EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 2, 2019, 622-648 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Direct product of finite intuitionistic anti fuzzy normal subrings over non-associative rings Nasreen Kausar Department of Mathematics, University of Agriculture Faisalabad, Pakistan Abstract. Shal et. al [17], have introduced the concept of intuitionistic fuzzy normal subrings over a non-associative ring. In this paper, we investigate the concept of intuitionistic anti fuzzy normal subrings over non-associative rings and give some properties of such subrings 2010 Mathematics Subject Classifications: 03F55, 08A72, 20N25 Key Words and Phrases: Direct product of (intuitionistic) fuzzy sets, direct product of (intu- itionistic anti) fuzzy LA-subrings, direct product of (intuitionistic anti) fuzzy normal LA-subrings. 1. Introduction In 1972, ageneralization of commutative semigroups has been established by Kazim et. al [9]. In ternary commutative law: abc = cba, they introduced the braces on the left side of this law and explored a new pseudo associative law, that is (ab)c = (cb)a. This law (ab)c = (cb)a, is called the left invertive law. A groupoid S is said to be a left almost semigroup (abbreviated as LA-semigroup) if it satisfies the left invertive law: (ab)c = (cb)a. In [7] (resp. [5]), a groupoid S is said to be medial (resp. paramedial) if (ab)(cd) = (ac)(bd) (resp. (ab)(cd) = (db)(ca)). In [9], an LA-semigroup is medial, but in general an LA-semigroup needs not to be paramedial. Every LA-semigroup with left identity is paramedial in [15] and also satisfies a(bc) = b(ac), (ab)(cd) = (dc)(ba). Kamran [8], extended the notion of LA-semigroup to the left almost group (LA-group). An LA-semigroup G is said to be a left almost group, if there exists left identity e ∈ G such that ea = a for all a ∈ G and for every a ∈ G there exists b ∈ G such that ba = e. Shah et. al [18], discussed the left almost ring (LA-ring) of finitely nonzero functions which is a generalization of commutative semigroup ring. By a left almost ring, we mean a non-empty set R with at least two elements such that (R,+) is an LA-group, (R, ·) is an LA-semigroup, both left and right distributive laws hold. For example, from a commutative ring (R,+, ·) , we can always obtain an LA-ring (R,⊕, ·) by defining for all DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3427 Email address: kausar.nasreen@gmail.com (K. Nasreen) http://www.ejpam.com 622 c© 2019 EJPAM All rights reserved. K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 623 a, b ∈ R, a⊕b = b−a and a ·b is same as in the ring. In fact an LA-ring is a non-associative and non-commutative ring. A non-empty subset A of an LA-ring R is an LA-subring of an LA-ring R if a − b and ab ∈ A for all a, b ∈ A. A is a left (resp. right) ideal of R if (A,+) is an LA-group and RA ⊆ A (resp. AR ⊆ A). A is called an ideal of R if it is both a left ideal and a right ideal of R. After the introduction of fuzzy set by Zadeh [22], several researchers explored on the generalization of the notion of fuzzy set. The concept of intuitionistic fuzzy set was introduced by Atanassov [1, 2], as a generalization of the notion of fuzzy set. Sherwood [20], introduced the concept of product of fuzzy subgroups. After this, further study on this concept continued by Osman [11, 12] and Ray [16]. Zaid [23], gave the idea of normal fuzzy subgroups. An intuitionistic fuzzy set (briefly, IFS) A in a non-empty set X is an object having the form A = {(x, µA(x), γA(x)) : x ∈ X}, where the functions µA : X → [0, 1] and γA : X → [0, 1] denote the degree of membership and the degree of nonmembership, respectively and 0 ≤ µA(x) + γA(x) ≤ 1 for all x ∈ X [1, 2]. An intuitionistic fuzzy set A = {(x, µA(x), γA(x)) : x ∈ X} in X can be identified to be an ordered pair (µA, γA) in IX × IX , where IX is the set of all functions from X to [0, 1]. For the sake of simplicity, we will use the symbol A = (µA, γA) for the IFS A = {(x, µA(x), γA(x)) : x ∈ X}. Intuitionistic fuzzy subrings and intuitionistic fuzzy ideals of a ring have been defined in [3, 6]. Palaniappan et al [13, 14], explored the notions of homomorphism, antihomo- morphism of intuitionistic fuzzy normal subrings and also discussed some properties of intuitionistic fuzzy normal subrings. Moreover intuitionistic fuzzy ring and its homomor- phism image have been investigated by Yan [21]. Shal et al [17], introduced the concept of intuitionistic fuzzy normal subrings over a non-associative ring (LA-ring). We define the direct product of intuitionistic fuzzy sets A1 and A2 of LA-rings R1 and R2, respectively and investigate the some basic properties of intuitionistic anti fuzzy normal LA-subrings of an LA-ring R1 ×R2. We define the direct product of intuitionistic fuzzy sets A1, A2, ..., An of LA-rings R1, R2, ..., Rn, respectively and examine the some fundamental properties of intuitionistic anti fuzzy normal LA-subrings of an LA-ring R1 ×R2 × ...×Rn. Specifically we show that: Let X = A×B and Y = C×D be two LA-subrings of an LA-ring R1×R2. Then X∩Y is an LA-subring of an LA-ring R1 × R2 if and only if the intuitionistic anti characteristic function χZ = 〈µχZ , γχZ 〉 of Z = X ∩ Y is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Let A = A1×A2× ...×An and B = B×B2× ...×Bn be two LA-subrings of an LA-ring R1 ×R2 × ...×Rn. Then A ∩B is an LA-subring of an LA-ring R1 ×R2 × ...×Rn if and only if the intuitionistic anti characteristic function χZ = 〈µχZ , γχZ 〉 of Z = A ∩ B is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Let A and B be intuitionistic fuzzy sets of LA-rings R1 and R2 with left identities e1 and e2, respectively and A × B be an intuitionistic anti fuzzy normal LA-subring of an K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 624 LA-ring R1 ×R2. Then the following conditions are true. 1. If µA (x) ≥ µB (e2) and γA (x) ≤ γB (e2) , for all x ∈ R1, then A is an intuitionistic anti fuzzy normal LA-subring of R1. 2. If µB (x) ≥ µA (e1) and γB (x) ≤ γA (e1) , for all x ∈ R2, then B is an intuitionistic anti fuzzy normal LA-subring of R2. 