi(x)) ∨ sup(νAi(z))}|i ∈ V, x, z ∈ M}  = {{sup(νAi(x)|i ∈ V, x ∈ M)} ∨ {sup(νAi(z)|i ∈ Λ, z ∈ M)}}  = {VνAi(x) ∨ VνAi(z)}.Hence, ∩Ai = {ΛµAi, VνAi} is an intuitionistic fuzzy bi-ideal of M.Theorem 3.6.: If A is an intuitionistic fuzzy bi-ideal of M then A′ is also an intuitionistic fuzzy bi-ideal of M.Proof: Let x, y ∈ M. We have, µA' (x−y) = 1 − µA(x−y)         =1 − {µA(x) ∧ µA(y)}, nA'(x−y) = 1 − nA(x−y)         = 1 − {nA(x) ∨ nA(y)}.Let x, y ∈ M. We have, µA' (y + x − y) = 1 − µA(y + x − y)                  = 1 − µA(x)                  = µA' (x), nA' (y + x − y) = 1 − nA(y + x − y)                  = 1 − nA(x)                  = nA' n(x).Let x, y, z ∈ M and α, β ∈ G. We haveµA' ((x α y β z) ∧ (x a (y + z) – x a z)) = 1 − µA((x α y β z) ∧ (x a (y + z) – x a z))  = 1−{µA(x) ∧ µA(z)}  = {1−µA(x) ∧ 1−µA(z)}  = { µA' (x) ∧ µA' (z)},nA' ((x α y β z) ∨ (x a (y + z) – x a z)) = 1 − nA((x α y β z) ∨ (x a (y + z) –x a z))  = 1 − {nA(x) ∨ nA(z)}  = {1 − nA(x) ∨ 1 − nA(z)}  = {nA' n(x) ∨ nA' n(z)}.Therefore, A′ is also an intuitionistic fuzzy bi-ideal of M.Theorem 3.7: An IFS A of M is an intuitionistic fuzzy bi-ideal of M if and only if the level setsU(µA; t) = {x ∈ M|µ(x) ≥ t} andL(nA; t) = {x ∈ M: nA(x) ≤ t} are a bi-ideal of M when it is non-empty.Proof: Let A be an intuitionistic fuzzy bi-ideal of M.Then mA(x−y) ≥ {mA(x)  ∧ mA(y)}.x, y ∈ U(mA; t)⇒ mA(x) ≥ t, mA(y) ≥ t                  mA(x−y) ≥ {mA(x)  ∧ mA(y)} ≥ t                  mA(x−y) ≥ t                  ⇒ x−y ∈ U(mA; t).mA(x−y) ≥ {mA(x)  ∧ mA(y)}.x, y ∈ L(nA; t)⇒ nA(x) ≤ t, nA(y) ≤ t                  nA(x − y) ≤ {nA(x) ∨ nA(y)} ≤ t                  nA(x − y) ≤ t                  ⇒ x − y ∈ L(nA; t).Let mA(y + x−y) ≥ mA(x). x, y ∈ U(mA; t)⇒ mA(x) ≥ t, mA(y) ≥ t mA(y + x−y) ≥ mA(x) ≥ t mA(y + x−y) ≥ t ⇒ y + x−y ∈ U(mA; t).Let nA(y+ x−y) ≤ nA(x).x, y ∈ L(nA; t)⇒ nA(x) ≤ t, nA(y) ≤ t nA(y + x−y) ≤ nA(x) ≤ t nA(y + x−y) ≤ t ⇒ y + x−y ∈ L(nA; t).Also, let,mA((x α y β z) ∨ (x a (y + z) – x a z)) ≥ {mA(x)  ∧ mA(z)}.x, y, z ∈ U(mA; t), α, β ∈ G ⇒ mA(x) ≥ t, mA(y) ≥ t, mA(z) ≥ tmA((x α y β z) ∧ (x a (y + z) – x a z)) ≥ {mA(x)  ∧ mA(z)} ≥ tmA ((x α y β z) ∧ (x a (y + z) – x a z)) ≥ t⇒ (x α y β z), (x a (y + z) – x a z) ∈ U (mA; t).Thus, U (mA; t) is a bi-ideal of M.nA((x α y β z) ∨ (x a (y + z) – x a z)) ≤ {nA(x) ∨ nA(z)}.x, y, z ∈ L(nA; t), α, β ∈ G ⇒ nA(x) ≤ t, nA(y) ≤ t, nA(z) ≤ tnA((x α y β z) ∨ (x a (y + z) – x