On Intuitionitic Fuzzy Maximal Ideals of. Gamma Near-Rings

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1 International Journal of Algebra, Vol. 5, 2011, no. 28, On Intuitionitic Fuzzy Maximal Ideals of Gamma Near-Rings D. Ezhilmaran and * N. Palaniappan Assistant Professor, School of Advanced Sciences, VIT University, Vellore , Tamilnadu, India. ezhil.devarasan@yahoo.com Professor, School of Mathematics, *Alagappa University, Karaikudi , Tamilnadu, India. palaniappan.nallappan@gmail.com Abstract In this paper, we introduce the notion of intuitionistic fuzzy maximal ideals and complete normal intuitionistic fuzzy ideals in Γ-near-rings and some related properties are investigated. Mathematics Subject Classification: 06F35, 03G25, 03E72, 16Y30 Keywords: intuitionistic fuzzy maximal ideals and complete normal intuitionistic fuzzy ideals 1. Introduction -Near-rings were defined by Bh.Satyanarayana [8] and G.L.Booth [2, 3] studied the ideal theory in -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. After the introduction of the concept of fuzzy sets by L. A. Zadeh [11], several researches were conducted on the generalization of the notion of fuzzy sets. Y.B. Jun [5, 6] introduced the notion of fuzzy left (resp.right) ideals, normal fuzzy ideals and fuzzy maximal ideals of -Near-rings, and studied some of their properties. The idea of intuitionistic fuzzy set was first published by K.T.Atanassov [1], as a generalization of the notion of fuzzy set. In this paper, we introduce the notion of intuitionistic fuzzy maximal ideals and complete normal intuitionistic fuzzy ideals in Γ-near-rings and some related properties are investigated.

2 1406 D. Ezhilmaran and N. Palaniappan 2. Preliminaries paper. In this section we include some elementary aspects that are necessary for this 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) (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 A -near-ring is a triple (M, +, ) where (i) (M, +) is a group, (ii) is a nonempty set of binary operators on M such that for each, ( M, +, ) is a near ring, (iii) x (y z) = (x y) z for all x, y, z M and,. Definition 2.3 A subset A of a -near-ring M is called a left (resp. right) ideal of M if (i) (A, +) is a normal divisor of (M, +), (ii) u (x + v) -u v A ( resp. x u A ) for all x A, and u, v M. Definition 2.4 A fuzzy set in a -near-ring M is called a fuzzy left (resp.right) ideal of M if (i) (x - y) min{ (x), (y) }, (ii) (y + x - y) (x), for all x, y M. (iii) (u (x + v) - u v) (x) (resp. (x u) (x)) for all x, u, v M and. 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, A (x), A (x) >/ x X }, where the functions A : X [ 0, 1] and 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 A (x) + A (x) 1. Notation. For the sake of simplicity, we shall use the symbol A= < A, A > for the IFS A = {< x, A (x), A (x) > /x X}. Definition 2.6[1] Let X be a non empty set and let A = < A, A > and B = < B, B > be IFSs in X. Then (1) A B iff A B and A B. (2) A = B iff A B and B A. (3) A c = < A, A, >.

