EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 3, 2019, 906-943 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Intuitionistic Fuzzy Ideals with Thresholds (α, β] in LA-rings Nasreen Kausar1,∗, Badar ul Islam2, Syed Amjad Ahmad3, Muhammad Azam Waqar4 1 Department of Mathematics, University of Agriculture, FSD, Pakistan 2 Department of Electrical Engineering, NFC IEFR FSD, Pakistan 3 Department of Mechanical Engineering, NFC IEFR FSD, Pakistan 4 Department of School of Business Manegement, NFC IEFR FSD, Pakistan Abstract. In this paper, we give characterizations of regular (intra-regular, both regular and intra-regular) LA-rings by the properties of intuitionistic fuzzy (left, right, quasi-, bi-, generalized bi-) ideals with thresholds (α, β]. 2010 Mathematics Subject Classifications: 03F55, 08A72, 20N25 Key Words and Phrases: Intuitionistic fuzzy left (right, interior, quasi-, bi-, generalized bi-) ideals with thresholds (α, β], regular (intra-regular) LA-rings. 1. Introduction In ternary operations, the commutative law is given by abc = cba. Kazim et al [18], have generalized this notion by introducing the paranthesis on the left side of this equation to get 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 called a left almost semigroup (abbreviated as LA- semigroup) if it satisfies the left invertive law. An LA-semigroup is a midway structure between a commutative semigroup and a groupoid. Ideals in LA-semigroups have been investigated by Protic et al [24]. In [12] (resp. [8]), a groupoid S is said to be medial (resp. paramedial) if (ab)(cd) = (ac)(bd) (resp. (ab)(cd) = (db)(ca)). In [18], an LA-semigroup is medial, but in general an LA-semigroup needs not to be paramedial. Every LA-semigroup with left identity is paramedial by Protic et al [24] and also satisfies a(bc) = b(ac), (ab)(cd) = (dc)(ba). Kamran [14], extended the notion of LA-semigroup to the left almost group (LA- group). An LA-semigroup G is called a left almost group, if there exists a left identity ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3441 Email addresses: kausar.nasreen@gmail.com (K. Nasreen), badar.utp@gmail.com (I. Badar) samjadahmad67@yahoo.com (S. A. Ahmad ), azamwaqar4@gmail.com (W. Azam) http://www.ejpam.com 906 c© 2019 EJPAM All rights reserved. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 907 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 [25], discussed the left almost ring (abbreviated as 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 a, b ∈ R, a ⊕ b = b − a and a · b is same as in the ring. Although the structure is non-associative and non-commutative, nevertheless, it possesses many interesting properties which we usually find in associative and commutative algebraic structures. A non-empty subset A of R is called an LA-subring of R if a − b and ab ∈ A for all a, b ∈ A. A is called 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. A non-empty subset A of R is called an interior ideal of R if (A,+) is an LA-group and (RA)R ⊆ A. A non-empty subset A of R is called a quasi-ideal of R if (A,+) is an LA-group and AR∩RA ⊆ A. An LA-subring A of R is called a bi-ideal of R if (AR)A ⊆ A. A non-empty subset A of R is called a generalized bi-ideal of R if (A,+) is an LA-group and (AR)A ⊆ A. We will introduce the concept intuitionistic fuzzy left (resp. right, interior, quasi-, bi-, generalized bi-) ideals with thresholds (α, β] of an LA-ring R. We will establish a study by describing the different properties in terms of such ideals, which will be very useful for the characterizations of regular (intra-regular, both regular and intra-regular) LA-rings in terms of intuitionistic fuzzy left (right, quasi-, bi-, generalized bi-) ideals with thresholds (α, β]. 2. Intuitionistic Fuzzy Ideals with Thresholds (α, β] After the introduction of fuzzy set by Zadeh [31], 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. Liu [20], introduced the concept of fuzzy subrings and fuzzy ideals of a ring. Many authors have explored the theory of fuzzy rings (for example [11, 19, 21, 22, 29]). Gupta et al [11], gave the idea of intrinsic product of fuzzy subsets of a ring. Kuroki [19], characterized regular (intra-regular, both regular and intra-regular) rings in terms of fuzzy left (right, quasi, bi-) ideals. 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 shall use the symbol A = (µA, γA) for the IFS K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 908 A = {(x, µA(x), γA(x)) : x ∈ X}. Banerjee et al [3] and Hur et al [10], initiated the notion of intuitionistic fuzzy sub- rings and intuitionistic fuzzy ideals of a ring. Subsequently many authors studied the intuitionistic fuzzy subrings and intuitionistic fuzzy ideals of a ring by describing the dif- ferent properties (see [9]). Shah et al [26], have initiated the concept of intuitionistic fuzzy normal LA-subrings of an LA-ring. Bhakat et al [4–6], introduced the notion of (α, β)-fuzzy subgroups. It is a general- ization of Rosenfeld fuzzy subgroups as (∈,∈ ∨q)-fuzzy subgroups. Then many authors studied the algebraic structures by employing the idea of (α, β)-fuzzy subsets (for exam- ple [7, 13, 23]). Yuan et al [30], initiated the concept of fuzzy subgroups with thresholds. Shabir et al [28], gave the idea of fuzzy ideals with thresholds in semigroups. Now we initiate the concept of intuitionistic fuzzy LA-subrings with thresholds (α, β] and intuitionistic fuzzy left (resp. right, interior, quasi-, bi-, generalized bi-) ideals with thresholds (α, β] of an LA-ring R. An IFS A = (µA, γA) of an LA-ring R is called an intuitionistic fuzzy LA-subring with thresholds (α, β] of R if (1) max{µA (x− y) , α} ≥ min{µA (x) , µA(y), β}, (2) min{γA (x− y) , (1− α)} ≤ max{γA (x) , γA(y), (1− β)}, (3) max{µA (xy) , α} ≥ min{µA (x) , µA (y) , β}, (4) min{γA (xy) , (1− α)} ≤ max{γA (x) , γA (y) , (1− β)} for all x, y ∈ R and α, β ∈ (0, 1] such that α < β. An IFS A = (µA, γA) of an LA-ring R is called an intuitionistic fuzzy left ideal with thresholds (α, β] of R if (1) max{µA (x− y) , α} ≥ min{µA (x) , µA(y), β}, (2) min{γA (x− y) , (1− α)} ≤ max{γA (x) , γA(y), (1− β)}, (3) max{µA (xy) , α} ≥ min{µA (y) , β}, (4) min{γA (xy) , (1 − α)} ≤ max{γA (y) , (1 − β)} for all x, y ∈ R and α, β ∈ (0, 1] such that α < β. An IFS A = (µA, γA) of an LA-ring R is called an intuitionistic fuzzy right ideal with thresholds (α, β] of R if (1) max{µA (x− y) , α} ≥ min{µA (x) , µA(y), β}, (2) min{γA (x− y) , (1− α)} ≤ max{γA (x) , γA(y), (1− β)}, (3) max{µA (xy) , α} ≥ min{µA (x) , β}, (4) min{γA (xy) , (1 − α)} ≤ max{γA (x) , (1 − β)} for all x, y ∈ R and α, β ∈ (0, 1] such that α < β. An IFS A = (µA, γA) of an LA-ring R is called an intuitionistic fuzzy ideal with thresholds (α, β] of R if it is both an intuitionistic fuzzy left ideal with thresholds (α, β] and an intuitionistic fuzzy right ideal with thresholds (α, β] of R. Every intuitionistic fuzzy left (resp. right, two-sided) ideal with thresholds (α, β] of R is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R, but converse is not true in general. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 909 Example 1. Let R = {0, 1, 2, 3, 4, 5, 6, 7}. Define + and · in R as follows : + 0 1 2 3 4 5 6 7 0 0 1 2 3 4 5 6 7 1 2 0 3 1 6 4 7 5 2 1 3 0 2 5 7 4 6 3 3 2 1 0 7 6 5 4 4 4 5 6 7 0 1 2 3 5 6 4 7 5 2 0 3 1 6 5 7 4 6 1 3 0 2 7 7 6 5 4 3 2 1 0 and · 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 0 4 4 0 0 4 4 0 2 0 4 4 0 0 4 4 0 3 0 0 0 0 0 0 0 0 4 0 3 3 0 0 3 3 0 5 0 7 7 0 0 7 7 0 6 0 7 7 0 0 7 7 0 7 0 3 3 0 0 3 3 0 Then R is an LA-ring and A = (µA, γA) be an IFS of an LA-ring R. We define (α = 0.1, β = 0.7) µA(0) = µA(4) = 0.7, µA(1) = µA(2) = µA(3) = µA(5) = µA(6) = µA(7) = 0.1 and γA(0) = γA(4) = 0.1, γA(1) = γA(2) = γA(3) = γA(5) = γA(6) = γA(7) = 0.7. Since max{µA(41), α} = max{µA(3), α} = max{0.1, 0.1} = 0.1. min{µA(4), β} = min{0.7, 0.7} = 0.7. ⇒ max{µA(41), α} � min{µA(4), β}. and min{γA(41), (1− α)} = min{γA(3), (1− α)} = min{0.7, 0.9} = 0.7. max{γA(4), (1− β)} = max{0.1, 0.3} = 0.3. ⇒ min{γA(41), (1− α)} � max{γA(4), (1− β)}. Then A = (µA, γA) is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R, but not an intuitionistic fuzzy right ideal with thresholds (α, β] of R. An IFS A = (µA, γA) of an LA-ring R is called an intuitionistic fuzzy interior ideal with