a z)) ≤ {nA(x) ∨ nA(z)} ≤ tnA((x α y β z) ∨ (x a (y + z) – x a z)) ≤ t⇒ (x α y β z), (x a (y + z) – x a z) ∈ L(nA; t).Thus, L(nA; t) is a bi-ideal of M.Conversely, if U (mA; t) is a bi-ideal of M let t = {mA(x)  ∧ mA(y)}. Thenx, y ∈ U (mA; t), ⇒ x − y ∈ U (mA; t)⇒ mA(x−y) ≥ t⇒ mA(x−y) ≥ {mA(x)  ∧ mA(y)}.Also, x, y ∈ U (mA; t), ⇒ y + x − y ∈ U (mA; t)⇒ mA(y + x−y) ≥ mA(x).If L(nA; t) is a bi-ideal of M let t = {nA(x) ∨ nA(y)}. Thenx, y ∈ L(nA; t), ⇒ x − y ∈ L(nA; t)⇒ nA(x − y) ≤ t⇒ nA(x − y) ≤ {nA(x) ∨ nA(y)}.Also, x, y ∈ L(nA; t), ⇒ y + x − y ∈ L(nA; t)⇒ nA(y + x − y) ≤ nA(x).Next, define t = {mA(x)  ∧ mA(z)}. Thenx, y, z ∈ U (µ; t), α, β ∈ G⇒(x α y β z), (x a (y + z) – x a z) ∈ U (mA; t)⇒ mA((x α y β z) ∧ (x a (y + z) – x a z)) ≥ t⇒ mA((x α y β z) ∧ (x a (y + z) – x a z)) ≥ {µ(x)  ∧ µ(z)}.Next, define t = {nA(x) ∨ nA(z)}. Thenx, y, z ∈ U (µ; t), α, β ∈ G ⇒ (x α y β z), (x a (y + z) – x a z) ∈ L(nA; t)⇒ nA((x α y β z) ∨ (x a (y + z) – x a z)) ≤ t⇒ nA((x α y β z) ∨ (x a (y + z) – x a z)) ≤ {nA(x) ∨ nA(z)}.Consequently, A is an intuitionistic fuzzy bi-ideal of M.Theorem 3.8: Let A be an intuitionistic fuzzy bi-ideal of M. If M is completely regular, then mA(a) = mA(a α a), nA(a) = nA(a α a) for all a ∈ M and α ∈ G.Proof: Straight forward.Let f be mappings from a set X to Y, and A be IFS on Y. Then the preimage of µ under f, denoted by f−1(A), is defined by:f−1(mA(x)) = mA(f(x)), f−1(nA(x)) = nA(f(x)) for all x ∈ X.Theorem 3.9: Let the pair of mappings f: M → N be a homomorphism of G-near-rings.If µ is an intuitionistic fuzzy bi-ideal of N, then the preimage f−1 (A) of A under f is an intuitionistic fuzzy bi-ideal of M.Proof: Let x, y ∈ M. Then we havef−1 (mA)(x−y) = mA(f(x−y))= mA(f(x)−f(y))Innovare Journal of Eng. & Tech, Vol 5, Issue 1, 2017, 1-4 Ezhilmaran and Dhandapani 4≥ {mA(f(x))  ∧ mA(f(y))}= {f−1 (mA(x))  ∧ f−1 (mA(y))}.f−1 (nA)(x−y) = nA(f(x−y))n= nA(f(x)−f(y))≤ {nA(f(x)) ∨ nA(f(y))}= {f−1 (nA(x)) ∨ f−1 (nA(y))}.Let x, y ∈ M. Then we havef−1 (mA)(y + x−y) = mA(f(y + x−y))≥ mA(f (x))= f−1 (mA(x)).f−1 (nA)(y + x−y) = nA(f (y + x−y))≤ nA(f(x))= f−1 (nA(x)).Let x, y, z ∈ M and α, β ∈ G. Then: f−1(mA)((x α y β z) ∧ (x a (y + z) – x a z)) = mA(f ((x α y β z) ∧ (x a (y + z) – x a z)))= mA((f(x α y β z)) ∧ (f(x a (y + z) – x a z)))≥ mA(f(x))  ∧ mA(f(z))}={f−1 (mA(x))  ∧ f−1 (mA(z))}.Therefore, f−1 (mA) is an intuitionistic fuzzy bi-ideal of M.f−1(nA)((x α y β z) ∨ (x a (y + z) – x a z)) = nA(f((x α y β z) ∨ (x a (y + z) – x a z)))= nA((f(x α y β z)) ∨ (f(x a (y + z) – x a z)))≤ {nA(f(x)) ∨ nA(f (z))}= {f−1 (nA (x)) ∨ f−1 (nA(z))}.Therefore, f−1 (nA) is an intuitionistic fuzzy bi-ideal of M.Therefore, f−1 (A) is an intuitionistic fuzzy bi-ideal of M.1. Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20(1):87-96.2. Anthony JM, Sherwood H. Fuzzy groups redefined. J Math Anal Appl 1979;69:124-30.3. Ali IS, Raja K. Fuzzy logic based ZSI using PMSG for WECS. Innov J Eng Technol 2013;1(1):12-6.4. Barnes WE. On the rings of Nobusawa. Pac J Math 1966;18(3):411-22.5. Booth GL. A note on G-near-rings. Stud Sci Math Hung 1988;23:471-5.6. Chinram R. Generalized transformation semigroups whose sets of quasi-ideals and bi-ideals coincide. Kyungpook Math J 2005;45(2):161-6.7. Dixit VN, Kumar R, Ajmal N. On fuzzy rings. Fuzzy Sets Syst 1992;49(2):205-13.8. Jun YB, Lee CY. Fuzzy G-rings. Pusan Kyongnam Math J 1992;8:163-70.9. Jun YB, Sapanci M, Ozturk MA. Fuzzy ideals in gamma near-rings. Turk J Math 1998;22(1998):449-59.10. Kin KH, Ozturk MA. Fuzzy maximal ideals of gamma near-rings. Turk J Math 2001;25(2001):457-63.11. Kuroki N. Fuzzy bi-ideals in semigroups. Comment Math Univ St Paul 1980;28(1):17-21.12. Kuroki N. On fuzzy ideals and fuzzy bi-ideals in semigroups. Fuzzy Sets Syst 1981;5(2):203-15.13. Liu W. Fuzzy invariant subgroups and fuzzy ideals. Fuzzy Sets Syst 1982;8(2):133-9.14. Palaniappan N, Veerappan PS, Ezhilmaran D. Some properties of intuitionistic fuzzy ideals in G-near-rings. J Indian Acad Math 2009;31(2):617-24.15. Rosenfeld A. Fuzzy groups. J Math Anal Appl 1971;35(3):512-7.16. Satyanarayana BH. Contributions to Near-Ring Theory Doctoral Thesis. India: Nagarjuna University; 1984.17. Satyanarayana BH. A note on G-near-rings. Indian J Math 1999;41:427-33.18. Chelvam TT, Meenakumari N. Bi-Ideals of gamma near rings. Southeast Asian Bull Math 2004;27(6):983-8.19. Thakur S, Raw SN, Sharma R. Design of a fuzzy model for thalassemia disease diagnosis: Using Mamdani type fuzzy inference system (FIS). Int J Pharm Pharm Sci 2016;8:356-61.20. Zadeh LA. Fuzzy sets. Inf Control 1965;8:338-53.REFERENCES

ON INTUITIONISTIC FUZZY BI-IDEALS IN GAMMA NEAR RINGS

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ON INTUITIONISTIC FUZZY BI-IDEALS IN GAMMA NEAR RINGS

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