2. Direct Product of Intuitionistic Anti Fuzzy Normal LA-subrings We define the direct product of intuitionistic fuzzy sets A1, A2 of LA-rings R1, R2, re- spectively and examine the some fundamental properties of direct product of intuitionistic anti fuzzy normal LA-subrings of an LA-ring R1 ×R2. Let µ1 and µ2 be fuzzy subsets of LA-rings R1 and R2, respectively. The direct product of fuzzy subsets µ1 and µ2 is denoted by µ1 × µ2 and defined by (µ1 × µ2)(x1, x2) = min{µ1(x1), µ2 (x2)}. A fuzzy subset µ1 × µ2 of an LA-ring R1 ×R2 is to be a fuzzy LA-subring of R1 ×R2 if 1. (µ1 × µ2)(x− y) ≥ min{µ1(x), µ2(y)}, 2. (µ1 × µ2)(xy) ≥ min{µ1(x), µ2(y)} for all x = (x1, x2) , y = (y1, y2) ∈ R1 ×R2. A fuzzy subset µ1 × µ2 of an LA-ring R1 × R2 is to be an anti fuzzy LA-subring of R1 ×R2 if 1. (µ1 × µ2)(x− y) ≤ max{µ1(x), µ2(y)} 2. (µ1 × µ2)(xy) ≤ max{µ1(x), µ2(y)} for all x = (x1, x2) , y = (y1, y2) ∈ R1 ×R2. A fuzzy LA-subring of an LA-ring R1 × R2 is to be a fuzzy normal LA-subring of R1 × R2 if (µ1 × µ2)(xy) = (µ1 × µ2)(yx) for all x = (x1, x2) , y = (y1, y2) ∈ R1 × R2. Similarly for anti fuzzy normal LA-subring. Let A and B be intuitionistic fuzzy sets of LA-rings R1 and R2, respectively. The direct product ofA andB is denoted byA×B and defined byA×B = {((x, y), µA×B (x, y) , γA×B (x, y)) | for all x ∈ R1 and y ∈ R2}, where µA×B(x, y) = max{µA(x), µB(y)} and γA×B(x, y) = min{γA(x), γB(y)}. An intuitionistic fuzzy set (IFS) A× B = (µA×B, γA×B) of an LA-ring R1 × R2 is an intuitionistic anti fuzzy LA-subring (IAFLSR) of R1 ×R2 if 1. µA×B(x− y) ≤ max{µA×B(x), µA×B(y)}, 2. µA×B(xy) ≤ max{µA×B(x), µA×B(y)}, 3. γA×B(x− y) ≥ min{γA×B(x), γA×B(y)}, 4. γA×B(xy) ≥ min{γA×B(x), γA×B(y)}, for all x = (x1, x2) , y = (y1, y2) ∈ R1 ×R2. An intuitionistic anti fuzzy LA-subring A×B = (µA×B, γA×B) of an LA-ring R1×R2 is an intuitionistic anti fuzzy normal LA-subring (IAFNLSR) of R1 × R2 if µA×B(xy) = µA×B(yx) and γA×B(xy) = γA×B(yx) for all x = (x1, x2) , y = (y1, y2) ∈ R1 ×R2. Let A × B be a non-empty subset of an LA-ring R1 × R2. The intuitionistic anti characteristic function of A×B is denoted by χA×B = 〈µχA×B , γχA×B 〉 and defined by µχA×B (x) = { 0 if x ∈ A×B 1 if x /∈ A×B and γχA×B (x) = { 1 if x ∈ A×B 0 if x /∈ A×B K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 625 Lemma 1. [17, Lemma 4.2] If A and B are LA-subrings of LA-rings R1 and R2, respectively, then A × B is an LA-subring of an LA-ring R1 × R2 under the same operations defined as in R1 ×R2. Proposition 1. Let A and B be LA-subrings of LA-rings R1 and R2, respectively. Then A × B is an LA-subring of an LA-ring R1 × R2 if and only if the intuitionistic anti characteristic function χC = 〈µχC , γχC 〉 of C = A × B is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Proof. Let C = A×B be an LA-subring of an LA-ring R1 ×R2 and a = (a1, a2), b = (b1, b2) ∈ R1×R2. If a, b ∈ C = A×B, then by definition of intuitionistic anti characteristic function µχC (a) = 0 = µχC (b) and γχC (a) = 1 = γχC (b). Since a− b and ab ∈ C, C being an LA-subring of R1 ×R2. This implies that µχC (a− b) = 0 = 0 ∨ 0 = µχC (a) ∨ µχC (b), µχC (ab) = 0 = 0 ∨ 0 = µχC (a) ∨ µχC (b), γχC (a− b) = 1 = 1 ∧ 1 = γχC (a) ∧ γχC (b), γχC (ab) = 1 = 1 ∧ 1 = γχC (a) ∧ γχC (b). Thus µχC (a− b) ≤ max{µχC (a), µχC (b)}, µχC (ab) ≤ max{µχC (a), µχC (b)}, γχC (a− b) ≥ min{γχC (a), γχC (b)}, γχC (ab) ≥ min{γχC (a), γχC (b)}. As ab and ba ∈ C, by definition we have µχC (ab) = 0 = µχC (ba) and γχC (ab) = 1 = γχC (ba), i.e., µχC (ab) = µχC (ba) and γχC (ab) = γχC (ba). Similarly, we have µχC (a− b) ≤ max{µχC (a), µχC (b)}, µχC (ab) ≤ max{µχC (a), µχC (b)}, γχC (a− b) ≥ min{γχC (a), γχC (b)}, γχC (ab) ≥ min{γχC (a), γχC (b)}, γχC (ab) = γχC (ba), γχC (ab) = γχC (ba), when a, b /∈ C. Hence the intuitionistic anti characteristic function χC = 〈µχC , γχC 〉 of C is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Conversely, suppose that the intuitionistic anti characteristic function χC = 〈µχC , γχC 〉 of C = A × B is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 × R2. Let a, b ∈ C = A×B, then by definition, we have µχC (a) = 0 = µχC (b) and γχC (a) = 1 = γχC (b). By our supposition µχC (a− b) ≤ µχC (a) ∨ µχC (b) = 0 ∨ 0 = 0, µχC (ab) ≤ µχC (a) ∨ µχC (b) = 0 ∨ 0 = 0, γχC (a− b) ≥ γχC (a) ∧ γχC (b) = 1 ∧ 1 = 1, K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 626 γχC (ab) ≥ γχC (a) ∧ γχC (b) = 1 ∧ 1 = 1. Thus µχC (a− b) = 0 = µχC (ab) and γχC (a− b) = 1 = γχC (ab), i.e., a− b and ab ∈ C. Hence C is an LA-subring of an LA-ring R1 ×R2. Lemma 2. If X = A× B and Y = C ×D are two LA-subrings of an LA-ring R1 × R2, then their intersection X ∩ Y is also an LA-subring of an LA-ring R1 ×R2. Proof. Straight forward. Theorem 1. Let X = A×B and Y = C×D be two LA-subrings of an LA-ring R1×R2. Then X ∩ Y is an LA-subring of an LA-ring R1 × R2 if and only if the intuitionistic anti characteristic function χZ = 〈µχZ , γχZ 〉 of Z = X ∩ Y is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Proof. Let Z = X ∩ Y be an LA-subring of an LA-ring R1 ×R2 and a = (a1, a2), b = (b1, b2) ∈ R1×R2. If a, b ∈ Z = X∩Y, then by definition of intuitionistic anti characteristic function µχZ (a) = 0 = µχZ (b) and γχZ (a) = 1 = γχZ (b). Since a− b and ab ∈ Z, Z being an LA-subring of an LA-ring R1 ×R2. This means that µχZ (a− b) = 0 = 0 ∨ 0 = µχZ (a) ∨ µχZ (b), µχZ (ab) = 0 = 0 ∨ 0 = µχZ (a) ∨ µχZ (b), γχZ (a− b) = 1 = 1 ∧ 1 = γχZ (a) ∧ γχZ (b), γχZ (ab) = 1 = 1 ∧ 1 = γχZ (a) ∧ γχZ (b). Thus µχZ (a− b) ≤ max{µχZ (a), µχZ (b)}, µχZ (ab) ≤ max{µχZ (a), µχZ (b)}, γχZ (a− b) ≥ min{γχZ (a), γχZ (b)}, γχZ (a− b) ≥ min{γχZ (a), γχZ (b)}. As ab and ba ∈ Z, by definition we get µχZ (ab) = 0 = µχZ (ba) and γχZ (ab) = 1 = γχZ (ba), i.e., µχZ (ab) = µχZ (ba) and γχZ (ab) = γχZ (ba). Similarly, we have µχZ (a− b) ≤ max{µχZ (a), µχZ (b)}, µχZ (ab) ≤ max{µχZ (a), µχZ (b)}, γχZ (a− b) ≥ min{γχZ (a), γχZ (b)}, γχZ (ab) ≥ min{γχZ (a), γχZ (b)}, γχZ (ab) = γχZ (ba), γχZ (ab) = γχZ (ba), when a, b /∈ Z. Hence the intuitionistic anti characteristic function χZ = 〈µχZ , γχZ 〉 of Z is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Conversely, assume that the intuitionistic anti characteristic function χZ = 〈µχZ , γχZ 〉 of Z = X ∩ Y is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 × R2. K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 627 Let a, b ∈ Z = X ∩ Y, then by definition, we have µχZ (a) = 0 = µχZ (b) and γχZ (a) = 1 = γχZ (b). By our assumption µχZ (a− b) ≤ µχZ (a) ∨ µχZ (b) = 0 ∨ 0 = 0, µχZ (ab) ≤ µχZ (a) ∨ µχZ (b) = 0 ∨ 0 = 0, γχZ (a− b) ≥ γχZ (a) ∧ γχZ (b) = 1 ∧ 1 = 1, γχZ (ab) ≥ γχZ (a) ∧ γχZ (b) = 1 ∧ 1 = 1. Thus µχZ (a− b) = 0 = µχZ (ab) and γχZ (a− b) = 1 = γχZ (ab), i.e., a− b and ab ∈ Z. Hence Z is an LA-subring of an LA-ring R1 ×R2. Corollary 1. Let {Ci}i∈I = {Ai ×Bi}i∈I be a family of LA-subrings of an LA-ring R1 × R2. Then C = ∩Ci is an LA-subring of an LA-ring R1 × R2 if and only if the intuitionistic anti characteristic function χC = 〈µχc , γχc〉 of C = ∩Ci is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Lemma 3. If A and B are intuitionistic anti fuzzy normal LA-subrings of LA-rings R1 and R2, respectively, then A×B is also an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Proof. Let A = {(x, µA(x), γA(x)) | x ∈ R1} and B = {(y, µB(y), γB(y)) | y ∈ R2} be intuitionistic anti fuzzy normal LA-subrings of LA-rings R1 and R2, respectively. Now A×B = {((x, y), µA×B(x, y), γA×B(x, y)) | for all x ∈ R1 and y ∈ R2}, where µA×B(x, y) = max{µA(x), µB(y) and γA×B(x, y) = min{γA(x), γB(y)}. We have to show that A × B is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Now µA×B((a, b)− (c, d)) = µA×B(a− c, b− d) = max{µA(a− c), µB(b− d)} = µA(a− c) ∨ µB(b− d) ≤ {µA(a) ∨ µA(c)} ∨ {µB(b) ∨ µB(d)} = µA(a) ∨ {µA(c) ∨ µB(b)} ∨ µB(d) = µA(a) ∨ {µB(b) ∨ µA(c)} ∨ µB(d) = {µA(a) ∨ µB(b)} ∨ {µA(c) ∨ µB(d)} = µA×B(a, b) ∨ µA×B(c, d) and µA×B((a, b) ◦ (c, d)) = µA×B(a ◦ c, b ◦ d) = max{µA(a ◦ c), µB(b ◦ d)} = µA(a ◦ c) ∨ µB(b ◦ d) K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 628 ≤ {µA(a) ∨ µA(c)} ∨ {µB(b) ∨ µB(d)} = µA(a) ∨ {µA(c) ∨ µB(b)} ∨ µB(d) = µA(a) ∨ {µB(b) ∨ µA(c)} ∨ µB(d) = {µA(a) ∨ µB(b)} ∨ {µA(c) ∨ µB(d)} = µA×B(a, b) ∨ µA×B(c, d). Thus µA×B((a, b)− (c, d)) ≤ µA×B(a, b) ∨ µA×B(c, d) and µA×B((a, b) ◦ (c, d)) ≤ µA×B(a, b) ∨ µA×B(c, d). Similarly, we have γA×B((a, b)− (c, d)) ≥ γA×B(a, b) ∧ γA×B(c, d) and γA×B((a, b) ◦ (c, d)) ≥ γA×B(a, b) ∧ γA×B(c, d). Therefore A×B is an intuitionistic anti fuzzy LA-subring of an LA-ring R1×R2. Now µA×B((a, b) ◦ (c, d)) = µA×B(ac, bd) = max{µA(ac), µB(bd)} = max{µA(ca), µB(db)} = µA×B(ca, db) = µA×B((c, d) ◦ (a, b)). Similarly, γA×B((a, b) ◦ (c, d)) = γA×B((c, d) ◦ (a, b)). Hence A× B is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Proposition 2. If X = A × B and Y = C ×D are two intuitionistic anti fuzzy normal LA-subrings of an LA-ring R1×R2, then their intersection X ∩Y is also an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Proof. Let X = A×B = {((x1, x2), µA×B(x1, x2), γA×B(x1, x2)) | for all (x1, x2) ∈ R1× R2} and Y = C ×D = {((y1, y2), µC×D(y1, y2), γC×D(y1, y2)) | for all (y1, y2) ∈ R1 ×R2} be two intuitionistic anti fuzzy normal LA-subrings of an LA-ring R1×R2. Let Z = X∩Y and Z = {((z1, z2), µZ(z1, z2), γZ(z1, z2)) | (z1, z2) ∈ R1 ×R2}, where µZ(z1, z2) = µX∩Y (z1, z2) = max{µX(z1, z2), µY (z1, z2)} and γZ(z1, z2) = γX∩Y (z1, z2) = min{γX(z1, z2), γY (z1, z2)}. Now µZ((z1, z2)− (z3, z4)) = µX∩Y ((z1, z2)− (z3, z4)) = max{µX((z1, z2)− (z3, z4)), µY ((z1, z2)− (z3, z4))} ≤ {µX(z1, z2) ∨ µX(z3, z4)} ∨ {µY (z1, z2) ∨ µY (z3, z4)} K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 629 = {µX(z1, z2) ∨ {µX(z3, z4) ∨ µY (z1, z2)} ∨ µY (z3, z4)} = {µX(z1, z2) ∨ {µY (z1, z2) ∨ µX(z3, z4)} ∨ µY (z3, z4)} = {µX(z1, z2) ∨ µY (z1, z2)} ∨ {µX(z3, z4) ∨ µY (z3, z4)} = max{µX∩Y (z1, z2), µX∩Y (z3, z4)} = max{µZ(z1, z2), µZ(z3, z4)} and µZ((z1, z2) ◦ (z3, z4)) = µX∩Y ((z1, z2) ◦ (z3, z4)) = max{µX((z1, z2) ◦ (z3, z4)), µY ((z1, z2) ◦ (z3, z4))} ≤ {µX(z1, z2) ∨ µX(z3, z4)} ∨ {µY (z1, z2) ∨ µY (z3, z4)} = {µX(z1, z2) ∨ {µX(z3, z4) ∨ µY (z1, z2)} ∨ µY (z3, z4)} = {µX(z1, z2) ∨ {µY (z1, z2) ∨ µX(z3, z4)} ∨ µY (z3, z4)} = {µX(z1, z2) ∨ µY (z1, z2)} ∨ {µX(z3, z4) ∨ µY (z3, z4)} = max{µX∩Y (z1, z2), µX∩Y (z3, z4)} = max{µZ(z1, z2), µZ(z3, z4)}. Thus µZ((z1, z2)− (z3, z4)) ≤ max{µZ(z1, z2), µZ(z3, z4)} and µZ((z1, z2) ◦ (z3, z4)) ≤ max{µZ(z1, z2), µZ(z3, z4)}. Similarly, we have γZ((z1, z2)− (z3, z4)) ≥ min{γZ(z1, z2), γZ(z3, z4)} and γZ((z1, z2) ◦ (z3, z4)) ≥ min{γZ(z1, z2), γZ(z3, z4)}. Therefore Z = (µZ , γZ) is an intuitionistic anti fuzzy LA-subring of an LA-ring R1 ×R2. Now µZ((z1, z2) ◦ (z3, z4)) = µX∩Y (z1z3, z2z4) = max{µX(z1z3, z2z4), µY (z1z3, z2z4)} = max{µX(z3z1, z4z2), µY (z3z1, z4z2)} = µX∩Y (z3z1, z4z2) = µZ((z3, z4) ◦ (z1, z2)). Similarly, γZ((z1, z2) ◦ (z3, z4)) = γZ((z3, z4) ◦ (z1, z2)). Hence Z = X ∩ Y is an intu- itionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Corollary 2. If {Ci}i∈I = {Ai × Bi}i∈I is a family of intuitionistic anti fuzzy normal LA-subrings of an LA-ring R1 × R2, then C = ∩Ci is also an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 630 Theorem 2. If X = A×B and Y = C×D are intuitionistic anti fuzzy normal LA-subrings of LA-rings R′ = R1 × R2 and R′′ = R3 × R4, respectively, then Z = X × Y is also an intuitionistic anti fuzzy normal LA-subring of an LA-ring R′×R′′ = (R1×R2)×(R3×R4). Proof. Let X = A × B = {((x1, x2), µA×B(x1, x2), γA×B(x1, x2)) | for all (x1, x2) ∈ R1 × R2} and Y = C × D = {((y1, y2), µC×D(y1, y2), γC×D(y1, y2)) | for all (y1, y2) ∈ R3 × R4} be intuitionistic anti fuzzy normal LA-subrings of LA-rings R′ = R1 × R2 and R′′ = R3 × R4, respectively. Let Z = X × Y and Z = {((z′, z′′), µZ(z′, z′′), γZ(z′, z′′)) | (z′, z′′) = ((z1, z2), (z3, z4)) ∈ R′ ×R′′}, where µZ(z′, z′′) = µX×Y ((z1, z2), (z3, z4)) = max{µX(z1, z2), µY (z3, z4)}, and γZ(z′, z′′) = γX×Y ((z1, z2), (z3, z4)) = min{γX(z1, z2), γY (z3, z4)}. Now µZ(((z1, z2), (z3, z4))− ((z5, z6), (z7, z8))) = µX×Y (((z1, z2), (z3, z4))− ((z5, z6), (z7, z8))) = µX×Y (((z1, z2)− (z5, z6)), ((z3, z4)− (z7, z8))) = max{µX((z1, z2)− (z5, z6)), µY ((z3, z4)− (z7, z8))} ≤ max{(µX(z1, z2) ∨ µX(z5, z6)), (µY (z3, z4) ∨ µY (z7, z8))} = ((µX(z1, z2) ∨ µY (z5, z6)) ∨ (µX(z3, z4) ∨ µY (z7, z8))) = ((µX(z1, z2) ∨ µY (z3, z4)) ∨ (µX(z5, z6) ∨ µY (z7, z8))) = max{(µX(z1, z2) ∨ µY (z3, z4)), (µX(z5, z6) ∨ µY (z7, z8))} = max{µX×Y ((z1, z2), (z3, z4)), µX×Y ((z5, z6), (z7, z8))} = max{µZ((z1, z2), (z3, z4)), µZ((z5, z6), (z7, z8))}. and µZ(((z1, z2), (z3, z4)) ◦ ((z5, z6), (z7, z8))) = µX×Y (((z1, z2), (z3, z4)) ◦ ((z5, z6), (z7, z8))) = µX×Y (((z1, z2) ◦ (z5, z6)), ((z3, z4) ◦ (z7, z8))) = max{µX((z1, z2) ◦ (z5, z6)), µY ((z3, z4) ◦ (z7, z8))} ≤ max{(µX(z1, z2) ∨ µX(z5, z6)), (µY (z3, z4) ∨ µY (z7, z8))} = ((µX(z1, z2) ∨ µY (z5, z6)) ∨ (µX(z3, z4) ∨ µY (z7, z8))) = ((µX(z1, z2) ∨ µY (z3, z4)) ∨ (µX(z5, z6) ∨ µY (z7, z8))) = max{(µX(z1, z2) ∨ µY (z3, z4)), (µX(z5, z6) ∨ µY (z7, z8))} = max{µX×Y ((z1, z2), (z3, z4)), µX×Y ((z5, z6), (z7, z8))} = max{µZ((z1, z2), (z3, z4)), µZ((z5, z6), (z7, z8))}. Similarly γZ(((z1, z2), (z3, z4))− ((z5, z6), (z7, z8))) K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 631 ≥ min{γZ((z1, z2), (z3, z4)), γZ((z5, z6), (z7, z8))} and γZ(((z1, z2), (z3, z4)) ◦ ((z5, z6), (z7, z8))) ≥ min{γZ((z1, z2), (z3, z4)), γZ((z5, z6), (z7, z8))} Thus Z = (µZ , γZ) is an intuitionistic anti fuzzy LA-subring of an LA-ring R′ ×R′′. Now µZ(((z1, z2), (z3, z4)) ◦ ((z5, z6), (z7, z8))) = µX×Y (((z1, z2) ◦ (z5, z6)), ((z3, z4) ◦ (z7, z8))) = max{µX((z1, z2) ◦ (z5, z6)), µY ((z3, z4) ◦ (z7, z8))} = max{µX((z5, z6) ◦ (z1, z2)), µY ((z7, z8) ◦ (z3, z4))} = µX×Y (((z5, z6) ◦ (z1, z2)), ((z7, z8) ◦ (z3, z4))) = µZ(((z5, z6), (z7, z8)) ◦ ((z1, z2), (z3, z4))). Similarly γZ(((z1, z2), (z3, z4)) ◦ ((z5, z6), (z7, z8))) = γZ(((z5, z6), (z7, z8)) ◦ ((z1, z2), (z3, z4))). Hence Z = X × Y is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R′ ×R′′. Proposition 3. If an IFS A×B is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2, then �A×B = ( µA×B, µA×B ) (resp. ♦A×B = ( γA×B, γA×B ) ) is also an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Proof. Let A × B be an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 × R2. We have to show that �A × B = ( µA×B, µA×B ) is also an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Now µA×B((x1, x2)− (y1, y2)) = 1− µA×B((x1, x2)− (y1, y2)) ≥ 1−max {µA×B(x1, x2), µA×B(y1, y2)} = min {1− µA×B(x1, x2), 1− µA×B(y1, y2)} = min{µA×B(x1, x2), µA×B(y1, y2)}. and µA×B((x1, x2) ◦ (y1, y2)) = 1− µA×B((x1, x2) ◦ (y1, y2)) ≥ 1−max {µA×B(x1, x2), µA×B(y1, y2)} = min {1− µA×B(x1, x2), 1− µA×B(y1, y2)} = min{µA×B(x1, x2), µA×B(y1, y2)}. Thus �A×B = ( µA×B, µA×B ) is an intuitionistic anti fuzzy LA-subring of an LA-ring R1 ×R2. Now µA×B((x1, x2) ◦ (y1, y2)) = 1− µA×B((x1, x2) ◦ (y1, y2)) K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 632 = 1− µA×B((y1, y2) ◦ (x1, x2)) = µA×B((y1, y2) ◦ (x1, x2)). Hence �A × B = ( µA×B, µA×B ) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Corollary 3. An IFS A×B is an intuitionistic anti fuzzy normal LA-subring of an LA- ring R1 ×R2 if and only if �A×B = ( µA×B, µA×B ) (resp. ♦A×B = ( γA×B, γA×B ) ) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Theorem 3. An IFS A × B = (µA×B, γA×B) is an intuitionistic anti fuzzy normal LA- subring of an LA-ring R1 × R2 if and only if the fuzzy subsets µA×B and γA×B are anti fuzzy normal LA-subrings of an LA-ring R1 ×R2. Proof. Let A × B = (µA×B, γA×B) be an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. This implies that µA×B is an anti fuzzy normal LA-subring of an LA-ring R1 ×R2. We have to show that γA×B is also an anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Now γA×B((x1, x2)− (y1, y2)) = 1− γA×B((x1, x2)− (y1, y2)) ≤ 1−min{γA×B(x1, x2), γA×B(y1, y2)} = max{1− γA×B(x1, x2), 1− γA×B(y1, y2)} = max{γA×B(x1, x2), γA×B(y1, y2)}. and γA×B((x1, x2) ◦ (y1, y2)) = 1− γA×B((x1, x2) ◦ (y1, y2)) ≤ 1−min{γA×B(x1, x2), γA×B(y1, y2)} = max{1− γA×B(x1, x2), 1− γA×B(y1, y2)} = max{γA×B(x1, x2), γA×B(y1, y2)}. Thus γA×B is an anti fuzzy LA-subring of an LA-ring R1 ×R2. Now γA×B((x1, x2) ◦ (y1, y2)) = 1− γA×B((x1, x2) ◦ (y1, y2)) = 1− γA×B((y1, y2) ◦ (x1, x2)) = γA×B((y1, y2) ◦ (x1, x2)). Hence γA×B is an anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Conversely, suppose that µA×B and γA×B are anti fuzzy normal LA-subrings of an LA-ring R1 × R2. We have to show that A × B = (µA×B, γA×B) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Now 1− γA×B((x1, x2)− (y1, y2)) = γA×B((x1, x2)− (y1, y2)) ≤ max{γA×B(x1, x2), γA×B(y1, y2)} = max{1− γA×B(x1, x2), 1− γA×B(y1, y2)} K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 633 = 1−min{γA×B(x1, x2), γA×B(y1, y2)} and 1− γA×B((x1, x2) ◦ (y1, y2)) = γA×B((x1, x2) ◦ (y1, y2)) ≤ max{γA×B(x1, x2), γA×B(y1, y2)} = max{1− γA(x1, x2), 1− γA(y1, y2)} = 1−min{γA(x1, x2), γA(y1, y2)}. Thus A × B = (µA×B, γA×B) is an intuitionistic anti fuzzy LA-subring of an LA-ring R1 ×R2. Now 1− γA×B((x1, x2) ◦ (y1, y2)) = γA×B((x1, x2) ◦ (y1, y2)) = γA×B((y1, y2) ◦ (x1, x2)) = 1− γA((y1, y2) ◦ (x1, x2)). Hence A× B = (µA×B, γA×B) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Theorem 4. An IFS A × B = (µA×B, γA×B) is an intuitionistic anti fuzzy normal LA- subring of an LA-ring R1 ×R2 if and only if the fuzzy subsets µA×B and γA×B are fuzzy normal LA-subrings of an LA-ring R1 ×R2. Proof. Let A × B = (µA×B, γA×B) be an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1×R2. This means that γA×B is a fuzzy normal LA-subring of an LA-ring R1 × R2. We have to show that µA×B is also a fuzzy normal LA-subring of an LA-ring R1 ×R2. Now µA×B((x1, x2)− (y1, y2)) = 1− µA×B((x1, x2)− (y1, y2)) ≥ 1−max{µA×B(x1, x2), µA×B(y1, y2)} = min{1− µA×B(x1, x2), 1− µA×B(y1, y2)} = min{µA×B(x1, x2), µA×B(y1, y2)}. and µA×B((x1, x2) ◦ (y1, y2)) = 1− µA×B((x1, x2) ◦ (y1, y2)) ≥ 1−max{µA(x1, x2), µA(y1, y2)} = min{1− µA(x1, x2), 1− µA(y1, y2)} = min{µA×B(x1, x2), µA×B(y1, y2)}. Thus µA×B is a fuzzy LA-subring of an LA-ring R1 ×R2. µA×B((x1, x2) ◦ (y1, y2)) = 1− µA((x1, x2) ◦ (y1, y2)) = 1− µA((y1, y2) ◦ (x1, x2)) = µA×B((y1, y2) ◦ (x1, x2)). Hence µA×B is a fuzzy normal LA-subring of an LA-ring R1 ×R2. K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 634 Conversely, assume that µA×Band γA×B are fuzzy normal LA-subrings of an LA-ring R1 × R2. We have to show that A × B = (µA×B, γA×B) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Now 1− µA×B((x1, x2)− (y1, y2)) = µA×B((x1, x2)− (y1, y2)) ≥ min{µA×B(x1, x2), µA×B(y1, y2)} = min{1− µA×B(x1, x2), 1− µA×B(y1, y2)} = 1−max{µA×B(x1, x2), µA×B(y1, y2)} and 1− µA×B((x1, x2) ◦ (y1, y2)) = µA×B((x1, x2) ◦ (y1, y2)) ≥ min{µA×B(x1, x2), µA×B(y1, y2)} = min{1− µA×B(x1, x2), 1− µA×B(y1, y2)} = 1−max{µA×B(x1, x2), µA×B(y1, y2)}. Thus A × B = (µA×B, γA×B) is an intuitionistic anti fuzzy LA-subring of an LA-ring R1 ×R2. Now 1− µA×B((x1, x2) ◦ (y1, y2)) = µA×B((x1, x2) ◦ (y1, y2)) = µA×B((y1, y2) ◦ (x1, x2)) = 1− µA×B((y1, y2) ◦ (x1, x2)). Hence A× B = (µA×B, γA×B) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2. Lemma 4. Let A and B be intuitionistic fuzzy sets of LA-rings R1 and R2 with left identities e1 and e2, respectively. If A×B is an intuitionistic anti fuzzy LA-subring of an LA-ring R1 ×R2, then at least one of the following two statements must hold. 1. µA (x) ≥ µB (e2) and γA (x) ≤ γB (e2) , for all x ∈ R1. 