3 On intuitionitic fuzzy maximal ideals 1407 (4) A B = ( A B, A B ). (5) A B = ( A B, A B ). 3. Intuitionistic fuzzy maximal ideals of Γ-near-rings In what follows let M denote a Γ-near-ring unless otherwise specified. Definition 3.1[8] An IFS A = in is called an intuitionistic fuzzy left resp. right ideal of a near-ring M if i x y x y ii y + x y x ii u x + v) u v x resp. x u x iv x y x y v y + x y x (vi) u x + v) u v x resp. x u x for all x y, u, v and Definition 3.2 Let A be an intuitionistic fuzzy subset in a -near-ring M. For each pair t, s [0, 1], the set A < t, s > = {x X : A (x ) t and A (x) s} is called the level set of A, where t + s 1. Proposition 3.3. If A is an intuitionistic fuzzy left (resp. right) ideal of a -near-ring M, then A (0) A (x) and A (0) A (x) for all x M where 0 is the zero element of M. Proposition 3.4. Let M be a -near-ring and A be an intuitionistic fuzzy left (resp. right) ideal of M. Then the set M A = {x M. A (x) = A (0) and A (x) = A (0)}is a left (resp. right) ideal of M. Proof. Let A be an intuitionistic fuzzy left ideal and let x, y M A. A (x - y) x y = A (0) and A (x - y) x y = A (0) So A (x - y) = A (0) and A (x - y) = A (0) or x - y M A. For every y M and x M A, we have A (y + x - y) A (x) = A (0) and A (y + x - y) A (x) = A (0). Hence y +x - y M A which shows that M A is a normal divisor of M with respect to addition. Let x M A, and u, v M. Then A (u (x + v) - u v) A (x) = A (0) and A (u (x + v) - u v) A (x) = A (0) That is u (x +v) - u v) M A. Therefore M A is a left ideal of M. Similarly we have the desired result for the right case Definition 3.5 An intuitionistic fuzzy ideal A of M is said to be normal if µ A (0) = 1, A (0) = 0.

4 1408 D. Ezhilmaran and N. Palaniappan Example 3.6 Let R be the set of all integers then R is a ring. Take M = = R. Let a, b M, suppose a b is the product of a,, b R. Then M is a -near-ring. Define an IFS A = in R as follows. A (0) = 1 and A (±1) = A (±2) = A (±3) = = 0 and A (0) = 0 and A (±1) = A (±2) = A (±3) = = 1 where t, s [0, 1] and t + s 1. By routine calculation A is a normal intuitionistic fuzzy ideal of M. Lemma 3.7 Let A be an intuitionistic fuzzy subset of a -near-ring M defined by for all x M and the pair s, t [ 0, 1]. Then A is a normal intuitionistic fuzzy ideal of M and M A = A with t + s 1. Theorem 3.8. Let C and D be normal intuitionistic fuzzy ideals of M, then C D is an ideal. Proof. Straightforward. Lemma 3.9 If A is an intuitionistic fuzzy ideal of M satisfying A * (x) = (0, 1) for some x M, then µ A (x) = 0, A (x) = 1. Lemma 3.10 An intuitionistic fuzzy ideal A of M is normal if and only if A * = A. Using a given intuitionistic fuzzy ideal A of M, we will construct a new intuitionistic fuzzy ideal. Let t > 0 be a real number, and define a mapping A t : M [0, 1] by µ t A (x) = (µ A (x)) t, t A (x) = ( A (x)) t. for all x M, where (µ A (x)) t = 2t µ A(x), ( A (x)) t = 2t A(x) when 0 < t < 1. Theorem Let t > 0 be a real number. If A is a normal intuitionistic fuzzy ideal of M, then A t is also a normal intuitionistic fuzzy ideal of M and M A t = M A. Proof. For any x, y M, we have t A (x y) = ( A (x y)) t { A (x) A (y)} t = {( A (x)) t ( A (y)) t } = { t A (x) t A (y)}, t A (x y) = ( A (x y)) t { A (x) A (y)} t = {( A (x)) t ( A (y)) t t t } = { A (x) A (y)}. and t A (y + x y) = ( A (y + x y)) t ( A (x)) t = t A (x), t A (y + x y) = ( A (y + x y)) t ( A (x)) t = t A (x). Let x, u, v M and α Γ. Then t A (uα(x + v) uαv) = ( A (uα(x + v) uαv)) t ( A (x)) t = A t (x),