thresholds (α, β] of R if (1) max{µA (x− y) , α} ≥ min{µA (x) , µA(y), β}, (2) min{γA (x− y) , (1− α)} ≤ max{γA (x) , γA(y), (1− β)}, (3) max{µA ((xy)z) , α} ≥ min{µA (y) , β}, (4) min{γA ((xy)z) , (1−α)} ≤ max{γA (y) , (1−β)} for all x, y, z ∈ R and α, β ∈ (0, 1] such that α < β. An IFS A = (µA, γA) of an LA-ring R is called an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R if (1) max{µA (x− y) , α} ≥ min{µA (x) , µA(y), β}, (2) min{γA (x− y) , (1− α)} ≤ max{γA (x) , γA(y), (1− β)}, (3) max{µA(x), α} ≥ min{(µA ◦R) (x), (R ◦ µA) (x), β}, (4) min{γA(x), (1−α)} ≤ max{(γA ◦R) (x), (R ◦ γA) (x), (1−β)} for all x, y ∈ R and α, β ∈ (0, 1] such that α < β. An intuitionistic fuzzy LA-subring A = (µA, γA) with thresholds (α, β] of an LA-ring R is called an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R if K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 910 (1) max{µA ((xy)z) , α} ≥ min{µA (x) , µA (z) , β}, (2) min{γA ((xy)z) , (1 − α)} ≤ max{γA (x) , γA (z) , (1 − β)} for all x, y, z ∈ R and α, β ∈ (0, 1] such that α < β. An IFS A = (µA, γA) of an LA-ring R is called an intuitionistic fuzzy generalized bi-ideal with thresholds (α, β] of R if (1) max{µA (x− y) , α} ≥ min{µA (x) , µA(y), β}, (2) min{γA (x− y) , (1− α)} ≤ max{γA (x) , γA(y), (1− β)}, (3) max{µA ((xy)z) , α} ≥ min{µA (x) , µA (z) , β}, (4) min{γA ((xy)z) , (1 − α)} ≤ max{γA (x) , γA (z) , (1 − β)} for all x, y, z ∈ R and α, β ∈ (0, 1] such that α < β. Let A = (µA, γA) and B = (µB, γB) be two intuitionistic fuzzy sets of an LA-ring R, then the product of A and B is denoted by A ◦B = (µA ◦ µB, γA ◦ γB) and defined by: (µA ◦ µB)(x) =  ∨ x= n∑ i=1 aibi {∧ni=1{µA(ai) ∧ µB(bi)}} if x = n∑ i=1 aibi, ai, bi ∈ R 0 if x 6= n∑ i=1 aibi and (γA ◦ γB)(x) =  ∧ x= n∑ i=1 aibi {∨ni=1{γA(ai) ∨ γB(bi)}} if x = n∑ i=1 aibi, ai, bi ∈ R 1 if x 6= n∑ i=1 aibi Let A = (µA, γA) be an IFS of an LA-ring R and α, β ∈ (0, 1] such that α < β. We define an intuitionistic fuzzy set Aβα of R as follow: (µA)βα(x) = (µA(x) ∧ β) ∨ α and (γA)βα(x) = (γA(x) ∨ (1− β)) ∧ (1− α) for all x ∈ R. Let A = (µA, γA) and B = (µB, γB) be two intuitionistic fuzzy sets of an LA-ring R. We define intuitionistic fuzzy sets A ∧βα B, A ∨βα B, A ◦βα B and A−βα B of R as follows: (µA ∧βα µB)(x) = {(µA ∧ µB)(x) ∧ β} ∨ α and (γA ∨βα γB)(x) = {(γA ∨ γB)(x) ∨ (1− β)} ∧ (1− α). (µA ∨βα µB)(x) = {(µA ∨ µB)(x) ∧ β} ∨ α and (γA ∧βα γB)(x) = {(γA ∧ γB)(x) ∨ (1− β)} ∧ (1− α). (µA ◦βα µB)(x) = {(µA ◦ µB)(x) ∧ β} ∨ α and (γA ◦βα γB)(x) = {(γA ◦ γB)(x) ∨ (1− β)} ∧ (1− α). (µA −βα µB)(x) = {(µA − µB)(x) ∧ β} ∨ α K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 911 and (γA −βα γB)(x) = {(γA − γB)(x) ∨ (1− β)} ∧ (1− α), for all x ∈ R. Now we are giving the central properties of such ideals of an LA-ring R, which will be very helpful for further sections. Lemma 1. Let A and B be two intuitionistic fuzzy sets of an LA-ring R. Then the following properties holds. (1) A ∧βα B = Aβα ∧Bβ α. (2) A ∨βα B = Aβα ∨Bβ α. (3) A ◦βα B ≥ Aβα ◦Bβ α. If every element x of R is expressible as x = n∑ i=1 aibi, then A ◦βα B = Aβα ◦Bβ α. If χA = (µχA , γχA) is an intuitionistic characteristic function of A, then (χA)βα is defined as (µχA)βα(x) = { β if x ∈ A α if x /∈ A and (γχA)βα(x) = { α if x ∈ A β if x /∈ A Lemma 2. Let R be an LA-ring. Then the following properties hold. (1) (A ◦βα B) ◦βα C = (C ◦βα B) ◦βα A, (2) (A◦βαB)◦βα (C ◦βαD) = (A◦βαC)◦βα (B ◦βαD) for all intuitionistic fuzzy sets A,B,C and D of R. Proof. Let A = (µA, γA) , B = (µB, γB) and C = (µC , γC) be intuitionistic fuzzy sets of an LA-ring R. We have to show that (A ◦βα B) ◦βα C = (C ◦βα B) ◦βα A. Now ((A ◦βα B) ◦βα C)(x) = {((A ◦B) ◦ C)(x) ∧ β} ∨ α = {((C ◦B) ◦A)(x) ∧ β} ∨ α = ((C ◦βα B) ◦βα A)(x). In same lines, we can prove (2) . Proposition 1. Let R be an LA-ring with left identity e. Then the following assertions hold. (1) A ◦βα (B ◦βα C) = B ◦βα (A ◦βα C), (2) (A ◦βα B) ◦βα (C ◦βα D) = (D ◦βα B) ◦βα (C ◦βα A), (3) (A◦βαB)◦βα (C ◦βαD) = (D ◦βαC)◦βα (B ◦βαA) for all intuitionistic fuzzy sets A,B,C and D of R. Proof. Same as Lemma 2. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 912 Theorem 1. Let A and B be two non-empty subsets of an LA-ring R. Then the following conditions hold. (1) χA ◦βα χB = (χAB)βα. (2) χA ∨βα χB = (χA∪B)βα. (3) χA ∧βα χB = (χA∩B)βα. Proof. Straight forward. Theorem 2. Let A be a non-empty subset of an LA-ring R. Then the following properties hold. (1) A is an LA-subring of R if and only if χA is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. (2) A is a left (resp. right, two-sided) ideal of R if and only if χA is an intuitionistic fuzzy left (resp. right, two-sided) ideal with thresholds (α, β] of R. Proof. (1) Let A be an LA-subring of an LA-ring R and x, y ∈ R. If x, y /∈ A, then by definition of intuitionistic characteristic function µχA(x) = 0 = µχA(y) and γχA(x) = 1 = γχA(y). Thus µχA(x− y) ≥ min{µχA(x), µχA(y)} = min{µχA(x), µχA(y), β} ⇒ µχA(x− y) ≥ min{µχA(x), µχA(y), β} ⇒ max{µχA(x− y), α} ≥ min{µχA(x), µχA(y), β} and µχA(xy) ≥ min{µχA(x), µχA(y)} = min{µχA(x), µχA(y), β} ⇒ µχA(xy) ≥ min{µχA(x), µχA(y), β} ⇒ max{µχA(xy), α} ≥ min{µχA(x), µχA(y), β}. Similarly, we have min{γχA(x− y), (1− α)} ≤ max{γχA(x), γχA(y), (1− β)} and min{γχA(xy), (1− α)} ≤ max{γχA(x), γχA(y), (1− β)}. In same lines, we have max{µχA(x− y), α} ≥ min{µχA(x), µχA(y), β}, max{µχA(xy), α} ≥ min{µχA(x), µχA(y), β}, min{γχA(x− y), (1− α)} ≤ max{γχA(x), γχA(y), (1− β)}, min{γχA(xy), (1− α)} ≤ max{γχA(x), γχA(y), (1− β)}, when x, y ∈ A. Hence the intuitionistic characteristic function χA of A is an intuition- istic fuzzy LA-subring with thresholds (α, β] of R. Conversely, suppose that the intuitionistic characteristic function χA of A is an intu- itionistic fuzzy LA-subring with thresholds (α, β] of an LA-ring R. Let x, y ∈ A, then by definition µχA(x) = 1 = µχA(y) and γχA(x) = 0 = γχA(y). Since max{µχA(x− y), α} ≥ min{µχA(x), µχA(y), β} = β, K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 913 max{µχA(xy), α} ≥ min{µχA(x), µχA(y), β} = β, min{γχA(x− y), (1− α)} ≤ max{γχA(x), γχA(y), (1− β)} = 1− β, min{γχA(xy), (1− α)} ≤ max{γχA(x), γχA(y), (1− β)} = 1− β, χA being an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. Thus max{µχA(x− y), α} ≥ β and max{µχA(xy), α} ≥ β. min{γχA(x− y), (1− α)} ≤ 1− β and min{γχA(xy), (1− α)} ≤ 1− β. This implies that µχA(x− y) = 1 = µχA(xy) and γχA(x− y) = 0 = γχA(xy), i.e., x− y and xy ∈ A. Hence A is an LA-subring of R. (2) Let A be a left ideal of an LA-ring R and x, y ∈ R. If y /∈ A, then by definition of intuitionistic characteristic function µχA(y) = 0 and γχA(y) = 1. Thus µχA(xy) ≥ µχA(y) = min{µχA(y), β} ⇒ µχA(xy) ≥ min{µχA(y), β} ⇒ max{µχA(xy), α} ≥ min{µχA(y), β} and γχA(xy) ≤ γχA(y) = max{γχA(y), (1− β)} ⇒ γχA(xy) ≤ max{γχA(y), (1− β)} ⇒ min{γχA(xy), (1− α)} ≤ max{γχA(y), (1− β)}. Similarly, we have max{µχA(xy), α} ≥ min{µχA(y), β}, min{γχA(xy), (1− α)} ≤ max{γχA(y), (1− β)}, when y ∈ A. Therefore the intuitionistic characteristic function χA of A is an intu- itionistic fuzzy left ideal with thresholds (α, β] of R. Conversely, assume that the intuitionistic characteristic function χA of A is an intu- itionistic fuzzy left ideal with thresholds (α, β] of an LA-ring R. Let y ∈ A and z ∈ R, then by definition µχA(y) = 1 and γχA(y) = 0. Since max{µχA(zy), α} ≥ min{µχA(y), β} = β, min{γχA(zy), (1− α)} ≤ max{γχA(y), (1− β)} = 1− β, χA being an intuitionistic fuzzy left ideal with thresholds (α, β] of R. Thus max{µχA(zy), α} ≥ β and min{γχA(zy), (1− α)} ≤ 1− β. This implies that µχA(zy) = 1 and γχA(zy) = 0, i.e., zy ∈ A. Therefore A is a left ideal of R. Remark 1. (i) A is an additive LA-subgroup of R if and only if χA is an intuitionistic fuzzy additive LA-subgroup with thresholds (α, β] of R. (ii) A is an LA-subsemigroup of R if and only if χA is an intuitionistic fuzzy LA- subsemigroup with thresholds (α, β] of R. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 914 Theorem 3. Let A be an IFS of an LA-ring R. Then the following assertions hold. (1) A is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R if and only if A ◦βα A ⊆ Aβα and A−βα A ⊆ Aβα. (2) A is an intuitionistic fuzzy left (resp. right) ideal with thresholds (α, β] of R if and only if R ◦βα A ⊆ Aβα (resp. A ◦βα R ⊆ Aβα) and A−βα A ⊆ Aβα. Proof. (1) Suppose that A = (µA, γA) is an intuitionistic fuzzy LA-subring with thresholds (α, β] of an LA-ring R and x ∈ R. For A ◦βα A ⊆ Aβα. If (A ◦βα A)(x) = 0, then obvious A ◦βα A ⊆ Aβα, otherwise we have (µA ◦βα µA)(x) = {(µA ◦ µA)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 {µA (ai) ∧ µA (bi)}} ) ∧ β} ∨ α ≤ { ( ∨x=∑n i=1 aibi {∧ni=1µA (aibi)} ) ∧ β} ∨ α = {(µA(x) ∧ β)} ∨ α = (µA)βα(x). ⇒ µA ◦βα µA ⊆ (µA)βα. Similarly, we have γA ◦βα γA ⊇ (γA)βα. Thus A ◦βα A ⊆ Aβα. Now for A−βα A ⊆ Aβα. If (A−βα A)(x) = 0, then obvious A−βα A ⊆ Aβα, otherwise we have (µA −βα µA)(x) = {(µA − µA)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 ai−bi{∧ n i=1 {µA (ai) ∧ µA (bi)}} ) ∧ β} ∨ α ≤ { ( ∨x=∑n i=1 ai−bi{∧ n i=1µA (ai − bi)} ) ∧ β} ∨ α = {(µA(x) ∧ β)} ∨ α = (µA)βα(x). ⇒ µA −βα µA ⊆ (µA)βα. Similarly, we have γA −βα γA ⊇ (γA)βα. Thus A−βα A ⊆ Aβα. Conversely, assume that A◦βαA ⊆ Aβα and A−βαA ⊆ Aβα. Let x, y ∈ R such that a = xy. Now max{µA(xy), α} = max{µA(a), α} = max{min{µA(a), β}, α} = (µA)βα(a) ≥ (µA ◦βα µA)(a) = {(µA ◦ µA)(a) ∧ β} ∨ α = { ( ∨a=∑n i=1 aibi {∧ni=1 {µA (ai) ∧ µA (bi)}} ) ∧ β} ∨ α ≥ {(µA(x) ∧ µA (y)) ∧ β} ∨ α = (µA(x) ∧ µA (y)) ∧ β = min{µA(x), µA (y) , β}. ⇒ max{µA(xy), α} ≥ min{µA(x), µA (y) , β}. Similarly, we have min{γA(xy), (1 − α)} ≤ max{γA(x), γA (y) , (1 − β)}. Now we set a = x− y and max{µA(x− y), α} = max{µA(a), α} = max{min{µA(a), β}, α} K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 915 = µβα(a) ≥ (µA −βα µA)(a) = {(µA − µA)(a) ∧ β} ∨ α = { ( ∨a=∑n i=1 ai−bi{∧ n i=1 {µA (ai) ∧ µA (bi)}} ) ∧ β} ∨ α ≥ {(µA(x) ∧ µA (y)) ∧ β} ∨ α = (µA(x) ∧ µA (y)) ∧ β = min{µA(x), µA (y) , β}. ⇒ max{µA(x− y), α} ≥ min{µA(x), µA (y) , β}. Similarly, we have min{γA(x− y), (1−α)} ≤ max{γA(x), γA (y) , (1− β)}. Hence A is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. (2) Assume that A is an intuitionistic fuzzy left ideal with thresholds (α, β] of an LA-ring R and x ∈ R. If (R ◦βα A)(x) = 0, then obvious R ◦βα A ⊆ Aβα, otherwise we have (R ◦βα µA)(x) = {(R ◦ µA)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 {R (ai) ∧ µA (bi)}} ) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 {1 ∧ µA (bi)}} ) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1µA (bi)} ) ∧ β} ∨ α ≤ { ( ∨x=∑n i=1 aibi {∧ni=1µA (aibi)} ) ∧ β} ∨ α = (µA(x) ∧ β) ∨ α = (µ)βα(x). ⇒ R ◦βα µA ⊆ (µA)βα. Similarly, we have R ◦βα γA ⊇ (γA)βα. Thus R ◦βα A ⊆ Aβα. Conversely, suppose that R ◦βα A ⊆ Aβα. Let y, z ∈ R such that x = yz. Now max{µA(yz), α} = max{µA(x), α} = max{min{µA(x), β}, α} = (µA)βα(x) ≥ (R ◦βα µA)(x) = {(R ◦ µA)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 {R (ai) ∧ µA (bi)}} ) ∧ β} ∨ α ≥ ((R(y) ∧ µA (z)) ∧ β) ∨ α = (1 ∧ µA (z)) ∧ β = min{µA (z) , β}. ⇒ max{µA(yz), α} ≥ min{µA (z) , β}. Similarly, we have min{γA(yz), (1 − α)} ≤ max{γA (z) , (1 − β)}. Therefore A is an intuitionistic fuzzy left ideal with thresholds (α, β] of R. Lemma 3. If A and B are two intuitionistic fuzzy LA-subrings (resp. (left, right, two- sided) ideals) with thresholds (α, β] of an LA-ring R, then A∧βαB is also an intuitionistic fuzzy LA-subring (resp. (left, right, two-sided) ideal) with thresholds (α, β] of R. Proof. Let A = (µA, γA) and B = (µB, γB) be two intuitionistic fuzzy LA-subrings with thresholds (α, β] of an LA-ring R. We have to show that A∧βαB is also an intuitionistic K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 916 fuzzy LA-subring with thresholds (α, β] of R. Now max{(µA ∧βα µB)(x− y), α} = max{{{(µA ∧ µB)(x− y) ∧ β} ∨ α}, α} = {(µA ∧ µB)(x− y) ∧ β} ∨ α = {µA(x− y) ∧ µB(x− y) ∧ β} ∨ α ≥ {µA(x) ∧ µA(y) ∧ µB(x) ∧ µB(y) ∧ β} ∨ α = {µA(x) ∧ µB(x) ∧ µA(y) ∧ µB(y) ∧ β} ∨ α = {(µA ∧ µB)(x) ∧ (µA ∧ µB)(y) ∧ β ∧ β ∧ β} ∨ α = {((µA ∧ µB)(x) ∧ β) ∧ ((µA ∧ µB)(y) ∧ β) ∧ β} ∨ α = ({(µA ∧ µB)(x) ∧ β} ∨ α) ∧ ({(µA ∧ µB)(y) ∧ β} ∨ α) ∧ (β ∨ α) = (µA ∧βα µB)(x) ∧ (µA ∧βα µB)(y) ∧ β = min{(µA ∧βα µB)(x), (µA ∧βα µB)(y), β}. Thus max{(µA ∧βα µB)(x− y), α} ≥ min{(µA ∧βα µB)(x), (µA ∧βα µB)(y), β}. Similarly, we have max{(µA ∧βα µB)(xy), α} ≥ min{(µA ∧βα µB)(x), (µA ∧βα µB)(y), β}. In same lines we have min{(γA ∨βα γB)(x − y), (1 − α)} ≤ max{(γA ∨βα γB)(x), (γA ∨βα γB)(y), (1 − β)} and min{(γA ∨βα γB)(xy), (1− α)} ≤ max{(γA ∨βα γB)(x), (γA ∨βα γB)(y), (1− β)}. Hence A ∧βα B is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. Lemma 4. If A and B are two intuitionistic fuzzy LA-subrings with thresholds (α, β] of an LA-ring R, then A ◦βαB is also an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. Proof. Suppose that A = (µA, γA) and B = (µB, γB) are two intuitionistic fuzzy LA- subrings with thresholds (α, β] of an LA-ring R. We have to show that A ◦βα B is also an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. Now (µA ◦βα µB)2 = (µA ◦βα µB) ◦βα (µA ◦βα µB) = (µA ◦βα µA) ◦βα (µB ◦βα µB) ⊆ (µA)βα ◦βα (µB)βα = µA ◦βα µB and (γA ◦βα γB)2 = (γA ◦βα γB) ◦βα (γA ◦βα γB) = (γA ◦βα γA) ◦βα (γB ◦βα γB) ⊇ (γA)βα ◦βα (γB)βα = γA ◦βα γB. Since µB −βα µB ⊆ (µB)βα and γB −βα γB ⊇ (γB)βα, B = (µB, γB) being an intuitionistic fuzzy LA-subring with thresholds (α, β]. This implies that µA◦βα(µB−βαµB) ⊆ µA◦βαµB and γA◦βα(γB−βαγB) ⊇ γA◦βαγB, i.e., µA◦βαµB−βαµA◦βαµB ⊆ µA◦βαµB and γA◦βαγB−βαγA◦βαγB ⊇ γA ◦βα γB. Therefore A ◦βα B is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. Remark 2. If A is an intuitionistic fuzzy LA-subring with thresholds (α, β] of an LA-ring R, then A ◦βα A is also an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 917 Lemma 5. Let R be an LA-ring with left identity e. Then every intuitionistic fuzzy right ideal with thresholds (α, β] of R is an intuitionistic fuzzy ideal with thresholds (α, β] of R. Proof. Suppose that A = (µA, γA) is an intuitionistic fuzzy right ideal with thresholds (α, β] of an LA-ring R and x, y ∈ R. Thus max{µA (xy) , α} = max{µA ((ex) y) , α} = max{µA ((yx) e) , α} ≥ min{µA (yx) , β} ≥ min{µA (y) , β} and min{γA (xy) , (1− α)} = min{γA ((ex) y) , (1− α)} = min{γA ((yx) e) , (1− α)} ≤ max{γA (yx) , (1− β)} ≤ max{γA (y) , (1− β)}. Therefore A is an intuitionistic fuzzy ideal with thresholds (α, β] of R. Lemma 6. If A and B are two intuitionistic fuzzy left (resp. right) ideals with thresholds (α, β] of an LA-ring R with left identity e, then A ◦βα B is also an intuitionistic fuzzy left (resp. right) ideal with thresholds (α, β] of R. Proof. Let A = (µA, γA) and B = (µA, γA) be two intuitionistic fuzzy left ideals with thresholds (α, β] of an LA-ring R. We have to show that A ◦βα B is also an intuitionistic fuzzy left ideal with thresholds (α, β] of R. Since, µA ◦βα µB −βα µA ◦βα µB ⊆ µA ◦βα µB and γA ◦βα γB −βα γA ◦βα γB ⊇ γA ◦βα γB by the Lemma 4. Now R ◦βα (µA ◦βα µB) = (R ◦βα R) ◦βα (µA ◦βα µB) = (R ◦βα µA) ◦βα (R ◦βα µB) ⊆ µA ◦βα µB and R ◦βα (γA ◦βα γB) = (R ◦βα R) ◦βα (γA ◦βα γB) = (R ◦βα γA) ◦βα (R ◦βα γB) ⊇ γA ◦βα γB. Hence A ◦βαB is an intuitionistic fuzzy left ideal with thresholds (α, β] of R. Similarly, we can prove for right ideals. Remark 3. If A is an intuitionistic fuzzy left (resp. right) ideal with thresholds (α, β] of an LA-ring R with left identity e, then A◦βαA is an intuitionistic fuzzy ideal with thresholds (α, β] of R. Lemma 7. If A and B are two intuitionistic fuzzy ideals with thresholds (α, β] of an LA-ring R, then A ◦βα B ⊆ A ∧βα B. Proof. Let A = (µA, γA) and B = (µB, γB) be two intuitionistic fuzzy ideals with thresholds (α, β] of an LA-ring R and x ∈ R. If (A ◦βα B)(x) = 0, then obvious A ◦βα B ⊆ A ∧βα B, otherwise we have (µA ◦βα µB)(x) = {(µA ◦ µB)(x) ∧ β} ∨ α K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 918 = { ( ∨x=∑n i=1 aibi {∧ni=1 {µA (ai) ∧ µB (bi)}} ) ∧ β} ∨ α ≤ { ( ∨x=∑n i=1 aibi {∧ni=1 {µA (aibi) ∧ µB (aibi)}} ) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 {(µA ∧ µB) (aibi)}} ) ∧ β} ∨ α = {(µA ∧ µB) (x) ∧ β} ∨ α = (µA ∧βα µB)(x). ⇒ µA ◦βα µB ⊆ µA ∧βα µB. Similarly, we have γA ◦βα γB ⊇ γA ∨βα γB. Hence A ◦βα B ⊆ A ∧βα B. Remark 4. If A is an intuitionistic fuzzy ideal with thresholds (α, β] of an LA-ring R, then A ◦βα A ⊆ Aβα. Lemma 8. Let R be an LA-ring. Then A ◦βα B ⊆ A ∧βα B for every intuitionistic fuzzy right ideal A with thresholds (α, β] and every intuitionistic fuzzy left ideal B with thresholds (α, β] of R. Proof. Same as Lemma 7. Theorem 4. Let A be a non-empty subset of an LA-ring R. Then the following conditions are true. (1) A is an interior ideal of R if and only if χA is an intuitionistic fuzzy interior ideal