2. µB (x) ≥ µA (e1) and γB (x) ≤ γA (e1) , for all x ∈ R2. Proof. Let A × B be an intuitionistic anti fuzzy LA-subring of an LA-ring R1 × R2. By contraposition, suppose that none of the statements (i) and (ii) holds. Then we can find a and b in R1 and R2, respectively such that µA (a) ≤ µB (e2) and γA (a) ≥ γB (e2) , µB (b) ≤ µA (e1) and γB (b) ≥ γA (e1) . Thus µA×B(a, b) = max{µA(a), µB(b)} ≤ max{µA(e1), µB(e2)} = µA×B(e1, e2) and γA×B(a, b) = min{γA(a), γB(b)} K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 635 ≥ min(γA(e1), γB(e2)) = γA×B(e1, e2). This implies that A × B is not an intuitionistic anti fuzzy LA-subring of an LA- ring R1 × R2. Hence either µA (x) ≥ µB (e2) and γA (x) ≤ γB (e2) , for all x ∈ R1 or µB (x) ≥ µA (e1) and γB (x) ≤ γA (e1) , for all x ∈ R2. Theorem 5. Let A and B be intuitionistic fuzzy sets of LA-rings R1 and R2 with left identities e1 and e2, respectively and A × B is an intuitionistic anti fuzzy normal LA- subring of an LA-ring R1 ×R2. Then the following conditions are true. 1. If µA (x) ≥ µB (e2) and γA (x) ≤ γB (e2) , for all x ∈ R1, then A is an intuitionistic anti fuzzy normal LA-subring of R1. 2. If µB (x) ≥ µA (e1) and γB (x) ≤ γA (e1) , for all x ∈ R2, then B is an intuitionistic anti fuzzy normal LA-subring of R2. Proof. 1. Let µA (x) ≥ µB (e2) and γA (x) ≤ γB (e2) for all x ∈ R1, and y ∈ R1. We have to show that A is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1. Now µA(x− y) = µA(x+ (−y)) = max{µA(x+ (−y)), µB(e2 + (−e2))} = µA×B(x+ (−y), e2 + (−e2)) = µA×B((x, e2) + (−y,−e2)) = µA×B((x, e2)− (y, e2)) ≤ µA×B(x, e2) ∨ µA×B(y, e2) = max{max{µA(x), µB(e2)},max{µA(y), µB(e2)}} = µA(x) ∨ µA(y) and µA(xy) = max{µA(xy), µB(e2e2)} = µA×B(xy, e2e2) = µA×B((x, e2) ◦ (y, e2)) ≤ µA×B(x, e2) ∨ µA×B(y, e2) = max{max{µA(x), µB(e2)},max{µA(y), µB(e2)}} = µA(x) ∨ µA(y). Similarly, we have γA(x− y) ≥ min{γA(x), γA(y)} and γA(xy) ≥ min{γA(x), γA(y)}. Thus A is an intuitionistic anti fuzzy LA-subring of an LA-ring R1. Now µA(xy) = max{µA(xy), µB(e2e2)} K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 636 = µA×B (xy, e2e2) = µA×B ((x, e2) ◦ (y, e2)) = µA×B ((y, e2) ◦ (x, e2)) = µA×B(yx, e2e2) = max{µA(yx), µB(e2e2)} = µA(yx). Similarly, γB(xy) = γB(yx). Hence A is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1. 2. is same as 1. 3. Direct Product of Finite Intuitionistic Anti Fuzzy Normal LA-subrings We define the direct product of intuitionistic fuzzy sets A1, A2, ..., An of LA-rings R1, R2, ..., Rn, respectively and examine the some fundamental properties of direct product of intuitionistic anti fuzzy normal LA-subrings of an LA-ring R1 ×R2 × ...×Rn. Let µ1, µ2, ..., µn be fuzzy subsets of LA-rings R1, R2, ..., Rn, respectively. The direct product of fuzzy subsets µ1, µ2, ..., µn is denoted by µ1 × µ2 × ... × µn and defined by (µ1 × µ2 × ...× µn)(x1, x2, ..., xn) = min{µ1(x1), µ2 (x2) , ..., µn(xn)}. A fuzzy subset µ1 × µ2 × ... × µn of an LA-ring R1 × R2 × ... × Rn is to be a fuzzy LA-subring of R1 ×R2 × ...×Rn if 1. (µ1 × µ2 × ...× µn)(x− y) ≥ min{(µ1 × µ2 × ...× µn)(x), (µ1 × µ2 × ...× µn)(y)}, 2. (µ1 × µ2 × ...× µn)(xy) ≥ min{(µ1 × µ2 × ...× µn)(x), (µ1 × µ2 × ...× µn)(y)} for all x = (x1, x2, ..., xn) , y = (y1, y2, ..., yn) ∈ R1 ×R2 × ...×Rn. A fuzzy subset µ1×µ2× ...×µn of an LA-ring R1×R2× ...×Rn is to be an anti fuzzy LA-subring of R1 ×R2 × ...×Rn if 1. (µ1 × µ2 × ...× µn)(x− y) ≤ max{(µ1 × µ2 × ...× µn)(x), (µ1 × µ2 × ...× µn)(y)}, 2. µ1 × µ2 × ...× µn(xy) ≤ max{(µ1 × µ2 × ...× µn)(x), (µ1 × µ2 × ...× µn)(y)} for all x = (x1, x2, ..., xn) , y = (y1, y2, ..., yn) ∈ R1 ×R2 × ...×Rn. A fuzzy LA-subring of an LA-ring R1 × R2 × ... × Rn is said to be a fuzzy normal LA-subring of R1 × R2 × ... × Rn if (µ1 × µ2 × ... × µn)(xy) = (µ1 × µ2 × ... × µn)(yx) for all x = (x1, x2, ..., xn) , y = (y1, y2, ..., yn) ∈ R1 ×R2 × ...×Rn. Similarly for anti fuzzy normal LA-subring. Let A1, A2, ..., An be intuitionistic fuzzy sets of LA-rings R1, R2, ..., Rn, respectively. The direct product of intuitionistic fuzzy sets A1, A2..., An is denoted by A1×A2× ...×An and defined by A1 × A2 × ... × An = {(x, µA1×A2×...×An(x), γA1×A2×...×An(x)) | for all x = (x1, x2, ..., xn) ∈ R1 ×R2 × ...×Rn}, where µA1×A2×...×An(x1, x2, ..., xn) = max{µA1(x1), µA2(x2), ..., µAn(xn)} and γA1×A2×...×An(x1, x2, ..., xn) = min{γA1(x1), γA2(x2), ..., γAn(xn)}. K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 637 An intuitionistic fuzzy set (IFS) A1×A2× ...×An = (µA1×A2×...×An , γA1×A2×...×An) of an LA-ring R1×R2× ...×Rn is to be an intuitionistic anti fuzzy LA-subring (IAFLSR) of R1 ×R2 × ...×Rn if 1. µA1×A2×...×An(x− y) ≤ max{µA1×A2×...×An(x), µA1×A2×...×An(y)}, 2. µA1×A2×...×An(xy) ≤ max{µA1×A2×...×An(x), µA1×A2×...×An(y)}, 3. γA1×A2×...×An(x− y) ≥ min{γA1×A2×...×An(x), γA1×A2×...×An(y)}, 4. γA1×A2×...×An(xy) ≥ min{γA1×A2×...×An(x), γA1×A2×...×An(y)}, for all x = (x1, x2, ..., xn) , y = (y1, y2, ..., yn) ∈ R1 ×R2 × ...×Rn. An intuitionistic anti fuzzy LA-subringA1×A2×...×An = (µA1×A2×...×An , γA1×A2×...×An) of an LA-ring R1×R2× ...×Rn is said to be an intuitionistic anti fuzzy normal LA-subring (IAFNLSR) of R1 ×R2 × ...×Rn if 1. µA1×A2×...×An(xy) = µA1×A2×...×An(yx) 2. γA1×A2×...×An(xy) = γA1×A2×...×An(yx) for all x = (x1, x2, ..., xn) , y = (y1, y2, ..., yn) ∈ R1 ×R2 × ...×Rn. Let A1 ×A2 × ...×An be a non-empty subset of an LA-ring R = R1 ×R2 × ...×Rn. The intuitionistic anti characteristic function of A = A1 × A2 × ... × An is denoted by χA1×A2×...×An = 〈µχA1×A2×...×An , γχA1×A2×..