5 On intuitionitic fuzzy maximal ideals 1409 t A (uα(x + v) uαv) = ( A (uα(x + v) uαv)) t ( A (x)) t = t A (x). t Note that A (0) = ( A (0)) t = 1 t = 1 & t A (0) = ( A (0)) t = 0 t = 0. Hence A t is a normal intuitionistic fuzzy ideal of M. Now M t A = x M t A (x) = t A (0), t A (x) = t A (0)} = {x M ( A (x)) t = 1, ( A (x)) t = 0} = {x M A (x) = 1, A (x) = 0} = {x M A (x) = A (0), A (x) = ( A (0)} = M A. This completes the proof. Let I (M) (resp. N(M)) denote the set of all ideals (resp. normal intuitionistic fuzzy ideals) of M. We define functions φ : I(M) N (M) and ψ : N (M) I (M) by φ(a) = A + = < A, A > and ψ(a + ) = M A, respectively, for all A I(M) and A + N(M). Then ψφ = 1 I(M) and φψ(a + ) = φ(m A ) = A + M A A +, Theorem If A, B I(M), then A B = A B, A B = A B that is, φ(a B) = φ(a) φ(b). If δ, σ N(M), then M δ σ = M δ M σ, that is, ψ(δ σ) = ψ(δ) ψ(σ). Proof. Let x M. If x A B, then A B (x) = 1, A B (x) = 0. From x A and x B it follows that A (x) = 1 = B (x), A (x) = 0 = B (x) Hence A B (x) = 1 = { A (x) B (x)} = ( A B )(x), A B (x) = 0 = { A (x) B (x)} = ( A B )(x). If x A B, then x A or x B. Thus A B (x) = 0 = { A (x) B (x)} = ( A B )(x), A B (x) = 1 = { A (x) B (x)} = ( A B )(x). Therefore A B = A B, A B = A B and so φ(a B) = φ(a) φ(b) for all A, B I(M). Now let δ, σ N(M). Then M δ σ = {x M δ σ (x) = δ σ (0) and δ σ (x) = δ σ (0)} = {x M { δ (x) σ (x)} = 1 and { δ (x) σ (x)} = 0} = {x M δ (x) = 1= σ (x) and δ (x) = 0 = σ (x)} = {x M δ (x) = 1, δ (x) = 0} {x M σ (x) = 1, σ (x) = 0} ={x M δ (x) = δ (0), δ (x) = δ (0)} {x M σ (x) = σ (0), σ (x) = σ (0)} = M δ M σ that is, ψ(δ σ) = M δ σ = M δ M σ = ψ(δ) ψ(σ). This completes the proof. Definition 3.13 An intuitionistic fuzzy ideal A of M is said to be intuitionistic fuzzy maximal if it satisfies (i) A is non-constant, (ii) A * is a maximal element of (N (M), ).