×An 〉 and defined by µχA (x) = { 0 if x ∈ A 1 if x /∈ A and γχA (x) = { 1 if x ∈ A 0 if x /∈ A [17] If R1, R2 are LA-rings, then direct product R1 × R2 of R1 and R2 is an LA-ring with pointwise addition ‘+’ and multiplication ‘◦’ defined as (a, b) + (c, d) = (a+ c, b+ d) and (a, b) ◦ (c, d) = (ac, bd) , respectively for every (a, b) , (c, d) ∈ R1 × R2. Likewise the direct product R = ×i∈ΩRi of a family of LA-rings {Ri : i ∈ Ω} has the structure of an LA-ring with the operations of addition and multiplication defined as a+ b = (a1, a2, a3, ...) + (b1, b2, b3, ...) = (a1 + b1, a2 + b2, a3 + b3, ...) and a ◦ b = (a1, a2, a3, ...) ◦ (b1, b2, b3, ...) = (a1b1, a2b2, a3b3, ...) for all a = (a1, a2, ..., an), b = (b1, b2, ..., bn) ∈ R. Lemma 5. If A1, A2, ..., An are LA-subrings of LA-rings R1, R2, ..., Rn, respectively, then A1 × A2 × ... × An is an LA-subring of an LA-ring R1 × R2 × ... × Rn under the same operations defined as in [17]. Proof. Straight forward. Lemma 6. Let A1, A2, ..., An be LA-subrings of LA-rings R1, R2, ..., Rn, respectively. Then A1 × A2 × ... × An is an LA-subring of an LA-ring R1 × R2 × ... × Rn if and only if the intuitionistic anti characteristic function χA = 〈µχA , γχA〉 of A = A1×A2× ...×An is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 638 Proof. Let A = A1×A2×...×An be an LA-subring of an LA-ring R1×R2×...×Rn and a = (a1, a2, ..., an), b = (b1, b2, ..., bn) ∈ R1×R2× ...×Rn. If a, b ∈ A = A1×A2× ...×An, then by definition of intuitionistic anti characteristic function µχA(a) = 0 = µχA(b) and γχA(a) = 1 = γχA(b). Since a− b and ab ∈ A, A being an LA-subring. This implies that µχA(a− b) = 0 = 0 ∨ 0 = µχA(a) ∨ µχA(b), µχA(ab) = 0 = 0 ∨ 0 = µχA(a) ∨ µχA(b), γχA(a− b) = 1 = 1 ∧ 1 = γχA(a) ∧ γχA(b), γχA(ab) = 1 = 1 ∧ 1 = γχA(a) ∧ γχA(b). Thus µχA(a− b) ≤ max{µχA(a), µχA(b)}, µχA(ab) ≤ max{µχA(a), µχA(b)}, γχA(a− b) ≥ min{γχA(a), γχA(b)}, γχA(ab) ≥ min{γχA(a), γχA(b)}. As ab and ba ∈ A, so µχA(ab) = 0 = µχA(ba) and γχA(ab) = 1 = γχA(ba), i.e., µχA(ab) = µχA(ba) and γχA(ab) = γχA(ba). Similarly, we have µχA(a− b) ≤ max{µχA(a), µχA(b)}, µχA(ab) ≤ max{µχA(a), µχA(b)}, γχA(a− b) ≥ min{γχA(a), γχA(b)}, γχA(ab) ≥ min{γχA(a), γχA(b)}, µχA(ab) = µχA(ba), γχA(ab) = γχA(ba), when a, b /∈ A. Hence the intuitionistic anti characteristic function χA = 〈µχA , γχA〉 of A = A1 × A2 × ... × An is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Conversely, suppose that the intuitionistic anti characteristic function χA = 〈µχA , γχA〉 of A = A1 ×A2 × ...×An is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. We have to show that A = A1 ×A2 × ...×An is an LA-subring of an LA-ring R1 × R2 × ...× Rn. Let a, b ∈ A, where a = (a1, a, ..., an) and b = (b1, b2, ..., bn) , by definition, we have µχA(a) = 0 = µχA(b) and γχA(a) = 1 = γχA(b). By our supposition µχA(a− b) ≤ µχA(a) ∨ µχA(b) = 0 ∨ 0 = 0, µχA(ab) ≤ µχA(a) ∨ µχA(b) = 0 ∨ 0 = 0, γχA(a− b) ≥ γχA(a) ∧ γχA(b) = 1 ∧ 1 = 1, γχA(ab) ≥ γχA(a) ∧ γχA(b) = 1 ∧ 1 = 1. Thus µχA(a− b) = 0 = µχA(ab) and γχA(a− b) = 1 = γχA(ab), i.e., a− b and ab ∈ A. Hence A = A1 ×A2 × ...×An is an LA-subring of an LA-ring R1 ×R2 × ...×Rn. Lemma 7. If A = A1 × A2 × ... × An and B = B1 × B2 × ... × Bn are two LA-subrings of an LA-ring R1 ×R2 × ...×Rn, then their intersection A ∩B is also an LA-subring of an LA-ring R1 ×R2 × ...×Rn. K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 639 Proof. Straight forward. Theorem 6. Let A = A1×A2×...×An and B = B×B2×...×Bn be two LA-subrings of an LA-ring R1×R2× ...×Rn. Then A∩B is an LA-subring of an LA-ring R1×R2× ...×Rn if and only if the intuitionistic anti characteristic function χZ = 〈µχZ , γχZ 〉 of Z = A∩B is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Proof. Let Z = A ∩ B be an LA-subring of an LA-ring R1 × R2 × ... × Rn and a = (a1, a2, ..., an), b = (b1, b1, ..., bn) ∈ R1×R2×...×Rn. If a, b ∈ Z = A∩B, then by definition of intuitionistic anti characteristic function µχZ (a) = 0 = µχZ (b) and γχZ (a) = 1 = γχZ (b). Since a− b and ab ∈ Z, Z being an LA-subring. This means that µχZ (a− b) = 0 = 0 ∨ 0 = µχZ (a) ∨ µχZ (b), µχZ (ab) = 0 = 0 ∨ 0 = µχZ (a) ∨ µχZ (b), γχZ (a− b) = 1 = 1 ∧ 1 = γχZ (a) ∧ γχZ (b), γχZ (ab) = 1 = 1 ∧ 1 = γχZ (a) ∧ γχZ (b). Thus µχZ (a− b) ≤ max{µχZ (a), µχZ (b)}, µχZ (ab) ≤ max{µχZ (a), µχZ (b)}, γχZ (a− b) ≥ min{γχZ (a), γχZ (b)}, γχZ (ab) ≥ min{γχZ (a), γχZ (b)}. As ab and ba ∈ Z, this implies that µχZ (ab) = 0 = µχZ (ba) and γχZ (ab) = 1 = γχZ (ba), i.e., µχZ (ab) = µχZ (ba) and γχZ (ab) = γχZ (ba). Similarly, we have µχZ (a− b) ≤ max{µχZ (a), µχZ (b)}, µχZ (ab) ≤ max{µχZ (a), µχZ (b)}, γχZ (a− b) ≥ min{γχZ (a), γχZ (b)}, γχZ (ab) ≥ min{γχZ (a), γχZ (b)}, γχZ (ab) = γχZ (ba), γχZ (ab) = γχZ (ba), when a, b /∈ Z. Hence the intuitionistic anti characteristic function χZ = 〈µχZ , γχZ 〉 of Z is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Conversely, assume that the intuitionistic anti characteristic function χZ = 〈µχZ , γχZ 〉 of Z = A ∩ B is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 × R2 × ... × Rn. Let a, b ∈ Z = A ∩ B, by definition, we have µχZ (a) = 0 = µχZ (b) and γχZ (a) = 1 = γχZ (b). By our assumption µχZ (a− b) ≤ µχZ (a) ∨ µχZ (b) = 0 ∨ 0 = 0, µχZ (ab) ≤ µχZ (a) ∨ µχZ (b) = 0 ∨ 0 = 0, γχZ (a− b) ≥ γχZ (a) ∧ γχZ (b) = 1 ∧ 1 = 1, γχZ (ab) ≥ γχZ (a) ∧ γχZ (b) = 1 ∧ 1 = 1. Thus µχZ (a− b) = 0 = µχZ (ab) and γχZ (a− b) = 1 = γχZ (ab), i.e., a− b and ab ∈ Z. Hence Z is an LA-subring of an LA-ring R1 ×R2 × ...×Rn. K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 640 Corollary 4. Let {Bi}i∈I = {Ai1 ×Ai2 × ...×Ain}i∈I be a family of LA-subrings of an LA-ring R1 × R2 × ... × Rn. Then B = ∩Bi is an LA-subring of an LA-ring R1 × R2 × ... × Rn if and only if the intuitionistic anti characteristic function χB = 〈µχB , γχB 〉 of B = ∩Bi is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1×R2×...×Rn. Theorem 7. If A = A1×A2×...×An and B = B1×B2×...×Bn are two intuitionistic anti fuzzy normal LA-subrings of an LA-ring R1×R2× ...×Rn, then their intersection A∩B is also an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Proof. Let A = A1 × A2 × ... × An = {((a), µA1×A2×...×An(a), γA1×A2×...×A(a)) | for all a = (a1, a2, ..., an) ∈ R1 × R2 × ... × Rn} and B = B1 × B2 × ... × Bn = {((b), µB1×B2×...×Bn(b), γB1×B2×...×Bn(b)) | for all b = (b1, b2, ..., bn) ∈ R1 ×R2 × ...×Rn} be two intuitionistic anti fuzzy normal LA-subrings of an LA-ring R1 ×R2 × ...×Rn. Let Z = A ∩B and Z = {((z), µZ(z), γZ(z)) | for all z = (z1, z2, ..., zn) ∈ R1 ×R2 × ...×Rn}, where µZ(z1, z2, ..., zn) = µA∩B(z1, z2, ..., zn) = max{µA(z1, z2, ..., zn), µB(z1, z2, ..., zn)} and γZ(z1, z2, ..., zn) = γA∩B(z1, z2, ..., zn) = min{γA(z1, z2, ..., zn), γB(z1, z2, ..., zn)}. Let z = (z1, zn, ..., zn), w = (w1, w2, ..., wn) ∈ R1 ×R2 × ...