6 1410 D. Ezhilmaran and N. Palaniappan Lemma 3.14 [4] Let A be a non-constant normal intuitionistic fuzzy ideal of M, which is maximal in the poset of normal intuitionistic fuzzy ideals under set inclusion. Then A takes only the values 0 and 1 and A takes only the values 1 and 0. Theorem If A is an intuitionistic fuzzy maximal ideal of M, then (i) A is normal, (ii) A * takes only the values (0, 1) and (1, 0), (iii) AM A = A, [i.e. M µa = µ A and M νa = ν A ], (iv) M A is a maximal ideal of M. Proof. Let A be a intuitionistic fuzzy maximal ideal of M. Then A * is a non-constant * maximal element of the poset (N (M), ). It follows from Lemma 3.13 that µ A and * ν A takes only the values (0, 1) and (1, 0) respectively. Note that µ * A (x) = 1, ν * A (x) = 0 if and only if µ A (x) = µ A (0), ν A (x) = ν A (0) and µ * A (x) = 0, ν * A (x) = 1 if and only if µ A (x) = µ A (0) 1, ν A (x) = ν A (0). By Lemma 3.8, we have µ A (x) = 0, ν A (x) = 1. that is, µ A (0) = 1, A (0) = 0. Hence A is normal. This proves (i) and (ii). (iii) Clearly, AM A A and AM A, takes only the values (0, 1) and (1, 0) respectively. Let x M. If µ A (x) = 0, ν A (x) = 1 then obviously A AM A. If µ A (x) = 1, ν A (x) = 0, then x M A and so AM A (x) = 1. This shows that A AM A. (iv) M A is a proper ideal of M because A is non-constant. Let C be an ideal of M such that. M A C. Using (iii) and Theorem 3.4, we have A = AM A A C. Since A, A C N(M) and A = A * is a maximal element of N (M), it follows that either A = A C or A C = 1 where 1: M [0, 1] is an intuitionistic fuzzy set defined by µ 1 (x) = 1, ν I (x) = 0 for all x M. The later case implies that C = M. If A = A C, then M A = M AC = C by Lemma 3.14 This proves that M A is a maximal ideal of M. This completes the proof. Definition 3.16 A normal intuitionistic fuzzy ideal A of M is said to be complete if there exists c M such that µ A (c) = 0, ν A (c) = 1. Note that B = is a complete normal intuitionistic fuzzy ideal of M for every ideal B of M. Denote by C(M) the set of all complete normal intuitionistic fuzzy ideals of M. Note that C(M) N(M) and the restriction of the partial ordering of N(M) gives a partial ordering of C(M). Theorem Every non-constant maximal element of (N (M), ) is also a maximal element of (C(M ), ). Proof. Let A be a non-constant maximal element of (N (M), ). By Lemma 3.13, A takes only the values 0 and 1, and in fact µ A (0) = 1, ν A (0) = 0 and µ A (c) = 0, ν A (c) = 1 for some c ( 0) M. Hence A is complete. Assume that there exists B C(M) such that A B. It follows that A B in N(M). Since A is maximal in (N (M), )

7 On intuitionitic fuzzy maximal ideals 1411 and since B is non-constant, therefore A = B. Thus A is a maximal element of (C(M), ). Theorem Every intuitionistic fuzzy maximal ideal of M is complete normal. Proof. Let A be an intuitionistic fuzzy maximal ideal of M. By Theorem 3.14 and Lemma 3.9, A is normal and A = A * takes only the values 0 and 1. Since A is nonconstant and µ A (0) = 1, ν A (0) = 0, it is clear that there exists c( 0) M such that µ A (c) = 0, ν A (c) = 1. Hence A is complete. This completes the proof. References [1] K.Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems 20(1)(1986) [2] G.L.Booth, A note on -near-rings Stud. Sci. Math. Hung. 23(1988), [3] G.L.Booth Jacobson radicals of -near-rings Proceedings of the Hobart Conference, Longman Sci. & technical (1987) 1-12 [4] Y.B. Jun, K. H. Kim and M. A. Ozturk, On fuzzy ideals of gamma near-rings, J. Fuzzy Math. 9(1) (2001), [5] Y.B.Jun, M.Sapanci and M.A.Ozturk, Fuzzy ideals in Gamma near rings, Tr. J. of Mathematics 22(1998) [6] Y. B. Jun, K. H. Kim and M. A. Ozturk, Fuzzy Maximal ideals in gamma nearrings, Tr. J. of Mathematics 25 (2001), [7] W. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8(1982), [8] N. Palaniappan, P. S. Veerappan, D. Ezhilmaran, Some properties of intuitionistic fuzzy ideals in -near-rings, Journal of Indian Acad. Maths(2009), 31(2), [9] N. Palaniappan, P. S. Veerappan, D. Ezhilmaran, A Note on Characterization of intuitionistic fuzzy ideals in -near-rings, Int. J. of Comp. Sci. and Maths.(2011), 3(1), [10] Bh. Satyanarayana, Contributions to near-ring theory, Doctoral Thesis, Nagarjuna Univ [11] L.A.Zadeh. Fuzzy sets, Information and control, 8 (1965),

8 1412 D. Ezhilmaran and N. Palaniappan Received: December, 2010