×Rn. Now µZ(z − w) = µZ(z − w) = max{µA(z − w), µB(z − w)} ≤ {µA(z) ∨ µA(w)} ∨ {µB(z) ∨ µB(w)} = µA(z) ∨ {µA(w) ∨ µB(z)} ∨ µB(w) = µA(z) ∨ {µB(z) ∨ µA(w)} ∨ µB(w) = {µA(z) ∨ µB(z)} ∨ {µA(w) ∨ µB(w)} = max{µA∩B(z), µA∩B(w)} = max{µZ(z), µZ(w)} and µZ(z ◦ w) = µZ(z ◦ w) = max{µA(z ◦ w), µB(z ◦ w)} ≤ {µA(z) ∨ µA(w)} ∨ {µB(z) ∨ µB(w)} = µA(z) ∨ {µA(w) ∨ µB(z)} ∨ µB(w) = µA(z) ∨ {µB(z) ∨ µA(w)} ∨ µB(w) = {µA(z) ∨ µB(z)} ∨ {µA(w) ∨ µB(w)} = max{µA∩B(z), µA∩B(w)} = max{µZ(z), µZ(w)}. Thus µZ((z1, z2, ..., zn)− (w1, w2, ..., wn)) K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 641 ≤ max{µZ(z1, z2, ..., zn), µZ(w1, w2, ..., wn)} and µZ((z1, z2, ..., zn) ◦ (w1, w2, ..., wn)) ≤ max{µZ(z1, z2, ..., zn), µZ(w1, w2, ..., wn)}. Similarly, we have γZ((z1, z2, ..., zn)− (w1, w2, ..., wn)) ≥ min{γZ(z1, z2, ..., zn), γZ(w1, w2, ..., wn)} and γZ((z1, z2, ..., zn) ◦ (w1, w2, ..., wn)) ≥ min{γZ(z1, z2, ..., zn), γZ(w1, w2, ..., wn)} Thus Z = (µZ , γZ) is an intuitionistic anti fuzzy LA-subring of an LA-ring R1 ×R2 × ...×Rn. Now µZ((z1, z2, ..., zn) ◦ (w1, w2, ..., wn)) = µA∩B(z1w1, z2w2, ..., znwn) = max{µA(z1w1, z2w2, ..., znwn), µB(z1w1, z2w2, ..., znwn)} = max{µA(w1z1, w2z2, ..., wnzn), µB(w1z1, w2z2, ..., wnzn)} = µA∩B(w1z1, w2z2, ..., wnzn) = µZ((w1, w2, ..., wn) ◦ (z1, z2, ..., zn)). Similarly γZ((z1, z2, ..., zn) ◦ (w1, w2, ..., wn)) = γZ((w1, w2, ..., wn) ◦ (z1, z2, ..., zn)). Hence Z = A ∩ B is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Corollary 5. If {Bi}i∈I = {Ai1 ×Ai2 × ...×Ain}i∈I is a family of intuitionistic anti fuzzy normal LA-subrings of an LA-ring R1 × R2 × ... × Rn, then B = ∩Bi is also an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Proposition 4. If an IFS A = A1 × A2 × ... × An is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1×R2× ...×Rn, then �A = (µA, µA) (resp. ♦A = (γA, γA)) is also an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Proof. Let A1×A2×...×An be an intuitionistic anti fuzzy normal LA-subring of an LA- ringR1×R2×...×Rn.We have to show that �A1×A2×...×An = (µA1×A2×...×An , µA1×A2×...×An ) is also an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 × R2 × ... × Rn. Now µA1×A2×...×An ((x1, x2, ..., xn)− (y1, y2, ..., yn)) = 1− µA1×A2×...×An((x1, x2, ..., xn)− (y1, y2, ..., yn)) K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 642 ≥ 1−max {µA1×A2×...×An(x1, x2, ..., xn), µA1×A2×...×An(y1, y2, ..., yn)} = min {1− µA1×A2×...×An(x1, x2, ..., xn), 1− µA1×A2×...×An(y1, y2, ..., yn)} = min{µA1×A2×...×An (x1, x2, ..., xn), µA1×A2×...×An (y1, y2, ..., yn)} and µA1×A2×...×An ((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = 1− µA1×A2×...×An((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) ≥ 1−max {µA1×A2×...×An(x1, x2, ..., xn), µA1×A2×...×An(y1, y2, ..., yn)} = min {1− µA1×A2×...×An(x1, x2, ..., xn), 1− µA1×A2×...×An(y1, y2, ..., yn)} = min{µA1×A2×...×An (x1, x2, ..., xn), µA1×A2×...×An (y1, y2, ..., yn)}. Thus �A1 × A2 × ... × An = (µA1×A2×...×An , µA1×A2×...×An ) is an intuitionistic anti fuzzy LA-subring of an LA-ring R1 ×R2 × ...×Rn. Now = µA1×A2×...×An ((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = 1− µA1×A2×...×An((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = 1− µA1×A2×...×An((y1, y2, ..., yn) ◦ (x1, x2, ..., xn)) = µA1×A2×...×An ((y1, y2, ..., yn) ◦ (x1, x2, ..., xn)). Hence �A1 × A2 × ... × An = (µA1×A2×...×An , µA1×A2×...×An ) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Corollary 6. An IFS A = A1 × A2 × ...× An is an intuitionistic anti fuzzy normal LA- subring of an LA-ring R1×R2×...×Rn if and only if �A = (µA, µA) (resp. ♦A = (γA, γA)) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Theorem 8. An IFS A1×A2×...×An = (µA1×A2×...×An , γA1×A2×...×An) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 × R2 × ... × Rn if and only if the fuzzy subsets µA1×A2×...×An and γA1×A2×...×An are anti fuzzy normal LA-subrings of an LA-ring R1 ×R2 × ...×Rn. Proof. Let A1×A2× ...×An = (µA1×A2×...×An , γA1×A2×...×An) be an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1×R2× ...×Rn. This implies that µA1×A2×...×An is an anti fuzzy normal LA-subring of an LA-ring R1×R2× ...×Rn. We have to show that γA1×A2×...×An is also an anti fuzzy normal LA-subring of an LA-ring R1 × R2 × ...× Rn. Now γA1×A2×...×An ((x1, x2, ..., xn)− (y1, y2, ..., yn)) = 1− γA1×A2×...×An((x1, x2, ..., xn)− (y1, y2, ..., yn)) ≤ 1−min{γA1×A2×...×An(x1, x2, ..., xn), γA1×A2×...×An(y1, y2, ..., yn)} = max{1− γA1×A2×...×An(x1, x2, ..., xn), 1− γA1×A2×...×An(y1, y2, ..., yn)} = max{γA1×A2×...×An (x1, x2, ..., xn), γA1×A2×...×An (y1, y2, ..., yn)}. and γA1×A2×...×An ((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 643 = 1− γA1×A2×...×An((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) ≤ 1−min{γA1×A2×...×An(x1, x2, ..., xn), γA1×A2×...×An(y1, y2, ..., yn)} = max{1− γA1×A2×...×An(x1, x2, ..., xn), 1− γA1×A2×...×An(y1, y2, ..., yn)} = max{γA1×A2×...×An (x1, x2, ..., xn), γA1×A2×...×An (y1, y2, ..., yn)}. Thus γA1×A2×...×An is an anti fuzzy LA-subring of an LA-ring R1×R2× ...×Rn. Now γA1×A2×...×An ((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = 1− γA1×A2×...×An((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = 1− γA1×A2×...×An((y1, y2, ..., yn) ◦ (x1, x2, ..., xn)) = γA1×A2×...×An ((y1, y2, ..., yn) ◦ (x1, x2, ..., xn)). Hence γA1×A2×...×An is an anti fuzzy normal LA-subring of an LA-ring R1×R2×...×Rn. Conversely, suppose that µA1×A2×...×An and γA1×A2×...×An are anti fuzzy normal LA- subrings of an LA-ring R1 × R2 × ... × Rn. We have to show that A1 × A2 × ... × An = (µA1×A2×...×An , γA1×A2×...×An) is an intuitionistic anti fuzzy normal LA-subring of an LA- ring R1 ×R2 × ...×Rn. Now 1− γA1×A2×...×An((x1, x2, ..., xn)− (y1, y2, ..., yn)) = γA1×A2×...×An ((x1, x2, ..., xn)− (y1, y2, ..., yn)) ≤ max{γA1×A2×...×An (x1, x2, ..., xn), γA1×A2×...×An (y1, y2, ..., yn)} = max{1− γA1×A2×...×An(x1, x2, ..., xn), 1− γA1×A2×...×An(y1, y2, ..., yn)} = 1−min{γA1×A2×...×An(x1, x2, ..., xn), γA1×A2×...×An(y1, y2, ..., yn)} and 1− γA1×A2×...×An((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = γA1×A2×...×An ((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) ≤ max{γA1×A2×...×An (x1, x2, ..., xn), γA1×A2×...×An (y1, y2, ..., yn)} = max{1− γA1×A2×...×An(x1, x2, ..., xn), 1− γA1×A2×...×An(y1, y2, ..., yn)} = 1−min{γA1×A2×...×An(x1, x2, ..., xn), γA1×A2×...×An(y1, y2, ..., yn)}. Thus A1×A2× ...×An = (µA1×A2×...×An , γA1×A2×...×An) is an intuitionistic anti fuzzy LA-subring of an LA-ring R1 ×R2 × ...×Rn. Now 1− γA1×A2×...×An((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = γA1×A2×...×An ((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = γA1×A2×...×An ((y1, y2, ..., yn) ◦ (x1, x2, ..., xn)) = 1− γA1×A2×...×An((y1, y2, ..., yn) ◦ (x1, x2, ..., xn)). Hence A1×A2× ...×An = (µA1×A2×...×An , γA1×A2×...×An) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 644 Theorem 9. An IFS A1×A2×...×An = (µA1×A2×...×An , γA1×A2×...×An) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 × R2 × ... × Rn if and only if the fuzzy subsets µA1×A2×...×An and γA1×A2×...×An are fuzzy normal LA-subrings of an LA-ring R1 ×R2 × ...×Rn. Proof. Let A1×A2× ...×An = (µA1×A2×...×An , γA1×A2×...×An) be an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1×R2× ...×Rn. This means that γA1×A2×...×An is a fuzzy normal LA-subring of an LA-ring R1 × R2 × ... × Rn. We have to show that µA1×A2×...×An is also a fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Now µA1×A2×...×An ((x1, x2, ..., xn)− (y1, y2, ..., yn)) = 1− µA1×A2×...×An((x1, x2, ..., xn)− (y1, y2, ..., yn)) ≥ 1−max{µA1×A2×...×An(x1, x2, ..., xn), µA1×A2×...×An(y1, y2, ..., yn)} = min{1− µA1×A2×...×An(x1, x2, ..., xn), 1− µA1×A2×...×An(y1, y2, ..., yn)} = min{µA1×A2×...×An (x1, x2, ..., xn), µA1×A2×...×An (y1, y2, ..., yn)} and µA1×A2×...×An ((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = 1− µA1×A2×...×An((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) ≥ 1−max{µA1×A2×...×An(x1, x2, ..., xn), µA1×A2×...×An(y1, y2, ..., yn)} = min{1− µA1×A2×...×An(x1, x2, ..., xn), 1− µA1×A2×...×An(y1, y2, ..., yn)} = min{µA1×A2×...×An (x1, x2, ..., xn), µA1×A2×...×An (y1, y2, ..., yn)}. Thus µA1×A2×...×An is a fuzzy LA-subring of an LA-ring R1 ×R2 × ...×Rn. Now µA1×A2×...×An ((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = 1− µA1×A2×...×An((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = 1− µA1×A2×...×An((y1, y2, ..., yn) ◦ (x1, x2, ..., xn)) = µA1×A2×...×An ((y1, y2, ..., yn) ◦ (x1, x2, ..., xn)). Hence µA1×A2×...×An is a fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Conversely, assume that µA1×A2×...×An and γA1×A2×...×An are fuzzy normal LA-subrings of an LA-ringR1×R2×...×Rn.We have to show thatA1×A2×...×An = (µA1×A2×...×An , γA1×A2×...×An) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Now 1− µA1×A2×...×An((x1, x2, ..., xn)− (y1, y2, ..., yn)) = µA1×A2×...×An ((x1, x2, ..., xn)− (y1, y2, ..., yn)) ≥ min{µA1×A2×...×An (x1, x2, ..., xn), µA1×A2×...×An (y1, y2, ..., yn)} = min{1− µA1×A2×...×An(x1, x2, ..., xn), 1− µA1×A2×...×An(y1, y2, ..., yn)} = 1−max{µA1×A2×...×An(x1, x2, ..., xn), µA1×A2×...×An(y1, y2, ..., yn)} and 1− µA1×A2×...×An((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = µA1×A2×...×An ((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) ≥ min{µA1×A2×...×An (x1, x2, ..., xn), µA1×A2×...×An (y1, y2, ..., yn)} K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 645 = min{1− µA1×A2×...×An(x1, x2, ..., xn), 1− µA1×A2×...×An(y1, y2, ..., yn)} = 1−max{µA1×A2×...×An(x1, x2, ..., xn), µA1×A2×...×An(y1, y2, ..., yn)}. Thus A1×A2× ...×An = (µA1×A2×...×An , γA1×A2×...×An) is an intuitionistic anti fuzzy LA-subring of an LA-ring R1 ×R2 × ...×Rn. Now 1− µA1×A2×...×An((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = µA1×A2×...×An ((x1, x2, ..., xn) ◦ (y1, y2, ..., yn)) = µA1×A2×...×An ((y1, y2, ..., yn) ◦ (x1, x2, ..., xn)) = 1− µA1×A2×...×An((y1, y2, ..., yn) ◦ (x1, x2, ..., xn)). Hence A1×A2× ...×An = (µA1×A2×...×An , γA1×A2×...×An) is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. Proposition 5. Let A = A1 ×A2 × ...×An and B = B1 ×B2 × ...×Bn be intuitionistic fuzzy sets of LA-rings R = R1×R2× ...×Rn and R′ = R′1×R′2× ...×R′n with left identities e = (e1, e2, ..., en) and e′ = (e1′, e2′, ..., en′), respectively. If A×B is an intuitionistic anti fuzzy LA-subring of an LA-ring R×R′, then at least one of the following two statements must hold. 1. µA (x) ≥ µB (e′) and γA (x) ≤ γB (e′) , for all x ∈ R. 2. µB (x) ≥ µA (e) and γB (x) ≤ γA (e) , for all x ∈ R′. Proof. Let A×B be an intuitionistic anti fuzzy LA-subring of an LA-ring R×R′. By contraposition, suppose that none of the statements (i) and (ii) holds. Then we can find a and b in R and R′, respectively such that µA (a) ≤ µB ( e′ ) and γA (a) ≥ γB ( e′ ) . µB (b) ≤ µA (e) and γB (b) ≥ γA (e) . Thus µA×B(a, b) = max{µA(a), µB(b)} ≤ max{µA(e), µB(e′)} = µA×B(e, e′) and γA×B(a, b) = min{γA(a), γB(b)} ≥ min{γA(e), γB(e′)} = γA×B(e, e′). Therefore A× B is not an intuitionistic anti fuzzy LA-subring of an LA-ring R × R′. Hence either µA (x) ≥ µB (e′) and γA (x) ≤ γB (e′) , for all x ∈ R1 or µB (x) ≥ µA (e) and γB (x) ≤ γA (e) , for all x ∈ R2. K. Nasreen / Eur. J. Pure Appl. Math, 12 (2) (2019), 622-648 646 Theorem 10. Let A = A1×A2× ...×An and B = B1×B2× ...×Bn be intuitionistic fuzzy sets of LA-rings R = R1 × R2 × ...× Rn and R′ = R′1 × R′2 × ...× R′n with left identities e = (e1, e2, ..., en) and e′ = (e1′, e2′, ..., en′), respectively and A×B is an intuitionistic anti fuzzy normal LA-subring of an LA-ring R×R′. Then the following conditions are true. 1. If µA (x) ≥ µB (e′) and γA (x) ≤ γB (e′), for all x ∈ R, then A is an intuitionistic anti fuzzy normal LA-subring of R. 2. If µB (x′) ≥ µA (e) and γB (x′) ≤ γA (e), for all x′ ∈ R′, then B is an intuitionistic anti fuzzy normal LA-subring of R′. Proof. 1. Let µA (x) ≥ µB (e′) and γA (x) ≤ γB (e′) for all x ∈ R, and y ∈ R. We have to show that A is an intuitionistic anti fuzzy normal LA-subring of R. Now µA(x− y) = µA(x+ (−y)) = max{µA(x+ (−y)), µB(e′ + (−e′))} = µA×B(x+ (−y), e′ + (−e′)) = µA×B((x, e′) + (−y,−e′)) = µA×B((x, e′)− (y, e′)) ≤ µA×B(x, e′) ∨ µA×B(y, e′) = max{max{µA(x), µB(e′)},max{µA(y), µB(e′)}} = µA(x) ∨ µA(y) and µA(xy) = max{µA(xy), µB(e′e′)} = µA×B(xy, e′e′) = µA×B((x, e′) ◦ (y, e′)) ≤ µA×B(x, e′) ∨ µA×B(y, e′) = max{max{µA(x), µB(e′)},max{µA(y), µB(e′)}} = µA(x) ∨ µA(y). Similarly, we have γA(x− y) ≥ min{γA(x), γA(y)} and γA(xy) ≥ min{γA(x), γA(y)}. Thus A is an intuitionistic anti fuzzy LA-subring of R. 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