with thresholds (α, β] of R. (2) A is a quasi-ideal of R if and only if χA is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R. (3) A is a bi-ideal of R if and only if χA is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. (4) A is a generalized bi-ideal of R if and only if χA is an intuitionistic fuzzy generalized bi-ideal with thresholds (α, β] of R. Proof. Let A be an interior ideal of an LA-ring R, this implies that A is an additive LA-subgroup. Then χA is an intuitionistic fuzzy additive LA-subgroup with thresholds (α, β] of R by the Remark 1. Let x, y, a ∈ R. If a /∈ A, then by definition of intuitionistic characteristic function µχA(a) = 0 and γχA(a) = 1. Thus µχA((xa)y) ≥ µχA(a) = min{µχA(a), β} ⇒ µχA((xa)y) ≥ min{µχA(a), β} ⇒ max{µχA((xa)y), α} ≥ min{µχA(a), β}. Similarly, we have min{γχA((xa)y), (1 − α)} ≤ max{γχA(a), (1 − β)}. In same lines, we have max{µχA((xa)y), α} ≥ min{µχA(a), β} and min{γχA((xa)y), (1− α)} ≤ max{γχA(a), (1− β)}, K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 919 when a ∈ A. Hence the intuitionistic characteristic function χA of A is an i ntuitionistic fuzzy interior ideal with thresholds (α, β] of R. Conversely, suppose that the intuitionistic characteristic function χA of A is an intu- itionistic fuzzy interior ideal with thresholds (α, β] of R, this means that χA is an intu- itionistic fuzzy additive LA-subgroup with thresholds (α, β] of R. Then A is an additive LA-subgroup of R by the Remark 1. Let t ∈ (RA)R, so t = (xa)y, where a ∈ A and x, y ∈ R. Then by definition µχA(a) = 1 and γχA(a) = 0. Since max{µχA((xa)y), α} ≥ min{µχA(a), β} = β and min{γχA((xa)y), (1− α)} ≤ max{γχA(a), (1− β)} = 1− β, χA being an intuitionistic fuzzy interior ideal with thresholds (α, β] of R. This implies that µχA((xa)y) ≥ β and γχA((xa)y) ≤ 1 − β, thus µχA((xa)y) = 1 and µχA((xa)y) = 0, i.e., (xa)y ∈ A. Hence A is an interior ideal of R. (2) Let A be a quasi-ideal of R, this implies that A is an additive LA-subgroup. Then χA is an intuitionistic fuzzy additive LA-subgroup with thresholds (α, β] of R by the Remark 1. Let x ∈ R and x /∈ A, then x /∈ RA or x /∈ AR. If x /∈ RA, then definition of intuitionistic characteristic function (R ◦ µχA)(x) = 0 and (R ◦ γχA)(x) = 1. Thus max{µχA(x), α} ≥ 0 = min{(µχA ◦R) (x), (R ◦ µχA) (x), β} and min{γχA(x), (1− α)} ≤ 1 = max{(γχA ◦R) (x), (R ◦ γχA) (x), (1− β)}. If x ∈ A, then max{µχA(x), α} = 1 ≥ min{(µχA ◦R) (x), R ◦ µχA(x), β} and min{γχA(x), (1− α)} = 0 ≤ max{(γχA ◦R) (x), R ◦ γχA(x), (1− β)}. Therefore the intuitionistic characteristic function χA of A is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R. Conversely, assume that the intuitionistic characteristic function χA of A is an intu- itionistic fuzzy quasi-ideal with thresholds (α, β] of R, this means that χA is an intu- itionistic fuzzy additive LA-subgroup with thresholds (α, β] of R. Then A is an additive LA-subgroup of R by the Remark 1. Let x be an element of AR ∩ RA, this means that x ∈ AR and RA. Since max{µχA(x), α} ≥ min{(µχA ◦R)(x), (R ◦ µχA)(x), β} = min{(µχA ◦ µχR)(x), (µχR ◦ µχA)(x), β} = min{µχAR(x), µχRA(x), β} = β. ⇒ max{µχA(x), α} ≥ β. Similarly, we have min{γχA(x), (1 − α)} ≤ 1 − β, thus µχA(x) = 1 and γχA(x) = 0, i.e., x ∈ A. Therefore A is a quasi-ideal of R. (3) Let A be a bi-ideal of R, this implies that A is an LA-subring of R. Then χA is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R by the Remark 1. Let K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 920 x, y, a ∈ R. If x, y /∈ A, then by definition of intuitionistic characteristic function µχA(x) = µχA(y) = 0 and γχA(x) = γχA(y) = 1. Thus µχA((xa)y) ≥ µχA(x) ∧ µχA(y) = min{µχA(x), µχA(y), β} ⇒ µχA((xa)y) ≥ min{µχA(x), µχA(y), β} ⇒ max{µχA((xa)y), α} ≥ min{µχA(x), µχA(y), β}. Similarly, we have min{γχA((xa)y), (1−α)} ≤ max{γχA(x), µχA(y), (1− β)}. In same lines we have max{µχA((xa)y), α} ≥ min{µχA(x), µχA(y), β} and min{γχA((xa)y), (1− α)} ≤ max{γχA(x), γχA(y), (1− β)}, when x, y ∈ A. ence the intuitionistic characteristic function χA of A is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. Conversely, suppose that the intuitionistic characteristic function χA of A is an intu- itionistic fuzzy bi-ideal with thresholds (α, β] of R, this means that χA is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. Then A is an LA-subring of R by the Re- mark 1. Let t ∈ (AR)A, so t = (xa)y, where x, y ∈ A and a ∈ R. Then the definition µχA(x) = µχA(y) = 1 and γχA(x) = γχA(y) = 0. As max{µχA((xa)y), α} ≥ min{µχA(x), µχA(y), β} = β and min{γχA((xa)y), (1− α)} ≤ max{γχA(x), γχA(y), (1− β)} = 1− β, χA being an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. This implies that µχA((xa)y) ≥ β and γχA((xa)y) ≤ 1 − β, thus µχA((xa)y) = 1 and µχA((xa)y) = 0, i.e., (xa)y ∈ A. Hence A is bi-ideal of R. Similarly, we can prove (4) . Theorem 5. Let A = (µA, γA) be an IFS of an LA-ring R. Then A is an intuitionistic fuzzy interior ideal with thresholds (α, β] of R if and only if (R ◦βα A) ◦βα R ⊆ Aβα and A−βα A ⊆ Aβα. Proof. Suppose that A = (µA, γA) is an intuitionistic fuzzy interior ideal with thresh- olds (α, β] of an LA-ring R and x ∈ R. If ((R◦βαA)◦βαR)(x) = 0, then obvious (R◦βαA)◦βαR ⊆ Aβα. Otherwise there exist ai, bi, ci, di ∈ R such that x = ∑n i=1 aibi and ai = ∑n i=1 cidi. Since A is an intuitionistic fuzzy interior ideal with thresholds (α, β] of R, this implies that max{µA((cidi)bi), α} ≥ min{µA(di), β} andmin{γA((cidi)bi), (1−α)} ≤ max{γA(di), (1− β)}. Now ((R ◦βα µA) ◦βα R)(x) = {((R ◦ µA) ◦R)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 {(R ◦ µA) (ai) ∧R (bi)}} ) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 {(R ◦ µA) (ai) ∧ 1}} ) ∧ β} ∨ α K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 921 = { ( ∨x=∑n i=1 aibi {∧ni=1(R ◦ µA) (ai)} ) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 ( ∨ai=∑n i=1 cidi {∧ni=1 {R (ci) ∧ µA (di)}} ) } ) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 ( ∨ai=∑n i=1 cidi {∧ni=1 {1 ∧ µA (di)}} ) } ) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 ( ∨ai=∑n i=1 cidi {∧ni=1µA (di)} ) } ) ∧ β} ∨ α = { ( ∨x=∑n i=1(cidi)bi {∧ni=1µA (di)} ) ∧ β} ∨ α ≤ { ( ∨x=∑n i=1(cidi)bi {∧ni=1µA ((cidi) bi)} ) ∧ β} ∨ α = {µA(x) ∧ β} ∨ α = (µA)βα(x). ⇒ (R ◦βα µA) ◦βα R ⊆ (µA)βα. Similarly, we have (R ◦βα γA) ◦βα R ⊇ (γA)βα. Hence (R ◦βα A) ◦βα R ⊆ Aβα. Conversely, assume that (R ◦βαA) ◦βαR ⊆ Aβα and x, y, z ∈ R such that a = (xy)z. Now max{µA((xy)z), α} = max{min{µA((xy)z), β}, α} = max{min{µA(a), β}, α} = (µA)βα(a) ≥ ((R ◦βα µA) ◦βα R)(a) = {((R ◦ µA) ◦R)(a) ∧ β} ∨ α = { ( ∨a=∑n i=1 aibi {∧ni=1 {(R ◦ µA) (ai) ∧R (bi)}} ) ∧ β} ∨ α ≥ {((R ◦ µA) (xy) ∧R (z)) ∧ β} ∨ α = {((R ◦ µA) (xy) ∧ 1) ∧ β} ∨ α = {(R ◦ µA) (xy) ∧ β} ∨ α = { ( ∨xy=∑n i=1 cidi {∧ni=1 {R (ai) ∧ µA (bi)}} ) ∧ β} ∨ α ≥ {(R (x) ∧ µA (y)) ∧ β} ∨ α = {(1 ∧ µA (y)) ∧ β} ∨ α = µA(y) ∧ β = min{µA(y), β}. ⇒ max{µA((xy)z), α} ≥ min{µA(y), β}. Similarly, we have min{γA((xy)z), (1− α)} ≤ max{γA(y), (1− β)}. Therefore A is an intuitionistic fuzzy interior ideal with thresholds (α, β] of R. Theorem 6. Let A = (µA, γA) be an intuitionistic fuzzy LA-subring with thresholds (α, β] of an LA-ring R. Then A is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R if and only if (A ◦βα R) ◦βα A ⊆ Aβα. Proof. Same as Theorem 5. Theorem 7. Let A = (µA, γA) be an IFS of an LA-ring R. Then A is an intuitionistic fuzzy generalized bi-ideal with thresholds (α, β] of R if and only if (A ◦βαR) ◦βαA ⊆ Aβα and A−βα A ⊆ Aβα. Proof. Same as Theorem 5. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 922 Lemma 9. If A and B are two intuitionistic fuzzy bi- (resp. generalized bi-, quasi-, interior) ideals with thresholds (α, β] of an LA-ring R, then A∧βαB is also an intuitionistic fuzzy bi- (resp. generalized bi-, quasi-, interior) ideal with thresholds (α, β] of R. Proof. Let A = (µA, γA) and B = (µB, γB) be two intuitionistic fuzzy bi-ideals with thresholds (α, β] of an LA-ringR.We have to show thatA∧βαB is also an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. Since A and B are intuitionistic fuzzy LA-subrings with thresholds (α, β] of R, then A ∧βα B is also an intuitionistic fuzzy LA-subring with thresholds (α, β] ofR by the Lemma 3. We have to show thatmax{(µA∧βαµB)((xa)y), α} ≥ min{(µA∧βαµB)(x), (µA∧βαµB)(y), β} and min{(γA∨βαγB)((xa)y), (1−α)} ≤ max{(γA∨βα γB)(x), (γA ∨βα γB)(y), (1− β)}. Now max{(µA ∧βα µB)((xa)y), α} = max{{{(µA ∧ µB)((xa)y) ∧ β} ∨ α}, α} = {(µA ∧ µB)((xa)y) ∧ β} ∨ α = {µA((xa)y) ∧ µB((xa)y) ∧ β} ∨ α ≥ {µA(x) ∧ µA(y) ∧ µB(x) ∧ µB(y) ∧ β} ∨ α = {µA(x) ∧ µB(x) ∧ µA(y) ∧ µB(y) ∧ β} ∨ α = {(µA ∧ µB)(x) ∧ (µA ∧ µB)(y) ∧ β ∧ β ∧ β} ∨ α = {((µA ∧ µB)(x) ∧ β) ∧ ((µA ∧ µB)(y) ∧ β) ∧ β} ∨ α = ({(µA ∧ µB)(x) ∧ β} ∨ α) ∧ ({(µA ∧ µB)(y) ∧ β} ∨ α) ∧ (β ∨ α) = (µA ∧βα µB)(x) ∧ (µA ∧βα µB)(y) ∧ β = min{(µA ∧βα µB)(x), (µA ∧βα µB)(y), β}. Thus max{(µA ∧βα µB)((xa)y), α} ≥ min{(µA ∧βα µB)(x), (µA ∧βα µB)(y), β}. Similarly, we have min{(γA ∨βα γB)((xa)y), (1− α)} ≤ max{(γA ∨βα γB)(x), (γA ∨βα γB)(y), (1− β)}. Hence A ∧βα B is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. Lemma 10. If A and B are two intuitionistic fuzzy bi- (resp. generalized bi-, interior) ideals with thresholds (α, β] of an LA-ring R with left identity e, then A ◦βα B is also an intuitionistic fuzzy bi- (resp. generalized bi-, interior) ideal with thresholds (α, β] of R. Proof. Let A = (µA, γA) and B = (µB, γB) be two intuitionistic fuzzy bi-ideals with thresholds (α, β] of an LA-ring R. We have to show that A ◦βα B is also an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. Since A and B are intuitionistic fuzzy LA- subrings with thresholds (α, β] of R, then A◦βαB is also an intuitionistic fuzzy LA-subring with thresholds (α, β] of R by the Lemma 4. Now ((µA ◦βα µB) ◦βα R) ◦βα (µA ◦βα µB) = ((µA ◦βα µB) ◦βα (R ◦βα R)) ◦βα (µA ◦βα µB) = ((µA ◦βα R) ◦βα (µB ◦βα R)) ◦βα (µA ◦βα µB) = ((µA ◦βα R) ◦βα µA) ◦βα ((µB ◦βα R) ◦βα µB) K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 923 ⊆ (µA)βα ◦βα (µB)βα = µA ◦βα µB. Similarly, we have ((γA ◦βα γB) ◦βα R) ◦βα (γA ◦βα γB) ⊇ γA ◦βα γB. Therefore A ◦βα B is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. Lemma 11. Every intuitionistic fuzzy ideal with thresholds (α, β] of an LA-ring R is an intuitionistic fuzzy interior ideal with thresholds (α, β] of R. The converse is not true in general. Proof. Let A = (µA, γA) be an intuitionistic fuzzy ideal with thresholds (α, β] of an LA-ring R and x, y, z ∈ R. Thus max{µA ((xy)z) , α} ≥ min{µA (xy) , β} ≥ min{µA (y) , β} and min{γA ((xy)z) , (1− α)} ≤ max{γA (xy) , (1− β)} ≤ max{γA (y) , (1− β)}. Hence A is an intuitionistic fuzzy interior ideal with thresholds (α, β] of R. Proposition 2. Let A = (µA, γA) be an IFS of an LA-ring R with left identity e. Then A is an intuitionistic fuzzy ideal with thresholds (α, β] of R if and only if A is an intuitionistic fuzzy interior ideal with thresholds (α, β] of R. Proof. Suppose that A = (µA, γA) is an intuitionistic fuzzy interior ideal with thresh- olds (α, β] of an LA-ring R and x, y ∈ R. Thus max{µA(xy), α} = max{µA((ex)y), α} ≥ min{µA(x), β} and min{γA(xy), (1− α)} = min{γA((ex)y), (1− α)} ≤ max{γA(x), (1− β)}. So A is an intuitionistic fuzzy right ideal with thresholds (α, β] of R. Therefore A is an intuitionistic fuzzy ideal with thresholds (α, β] of R by the Lemma 5. Converse is true by the Lemma 11. Lemma 12. Every intuitionistic fuzzy left (resp. right, two-sided) ideal with thresholds (α, β] of an LA-ring R is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. The converse is not true in general. Proof. Assume that A = (µA, γA) is an intuitionistic fuzzy right ideal with thresholds (α, β] of an LA-ring R and x, y, z ∈ R. Thus max{µA ((xy)z) , α} ≥ min{µA (xy) , β} ≥ min{µA (x) , β} and max{µA((xy)z), α} = max{µA((zy)x), α} ≥ min{µA(zy), β} ≥ min{µA(z), β}. This implies that max{µA((xy)z), α} ≥ min{µA(x), µA(z), β}. Similarly, we have min{γA((xy)z), (1− α)} ≤ max{γA(x), γA(z), (1− β)}. So A is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 924 Lemma 13. Every intuitionistic fuzzy bi-ideal with thresholds (α, β] of an LA-ring R is an intuitionistic fuzzy generalized bi-ideal with thresholds (α, β] of R. The converse is not true in general. Proof. Obvious. Lemma 14. Every intuitionistic fuzzy left (resp. right, two-sided) ideal with thresholds (α, β] of an LA-ring R is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R. The converse is not true in general. Proof. Let A = (µA, γA) be an intuitionistic fuzzy left ideal with thresholds (α, β] of an LA-ring R. Thus max{µA(x), α} ≥ min{(R ◦ µA)(x), β} ≥ min{(µA ◦R)(x), (R ◦ µA)(x), β} and min{γA(x), (1− α)} ≤ max{(R ◦ γA)(x), (1− β)} ≤ max{(γA ◦R)(x), (R ◦ γA)(x), (1− β)}. Hence A is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R. Proposition 3. Every intuitionistic fuzzy quasi-ideal with thresholds (α, β] of an LA-ring R is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. Proof. Suppose that A = (µA, γA) is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of an LA-ring R. Since µA◦βαµA ⊆ µA◦βαR and µA◦βαµA ⊆ R◦βαµA, this implies that µA ◦βα µA ⊆ µA ◦βαR∧R ◦βα µA ⊆ (µA)βα. Similarly we have, γA ◦βα γA ⊇ γA ◦βαR∨R ◦βα γA ⊇ (γA)βα. Therefore A is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. Proposition 4. Let A = (µA, γA) be an intuitionistic fuzzy right ideal with thresholds (α, β] and B = (µB, γB) be an intuitionistic fuzzy left ideal with thresholds (α, β] of an LA-ring R, respectively. Then A∧βαB is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R. Proof. We have to show that A∧βαB is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of an LA-ring R. Since max{(µA ∧βα µB)(x− y), α} ≥ min{(µA ∧βα µB)(x), (µA ∧βα µB)(y), β} and min{(γA ∨βα γB)(x− y), (1− α)} ≤ max{(γA ∨βα γB)(x), (γA ∨βα γB)(y), (1− β)}, by the Lemma 3 and ((µA ∧βα µB) ◦βα R) ∧ (R ◦βα (µA ∧βα µB)) K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 925 ⊆ (µA ◦βα R) ∧ (R ◦βα µB) ⊆ (µA)βα ∧ (µB)βα = µA ∧βα µB and ((γA ∨βα γB) ◦βα R) ∨ (R ◦βα (γA ∨βα γB)) ⊇ (γA ◦βα R) ∨ (R ◦βα γB) ⊇ (γA)βα ∨ (γB)βα = γA ∧βα γB. Thus A ∧βα B is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R. Lemma 15. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then every intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. Proof. Assume that A = (µA, γA) is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of an LA-ring R. This implies that A is an intuitionistic fuzzy LA-subring with thresholds (α, β] of R. We have to show that (µA◦βαR)◦βαµA ⊆ (µA)βα and (γA◦βαR)◦βαγA ⊇ (γA)βα. Now (µA ◦βα R) ◦βα µA ⊆ (R ◦βα R) ◦βα µA ⊆ R ◦βα µA and (µA ◦βα R) ◦βα µA ⊆ (µA ◦βα R) ◦βα R = (µA ◦βα R) ◦βα (e ◦βα R) = (µA ◦βα e) ◦βα (R ◦βα R) ⊆ (µA ◦βα e) ◦βα Rβα = (µA)βα ◦βα Rβα = µA ◦βα R. ⇒ (µA ◦βα R) ◦βα µA ⊆ µA ◦βα R ∧R ◦βα µA ⊆ (µA)βα. Similarly, we have (γA ◦βαR)◦βα γA ⊇ (γA)βα. So A is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. Proposition 5. If A and B are two intuitionistic fuzzy quasi-ideals with thresholds (α, β] of an LA-ring R with left identity e, such that (xe)R = xR for all x ∈ R, then A ◦βα B is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R. Proof. Let A and B be two intuitionistic fuzzy quasi-ideals with thresholds (α, β] of an LA-ring R, this implies that A and B be two intuitionistic fuzzy bi-ideals with thresholds (α, β] of R, by the Lemma 15. Then A ◦βα B is also an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R by the Lemma 10. 3. Regular LA-rings In this section, we characterize regular LA-rings by the properties of intuitionistic fuzzy left (right, quasi-, bi-, generalized bi-) ideals with thresholds (α, β]. An intuitionistic fuzzy ideal A = (µA, γA) with thresholds (α, β] of an LA-ring R is an intuitionistic fuzzy idempotent with thresholds (α, β] of R if A ◦βα A = Aβα. Lemma 16. Every intuitionistic fuzzy right ideal with thresholds (α, β] of a regular LA- ring R is an intuitionistic fuzzy ideal with thresholds (α, β] of R. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 926 Proof. Suppose that A = (µA, γA) is an intuitionistic fuzzy right ideal with thresholds (α, β] of R. Let x, y ∈ R, this implies that there exists a ∈ R, such that x = (xa)x. Thus max{µA(xy), α} = max{µA(((xa)x)y), α} = max{µA((yx)(xa)), α} ≥ min{µA(yx), β} ≥ min{µA(y), β} and min{γA(xy), (1− α)} = min{γA(((xa)x)y), (1− α)} = min{γA((yx)(xa)), (1− α)} ≤ max{γA(yx), (1− β)} ≤ max{γA(y), (1− β)}. Hence A is an intuitionistic fuzzy ideal with thresholds (α, β] of R. Lemma 17. Every intuitionistic fuzzy ideal with thresholds (α, β] of a regular LA-ring R is an intuitionistic fuzzy idempotent with thresholds (α, β]. Proof. Assume that A = (µA, γA) is an intuitionistic fuzzy ideal with thresholds (α, β] of R and A ◦βα A ⊆ Aβα. We have to show that Aβα ⊆ A ◦βα A. Let x ∈ R, this means that there exists a ∈ R such that x = (xa)x. Thus (µA ◦βα µA)(x) = {(µA ◦ µA)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 {µA (ai) ∧ µA (bi)}} ) ∧ β} ∨ α ≥ {{µA (xa) ∧ µA (x)} ∧ β} ∨ α = (µA (xa) ∨ α) ∧ (µA (x) ∨ α) ∧ (β ∨ α) ≥ (µA (x) ∧ β) ∧ µA (x) ∧ β = µA (x) ∧ β = (µA (x) ∧ β) ∨ α = (µA)βα(x). ⇒ (µA)βα ⊆ µA ◦βα µA. Similarly, we have (γA)βα ⊇ γA ◦βα γA. Therefore Aβα = A ◦βα A. Remark 5. Every intuitionistic fuzzy right ideal with thresholds (α, β] of a regular LA-ring R is an intuitionistic fuzzy idempotent with thresholds (α, β]. Proposition 6. Let A = (µA, γA) be an IFS of a regular LA-ring R. Then A is an intuitionistic fuzzy ideal with thresholds (α, β] of R if and only if A is an intuitionistic fuzzy interior ideal with thresholds (α, β] of R. Proof. Consider that A = (µA, γA) is an intuitionistic fuzzy interior ideal with thresh- olds (α, β] of R. Let x, y ∈ R, then there exists an element a ∈ R, such that x = (xa)x. Thus max{µA(xy), α} = max{µA(((xa)x)y), α} = max{µA((yx)(xa)), α} ≥ min{µA(x), β} and min{γA(xy), (1− α)} = min{γA(((xa)x)y), (1− α)} K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 927 = min{γA((yx)(xa)), (1− α)} ≤ max{γA(x), (1− β)}. Consequently A is an intuitionistic fuzzy right ideal with thresholds (α, β] of R. So A is an intuitionistic fuzzy ideal with thresholds (α, β] of R by the Lemma 16. Converse is true by the Lemma 11. Remark 6. The concept of intuitionistic fuzzy (interior, two-sided) ideals with thresholds (α, β] coincides in regular LA-rings. Proposition 7. Let R be a regular LA-ring. Then (A ◦βα R) ∧ (R ◦βα A) = Aβα for every intuitionistic fuzzy right ideal A with thresholds (α, β] of R. Proof. Suppose that A = (µA, γA) is an intuitionistic fuzzy right ideal with thresholds (α, β] of R. This implies that (A ◦βαR)∧ (R ◦βαA) ⊆ Aβα, because every intuitionistic fuzzy right ideal with thresholds (α, β] of R is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R by the Lemma 14. Let x ∈ R, this implies that there exists a ∈ R, such that x = (xa)x. Thus (µA ◦βα R)(x) = {(µA ◦R)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 {µA (ai) ∧R (bi)}} ) ∧ β} ∨ α ≥ {{µA (xa) ∧R (x)} ∧ β} ∨ α = {µA (xa) ∧ β} ∨ α = (µA (xa) ∨ α) ∧ (β ∨ α) ≥ (µA (x) ∧ β) ∧ β = µA (x) ∧ β = (µA (x) ∧ β) ∨ α = (µA)βα(x). ⇒ (µA)βα ⊆ µA ◦βα R. Similarly, we have (µA)βα ⊆ R ◦βα µA, i.e., (µA)βα ⊆ (µA ◦βαR)∧ (R ◦βα µA). In same lines, we have (γA)βα ⊇ (γA ◦βα R) ∨ (R ◦βα γA). Hence (A ◦βα R) ∧ (R ◦βα A) = Aβα. Lemma 18. Let R be a regular LA-ring. Then A ◦βα B = A ∧βα B for every intuitionistic fuzzy right ideal A = (µA, γA) with thresholds (α, β] and every intuitionistic fuzzy left ideal B = (µB, γB) with thresholds (α, β] of R. Proof. Since A ◦βα B ⊆ A ∧βα B, for every intuitionistic fuzzy right ideal A = (µA, γA) with thresholds (α, β] and every intuitionistic fuzzy left ideal B = (µB, γB) with thresholds (α, β] of R by the Lemma 8. Let x ∈ R, this means that there exists a ∈ R such that x = (xa)x. Thus (µA ◦βα µB)(x) = {(µA ◦ µB)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 aibi {∧ni=1 {µA (ai) ∧ µB (bi)}} ) ∧ β} ∨ α ≥ {{µA (xa) ∧ µB (x)} ∧ β} ∨ α = (µA (xa) ∨ α) ∧ (µB (x) ∨ α) ∧ (β ∨ α) K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 928 ≥ (µA (x) ∧ β) ∧ µB (x) ∧ β = µA (x) ∧ µB (x) ∧ β = (µA ∧ µB) (x) ∧ β = {(µA ∧ µB) (x) ∧ β} ∨ α = (µA ∧βα µB)(x). ⇒ µA ∧βα µB ⊆ µA ◦βα µB. Similarly, we have γA∨βαγB ⊇ γA◦βαγB, i.e., A∧βαB ⊆ A◦βαB. Therefore A◦βαB = A∧βαB. Lemma 19. Let R be an LA-ring with left identity e. Then Ra is the smallest left ideal of R containing a. Proof. Let x, y ∈ Ra and r ∈ R. This implies that x = r1a and y = r2a, where r1, r2 ∈ R. Now x− y = r1a− r2a = (r1 − r2)a ∈ Ra and rx = r(r1a) = (er)(r1a) = ((r1a)r)e = ((r1a)(er))e = ((r1e)(ar))e = (e(ar))(r1e) = (ar)(r1e) = ((r1e)r)a ∈ Ra. Since a = ea ∈ Ra. Thus Ra is a left ideal of R containing a. Let I be another left ideal of R containing a. Since ra ∈ I, where ra ∈ Ra, i.e., Ra ⊆ I. Hence Ra is the smallest left ideal of R containing a. Lemma 20. Let R be an LA-ring with left identity e. Then aR is a left ideal of R. Proof. Straight forward. Proposition 8. Let R be an LA-ring with left identity e. Then aR ∪ Ra is the smallest right ideal of R containing a. Proof. Let x, y ∈ aR ∪ Ra, this means that x, y ∈ aR or Ra. Since aR and Ra both are left ideals of R, so x − y ∈ aR and Ra, i.e., x − y ∈ aR ∪ Ra. We have to show that (aR ∪Ra)R ⊆ (aR ∪Ra). Now (aR ∪Ra)R = (aR)R ∪ (Ra)R = (RR)a ∪ (Ra)(eR) ⊆ Ra ∪ (Re)(aR) = Ra ∪R(aR) = Ra ∪ a(RR) ⊆ Ra ∪ aR = aR ∪Ra. ⇒ (aR ∪Ra)R ⊆ aR ∪Ra. Since a ∈ Ra, i.e., a ∈ aR ∪ Ra. Let I be another right ideal of R containing a. Since aR ∈ IR ⊆ I and Ra = (RR)a = (aR)R ∈ (IR)R ⊆ IR ⊆ I, i.e., aR ∪Ra ⊆ I. Therefore aR ∪Ra is the smallest right ideal of R containing a. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 929 Theorem 8. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is a regular. (2) A∧βαB = A◦βαB for every intuitionistic fuzzy right ideal A with thresholds (α, β] and every intuitionistic fuzzy left ideal B with thresholds (α, β] of R. (3) Cβα = (C ◦βα R) ◦βα C for every intuitionistic fuzzy quasi-ideal C with thresholds (α, β] of R. Proof. Consider that (1) holds and C = (µC , γC) be an intuitionistic fuzzy quasi- ideal with thresholds (α, β] of R. This implies that (C ◦βα R) ◦βα C ⊆ Cβα , because every intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R by the Lemma 15. Let x ∈ R, then there exists an element a ∈ R such that x = (xa)x. Thus ((µC ◦βα R) ◦βα µC)(x) = {((µC ◦R) ◦ µC)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 piqi {∧ni=1 {(µC ◦R) (pi) ∧ µC (qi)}} ) ∧ β} ∨ α ≥ {{(µC ◦R) (xa) ∧ µC (x)} ∧ β} ∨ α = ((µC ◦R) (xa) ∨ α) ∧ (µC (x) ∨ α) ∧ (β ∨ α) = ((µC ◦R) (xa) ∨ α) ∧ µC(x) ∧ β = ( ( ∨xa=∑n i=1mini {∧ni=1 {µC (mi) ∧R (ni)}} ) ∨ α) ∧ µC(x) ∧ β ≥ ({µC(x) ∧R(a)} ∨ α) ∧ µC(x) ∧ β = ({µC(x) ∧ 1} ∨ α) ∧ µC(x) ∧ β = (µC(x) ∨ α) ∧ µC(x) ∧ β = µC(x) ∧ β = (µC(x) ∧ β) ∨ α = (µC)βα(x). ⇒ (µC)βα ⊆ (µC ◦βα R) ◦βα µC . Similarly, we have (γC)βα ⊇ (γC ◦βα R) ◦βα γC . So Cβα = (C ◦βα R) ◦βα C, i.e., (1) implies (3) . Suppose that (3) holds. Let A be an intuitionistic fuzzy right ideal with thresholds (α, β] and B be an intuitionistic fuzzy left ideal with thresholds (α, β] of R. This implies that A and B be intuitionistic fuzzy quasi-ideals with thresholds (α, β] of R by the Lemma 14, so A ∧βα B be also an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R. Then by our supposition, A ∧βα B = ((A ∧βα B) ◦βα R) ◦βα (A ∧βα B) ⊆ (A ◦βα R) ◦βα B ⊆ A ◦βα B, i.e., A∧βαB ⊆ A◦βαB. Since A◦βαB ⊆ A∧βαB, so A◦βαB = A∧βαB, i.e., (3)⇒ (2). Assume that (2) is true and a ∈ R. Then Ra is a left ideal of R containing a by the Lemma 19 and aR∪Ra is a right ideal of R containing a by the Proposition 8. This means that χRa is an intuitionistic fuzzy left ideal with thresholds (α, β] and χaR∪Ra is an intuitionistic fuzzy right ideal with thresholds (α, β] of R, by the Theorem 2. Then by our assumption χaR∪Ra ∧βα χRa = χaR∪Ra ◦βα χRa, i.e., (χ(aR∪Ra)∩Ra) β α = (χ(aR∪Ra)Ra) β α by the Theorem 1. Thus (aR∪Ra)∩Ra = (aR∪Ra)Ra. Since a ∈ (aR∪Ra)∩Ra, i.e., a ∈ (aR∪Ra)Ra, so a ∈ K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 930 (aR)(Ra) ∪ (Ra)(Ra). This implies that a ∈ (aR)(Ra) or a ∈ (Ra)(Ra). If a ∈ (aR)(Ra), then a = (ax)(ya) = ((ya)x)a = (((ey)a)x)a = (((ay)e)x)a = ((xe)(ay))a = (a((xe)y))a for any x, y ∈ R. If a ∈ (Ra)(Ra), then (Ra)(Ra) = ((Re)a)(Ra) = ((ae)R)(Ra) = (aR)(Ra), i.e., a ∈ (aR)(Ra). Therefore a is a regular, i.e., R is a regular. So (2)⇒ (1) . Theorem 9. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is a regular. (2 ) Aβα = (A ◦βα R) ◦βα A for every intuitionistic fuzzy quasi-ideal A with thresholds (α, β] of R. (3) Bβ α = (B ◦βαR)◦βαB for every intuitionistic fuzzy bi-ideal B with thresholds (α, β] of R. (4) Cβα = (C ◦βα R) ◦βα C for every intuitionistic fuzzy generalized bi-ideal C with thresholds (α, β] of R. Proof. (1) ⇒ (4), is obvious. (4) ⇒ (3) , since every intuitionistic fuzzy bi-ideal with thresholds (α, β] of R is an intuitionistic fuzzy generalized bi-ideal with thresholds (α, β] of R by the Lemma 13. (3)⇒ (2) , since every intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R is an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R by the Lemma 15. (2)⇒ (1) , by the Theorem 8. Theorem 10. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is a regular. (2) A ∧βα I = (A ◦βα I) ◦βα A for every intuitionistic fuzzy quasi-ideal A with thresholds (α, β] and every intuitionistic fuzzy ideal I with thresholds (α, β] of R. (3) B ∧βα I = (B ◦βα I) ◦βα B for every intuitionistic fuzzy bi-ideal B with thresholds (α, β] and every intuitionistic fuzzy ideal I with thresholds (α, β] of R. (4) C ∧βα I = (C ◦βα I) ◦βα C for every intuitionistic fuzzy generalized bi-ideal C with thresholds (α, β] and every intuitionistic fuzzy ideal I with thresholds (α, β] of R. Proof. Suppose that (1) holds. Let C = (µC , γC) be an intuitionistic fuzzy generalized bi-ideal with thresholds (α, β] and I = (µI , γI) be an intuitionistic fuzzy ideal with thresh- olds (α, β] of R. Now (C ◦βα I) ◦βα C ⊆ (R ◦βα I) ◦βα R ⊆ I ◦βα R ⊆ Iβα and (C ◦βα I) ◦βα C ⊆ (C ◦βα R) ◦βα C ⊆ Cβα , i.e., (C ◦βα I) ◦βα C ⊆ Cβα ∧ Iβα = C ∧βα I. Let x ∈ R, this implies that there exists a ∈ R such that x = (xa)x. Now xa = ((xa)x)a = (ax)(xa) = x((ax)a). Thus ((µC ◦βα µI) ◦βα µC)(x) = {((µC ◦ µI) ◦ µC)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 piqi {∧ni=1 {(µC ◦ µI) (pi) ∧ µC (qi)}} ) ∧ β} ∨ α ≥ {{(µC ◦ µI) (xa) ∧ µC (x)} ∧ β} ∨ α = ((µC ◦ µI) (xa) ∨ α) ∧ (µC (x) ∨ α) ∧ (β ∨ α) K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 931 = ((µC ◦ µI) (xa) ∨ α) ∧ µC(x) ∧ β = ( ( ∨xa=∑n i=1mini {∧ni=1 {µC (mi) ∧ µI (ni)}} ) ∨ α) ∧ µC(x) ∧ β ≥ ({µC(x) ∧ µI((ax)a)} ∨ α) ∧ µC(x) ∧ β = (µC(x) ∨ α) ∧ (µI((ax)a) ∨ α) ∧ µC(x) ∧ β ≥ µC(x) ∧ (µI(x) ∧ β) ∧ µC(x) ∧ β = µC(x) ∧ µI(x) ∧ β = (µC ∧ µI)(x) ∧ β = {(µC ∧ µI)(x) ∧ β} ∨ α = (µC ∧βα µI)(x). ⇒ µC ∧βα µI ⊆ (µC ◦βα µI) ◦βα µC . Similarly, we have γC∨βαγI ⊇ (γC◦βαγI)◦βαγC . Therefore C∧βαI = (C◦βαI)◦βαC, i.e., (1)⇒ (4) . Since (4)⇒ (3) and (3)⇒ (2). Assume that (2) holds. Then A∧βαR = (A ◦βαR) ◦βαA, where R itself is an intuitionistic fuzzy two-sided ideal with thresholds (α, β] of R, i.e., Aβα = (A ◦βα R) ◦βα A. Hence R is a regular by the Theorem 8, i.e., (2)⇒ (1) . Theorem 11. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is a regular. (2) A ∧βα D ⊆ D ◦βα A for every intuitionistic fuzzy quasi-ideal A with thresholds (α, β] and every intuitionistic fuzzy right ideal D with thresholds (α, β] of R. (3) B∧βαD ⊆ D ◦βαB for every intuitionistic fuzzy bi-ideal B with thresholds (α, β] and every intuitionistic fuzzy right ideal D with thresholds (α, β] of R. (4) C∧βαD ⊆ D◦βαC for every intuitionistic fuzzy generalized bi-ideal C with thresholds (α, β] and every intuitionistic fuzzy right ideal D with thresholds (α, β] of R. Proof. (1)⇒ (4), is obvious. It is clear that (4)⇒ (3) and (3)⇒ (2) . Assume that (2) holds, this means that D ∧βα A = A∧βαD ⊆ D ◦βα A, where A is an intuitionistic fuzzy left ideal with thresholds (α, β] of R. Since D ◦βα A ⊆ D ∧βα A, so D ∧βα A = D ◦βα A. Therefore R is a regular by the Theorem 8, i.e., (2)⇒ (1) . Theorem 12. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is a regular. (2) A ∧βα D ∧βα L ⊆ (A ◦βα D) ◦βα L for every intuitionistic fuzzy quasi-ideal A with thresholds (α, β], every intuitionistic fuzzy right ideal D with thresholds (α, β] and every intuitionistic fuzzy left ideal L with thresholds (α, β] of R. (3) B∧βαD∧βαL ⊆ (B ◦βαD)◦βαL for every intuitionistic fuzzy bi-ideal B with thresholds (α, β], every intuitionistic fuzzy right ideal D with thresholds (α, β] and every intuitionistic fuzzy left ideal L with thresholds (α, β] of R. (4) C ∧βα D ∧βα L ⊆ (C ◦βα D) ◦βα L for every intuitionistic fuzzy generalized bi-ideal C with thresholds (α, β], every intuitionistic fuzzy right ideal D with thresholds (α, β] and every intuitionistic fuzzy left ideal L with thresholds (α, β] of R. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 932 Proof. Consider that (1) holds. Let C = (µC , γC) be an intuitionistic fuzzy generalized bi-ideal with thresholds (α, β], L = (µL, γL) be an intuitionistic fuzzy left ideal with thresholds (α, β] and D = (µD, γD) be an intuitionistic fuzzy right ideal with thresholds (α, β] of R. Let x ∈ R, then there exists an element a ∈ R such that x = (xa)x. Now x = (xa)x. xa = ((xa)x)a = (ax)(xa) = x((ax)a). (ax)a = (a((xa)x))a = ((xa)(ax))a = (a(ax))(xa) = x((a(ax))a) = x(((ea)(ax))a) = x(((xa)(ae))a) = x((((ae)a)x)a) = x((nx)a) = x((nx)(ea)) = x((ae)(xn)) = x(x((ae)n)) = x(xm). ⇒ xa = x((ax)a) = x(x(xm)) = (ex)(x(xm)) = ((xm)x)(xe). Thus ((µC ◦βα µD) ◦βα µL)(x) = {((µC ◦ µD) ◦ µL)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 piqi {∧ni=1 {(µC ◦ µD) (pi) ∧ µL (qi)}} ) ∧ β} ∨ α ≥ {{(µC ◦ µD) (xa) ∧ µL (x)} ∧ β} ∨ α = ((µC ◦ µD) (xa) ∨ α) ∧ (µL (x) ∨ α) ∧ (β ∨ α) = ((µC ◦ µD) (xa) ∨ α) ∧ µL(x) ∧ β = ( ( ∨xa=∑n i=1mini {∧ni=1 {µC (mi) ∧ µD (ni)}} ) ∨ α) ∧ µL(x) ∧ β ≥ ({µC((xm)x) ∧ µD(xe)} ∨ α) ∧ µL(x) ∧ β = (µC((xm)x) ∨ α) ∧ (µD(xe) ∨ α) ∧ µL(x) ∧ β ≥ (µC(x) ∧ µC(x) ∧ β) ∧ (µD(x) ∧ β) ∧ µL(x) ∧ β = µC(x) ∧ µD(x) ∧ µL(x) ∧ β = (µC(x) ∧ µD(x) ∧ µL(x) ∧ β) ∨ α = (µC ∧βα µD ∧βα µL)(x). ⇒ µC ∧βα µD ∧βα µL ⊆ (µC ◦βα µD) ◦βα µL. Similarly, we have γC∨βαγD∨βαγL ⊇ (γC ◦βαγD)◦βαγL. Hence C∧βαD∧βαL ⊆ (C◦βαD)◦βαL, i.e., (1) ⇒ (4) . It is clear that (4) ⇒ (3) and (3) ⇒ (2) . Assume that (2) holds. Then A∧βαR∧βα L ⊆ (A ◦βαR) ◦βα L, where A is an intuitionistic fuzzy right ideal with thresholds (α, β] of R, i.e., A ∧βα L ⊆ A ◦βα L. Since A ◦βα L ⊆ A ∧βα L, thus A ◦βα L = A ∧βα L. So R is regular by the Theorem 8, i.e., (2)⇒ (1) . 4. Intra-regular LA-rings In this section, we characterize intra-regular LA-rings in terms of intuitionistic fuzzy left (right, quasi-, bi-, generalized bi-) ideals with thresholds (α, β]. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 933 Lemma 21. Every intuitionistic fuzzy left (right) ideal with thresholds (α, β] of an intra- regular LA-ring R is an intuitionistic fuzzy ideal with thresholds (α, β] of R. Proof. Suppose that A = (µA, γA) is an intuitionistic fuzzy left ideal with thresh- olds (α, β] of R. Let x, y ∈ R, this implies that there exist ai, bi ∈ R, such that x =∑n i=1(aix 2)bi. Thus max{µA(xy), α} = max{µA(((aix 2)bi)y), α} = max{µA((ybi)(aix 2)), α} ≥ min{µA(ai(xx)), β} ≥ min{µA(xx), β} ≥ min{µA(x), β} and min{γA(xy), (1− α)} = min{γA(((aix 2)bi)y), (1− α)} = min{γA((ybi)(aix 2)), (1− α)} ≤ max{γA(ai(xx)), (1− β)} ≤ max{γA(xx), (1− β)} ≤ max{γA(x), (1− β)}. Hence A is an intuitionistic fuzzy ideal with thresholds (α, β] of R. Lemma 22. Let R be an intra-regular LA-ring with left identity e. Then every intu- itionistic fuzzy ideal with thresholds (α, β] of R is an intuitionistic fuzzy idempotent with thresholds (α, β]. Proof. Assume that A = (µA, γA) is an intuitionistic fuzzy ideal with thresholds (α, β] of R and A ◦βα A ⊆ Aβα. Let x ∈ R, this means that there exist ai, bi ∈ R, such that x = ∑n i=1(aix 2)bi. Now x = (aix 2)bi = (ai(xx))bi = (x(aix))bi = (x(aix))(ebi) = (xe)((aix)bi) = (aix)((xe)bi). Thus (µA ◦βα µA)(x) = {(µA ◦ µA)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 piqi {∧ni=1 {µA (pi) ∧ µA (qi)}} ) ∧ β} ∨ α ≥ {{µA (aix) ∧ µA ((xe)bi)} ∧ β} ∨ α = (µA (aix) ∨ α) ∧ (µA ((xe)bi) ∨ α) ∧ (β ∨ α) ≥ (µA (x) ∧ β) ∧ (µA (x) ∧ β) ∧ β = µA (x) ∧ β = (µA (x) ∧ β) ∨ α = (µA)βα(x). ⇒ (µA)βα ⊆ µA ◦βα µA. Similarly, we have (γA)βα ⊇ γA ◦βα γA. Therefore Aβα = A ◦βα A. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 934 Proposition 9. Let A be an IFS of an intra-regular LA-ring R with left identity e. Then A is an intuitionistic fuzzy ideal with thresholds (α, β] of R if and only if A is an intuitionistic fuzzy interior ideal with thresholds (α, β] of R. Proof. Consider that A = (µA, γA) is an intuitionistic fuzzy interior ideal with thresh- olds (α, β] of R. Let x, y ∈ R, then there exist elements ai, bi ∈ R, such that x =∑n i=1(aix 2)bi. Thus max{µA(xy), α} = max{µA(((aix 2)bi)y), α} = max{µA((ybi)(aix 2)), α} = max{µA((ybi)(ai(xx))), α} = max{µA((ybi)(x(aix))), α} = max{µA((yx)(bi(aix))), α} ≥ min{µA(x), β}. ⇒ max{µA(xy), α} ≥ min{µA(x), β}. Similarly, we have min{γA(xy), (1−α)} ≤ max{γA(x), (1−β)}, i.e., A is an intuition- istic fuzzy right ideal with thresholds (α, β] of R. Therefore A is an intuitionistic fuzzy ideal with thresholds (α, β] of R by the Lemma 21. Converse is true by the Lemma 11. Remark 7. The concept of intuitionistic fuzzy (interior, two-sided) ideals with thresholds (α, β] coincides in intra-regular LA-rings with left identity. Lemma 23. Let R be an intra-regular LA-ring with left identity e. Then B ∧βα A ⊆ A ◦βα B for every intuitionistic fuzzy left ideal A = (µA, γA) with thresholds (α, β] and every intuitionistic fuzzy right ideal B = (µB, γB) with thresholds (α, β] of R. Proof. Suppose that A = (µA, γA) is an intuitionistic fuzzy left ideal with thresholds (α, β] and B = (µB, γB) is an intuitionistic fuzzy right ideal with thresholds (α, β] of R. Let x ∈ R, this implies that there exist ai, bi ∈ R such that x = ∑n i=1(aix 2)bi. Now x = (aix 2)bi = (ai(xx))bi = (x(aix))bi = (x(aix))(ebi) = (xe)((aix)bi) = (aix)((xe)bi). Thus (µA ◦βα µB)(x) = {(µA ◦ µB)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 piqi {∧ni=1 {µA (pi) ∧ µB (qi)}} ) ∧ β} ∨ α ≥ {{µA(aix) ∧ µB ((xe)bi)} ∧ β} ∨ α = (µA (aix) ∨ α) ∧ (µB ((xe)bi) ∨ α) ∧ (β ∨ α) ≥ (µA (x) ∧ β) ∧ (µB (x) ∧ β) ∧ β = µA (x) ∧ µB (x) ∧ β = µB (x) ∧ µA (x) ∧ β = {(µB ∧ µA) (x) ∧ β = {(µB ∧ µA) (x) ∧ β} ∨ α = (µB ∧βα µA)(x). ⇒ µB ∧βα µA ⊆ µA ◦βα µB. Similarly, we have γB ∨βα γA ⊇ γA ◦βα γB. Hence B ∧βα A ⊆ A ◦βα B. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 935 Theorem 13. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is an intra-regular. (2) B∧βαA ⊆ A◦βαB for every intuitionistic fuzzy left ideal A with thresholds (α, β] and every intuitionistic fuzzy right ideal B with thresholds (α, β] of R. Proof. (1) ⇒ (2) , is true by the Lemma 23. Assume that (2) holds and a ∈ R. Then Ra is a left ideal of R containing a by the Lemma 19 and aR ∪ Ra is a right ideal of R containing a by the Proposition 8. This means that χRa is an intuitionistic fuzzy left ideal with thresholds (α, β] and χaR∪Ra is an intuitionistic fuzzy right ideal with thresholds (α, β] of R, by the Theorem 2. By our assumption χaR∪Ra ∧βα χRa ⊆ χRa ◦βα χaR∪Ra, i.e., (χ(aR∪Ra)∩Ra) β α ⊆ (χ(Ra)(aR∪Ra)) β α by the Theorem 1. Thus (aR∪Ra)∩Ra ⊆ Ra(aR∪Ra). Since a ∈ (aR ∪ Ra) ∩ Ra, i.e., a ∈ Ra(aR ∪ Ra) = (Ra)(aR) ∪ (Ra)(Ra). This implies that a ∈ (Ra)(aR) or a ∈ (Ra)(Ra). If a ∈ (Ra)(aR), then (Ra)(aR) = (Ra)((ea)(RR)) = (Ra)((RR)(ae)) = (Ra)(((ae)R)R) = (Ra)((aR)R) = (Ra)((RR)a) = (Ra)(Ra) = ((Ra)a)R = ((Ra)(ea))R = ((Re)(aa))R = (Ra2)R. Thus a ∈ (Ra2)R. If a ∈ (Ra)(Ra), then obvious a ∈ (Ra2)R. So a is an intra regular. Therefore R is an intra-regular, i.e., (2)⇒ (1) . Theorem 14. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is an intra-regular. (2) A ∧βα I = (A ◦βα I) ◦βα A for every intuitionistic fuzzy quasi-ideal A with thresholds (α, β] and every intuitionistic fuzzy ideal I with thresholds (α, β] of R. (3) B ∧βα I = (B ◦βα I) ◦βα B for every intuitionistic fuzzy bi-ideal B with thresholds (α, β] and every intuitionistic fuzzy ideal I with thresholds (α, β] of R. (4) C ∧βα I = (C ◦βα I) ◦βα C for every intuitionistic fuzzy generalized bi-ideal C with thresholds (α, β] and every intuitionistic fuzzy ideal I with thresholds (α, β] of R. Proof. Consider that (1) holds. Let C = (µC , γC) be an intuitionistic fuzzy gen- eralized bi-ideal with thresholds (α, β] and I = (µI , γI) be an intuitionistic fuzzy ideal with thresholds (α, β] of R. Now (C ◦βα I) ◦βα C ⊆ (R ◦βα I) ◦βα R ⊆ I ◦βα R ⊆ Iβα and (C ◦βα I) ◦βα C ⊆ (C ◦βα R) ◦βα C ⊆ Cβα , thus (C ◦βα I) ◦βα C ⊆ Cβα ∧ Iβα = C ∧βα I. Let x ∈ R, then there exist elements ai, bi ∈ R such that x = ∑n i=1(aix 2)bi. Now x = (aix 2)bi = (ai(xx))bi = (x(aix))b = (bi(aix))x. bi(aix) = bi(ai((aix 2)bi)) = bi((aix 2)(aibi)) = bi((aix 2)ci) = (aix 2)(bici) = (aix 2)di = (aix 2)(edi) = (die)(x 2ai) = mi(x 2ai) = x2(miai) = (xx)li = (lix)x = (lix)(ex) K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 936 = (xe)(xli) = x((xe)li). Thus ((µC ◦βα µI) ◦βα µC)(x) = {((µC ◦ µI) ◦ µC)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 piqi {∧ni=1 {(µC ◦ µI) (pi) ∧ µC (qi)}} ) ∧ β} ∨ α ≥ {{(µC ◦ µI) (bi(aix)) ∧ µC (x)} ∧ β} ∨ α = ((µC ◦ µI) (bi(aix)) ∨ α) ∧ (µC (x) ∨ α) ∧ (β ∨ α) = ((µC ◦ µI) (bi(aix)) ∨ α) ∧ µC(x) ∧ β = ( ( ∨bi(aix)=∑n i=1mini {∧ni=1 {µC (mi) ∧ µI (ni)}} ) ∨ α) ∧ µC(x) ∧ β ≥ ({µC(x) ∧ µI((xe)li)} ∨ α) ∧ µC(x) ∧ β = (µC(x) ∨ α) ∧ (µI((xe)li) ∨ α) ∧ µC(x) ∧ β ≥ µC(x) ∧ (µI(x) ∧ β) ∧ µC(x) ∧ β = µC(x) ∧ µI(x) ∧ β = (µC ∧ µI)(x) ∧ β = {(µC ∧ µI)(x) ∧ β} ∨ α = (µC ∧βα µI)(x). ⇒ µC ∧βα µI ⊆ (µC ◦βα µI) ◦βα µC . Similarly, we have γC ∨βα γI ⊇ (γC ◦βα γI) ◦βα γC . Hence C ∧βα I = (C ◦βα I) ◦βα C, i.e., (1) implies (4) . It is clear that (4) ⇒ (3) and (3) ⇒ (2) . Suppose that (2) holds. Let A be an intuitionistic fuzzy right ideal with thresholds (α, β] and I be an intuitionistic fuzzy two-sided ideal with thresholds (α, β] of R. Since every intuitionistic fuzzy right ideal with thresholds (α, β] of R is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R by the Lemma 14, This implies that A is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R. By our supposition A ∧βα I = (A ◦βα I) ◦βα A ⊆ (R ◦βα I) ◦βα A ⊆ I ◦βα A, i.e., A ∧βα I ⊆ I ◦βα A. So R is an intra-regular by the Theorem 13, i.e., (2)⇒ (1) . Theorem 15. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is an intra-regular. (2) A∧βαL ⊆ L◦βαA for every intuitionistic fuzzy quasi-ideal A with thresholds (α, β] and every intuitionistic fuzzy left ideal L with thresholds (α, β] of R. (3) B ∧βα L ⊆ L ◦βαB for every intuitionistic fuzzy bi-ideal B with thresholds (α, β] and every intuitionistic fuzzy left ideal L with thresholds (α, β] of R. (4) C∧βαL ⊆ L◦βαC for every intuitionistic fuzzy generalized bi-ideal C with thresholds (α, β] and every intuitionistic fuzzy left ideal L with thresholds (α, β] of R. Proof. Suppose that (1) holds. Let C = (µC , γC) be an intuitionistic fuzzy generalized bi-ideal with thresholds (α, β] and L = (µL, γL) be an intuitionistic fuzzy left ideal with K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 937 thresholds (α, β] of R. Let x ∈ R, this implies that there exist ai, bi ∈ R such that x = ∑n i=1(aix 2)bi. Now x = (ai(xx))bi = (x(aix))bi = (bi(aix))x. Thus (µL ◦βα µC)(x) = {(µL ◦ µC)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 piqi {∧ni=1 {µL (pi) ∧ µC (qi)}} ) ∧ β} ∨ α ≥ {{µL (bi(aix)) ∧ µC (x)} ∧ β} ∨ α = (µL (bi(aix)) ∨ α) ∧ (µC (x) ∨ α) ∧ (β ∨ α) ≥ (µL (x) ∧ β) ∧ µC (x) ∧ β = µL (x) ∧ µC (x) ∧ β = µC (x) ∧ µL (x) ∧ β = (µC ∧ µL) (x) ∧ β = {(µC ∧ µL) (x) ∧ β} ∨ α = (µC ∧βα µL)(x). ⇒ µC ∧βα µL ⊆ µL ◦βα µC . Similarly, we have γC ∨βα γL ⊇ γL ◦βα γC . Hence C ∧βαL ⊆ L◦βαC, i.e., (1) implies (4) . It is clear that (4) ⇒ (3) and (3) ⇒ (2) . Assume that (2) holds. Let A be an intuitionistic fuzzy right ideal with thresholds (α, β] and L be an intuitionistic fuzzy left ideal with thresholds (α, β] of R. Since every intuitionistic fuzzy right ideal with thresholds (α, β] of R is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R, this means that A is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R. By our assumption, A ∧βα L ⊆ L ◦βα A. Hence R is an intra-regular by the Theorem 13, i.e., (2)⇒ (1) . Theorem 16. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is an intra-regular. (2) A ∧βα L ∧βα D ⊆ (L ◦βα A) ◦βα D for every intuitionistic fuzzy quasi-ideal A with thresholds (α, β], every intuitionistic fuzzy left ideal L with thresholds (α, β] and every intuitionistic fuzzy right ideal D with thresholds (α, β] of R. (3) B∧βαL∧βαD ⊆ (L◦βαB)◦βαD for every intuitionistic fuzzy bi-ideal B with thresholds (α, β], every intuitionistic fuzzy left ideal L with thresholds (α, β] and every intuitionistic fuzzy right ideal D with thresholds (α, β] of R. (4) C ∧βα L ∧βα D ⊆ (L ◦βα C) ◦βα D for every intuitionistic fuzzy generalized bi-ideal C with thresholds (α, β], every intuitionistic fuzzy left ideal L with thresholds (α, β] and every intuitionistic fuzzy right ideal D with thresholds (α, β] of R. Proof. Assume that (1) holds. Let C = (µC , γC) be an intuitionistic fuzzy generalized bi-ideal with thresholds (α, β], L = (µL, γL) be an intuitionistic fuzzy left ideal with thresholds (α, β] and D = (µD, γD) be an intuitionistic fuzzy right ideal with thresholds (α, β] of R. Let x ∈ R, this means that there exist ai, bi ∈ R such that x = ∑n i=1(aix 2)bi. Now x = (ai(xx))bi = (x(aix))bi = (bi(aix))x and bi(aix) = bi(ai((aix 2)bi)) = bi((aix 2)(aibi)) K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 938 = bi((aix 2)ci) = (aix 2)(bici) = (aix 2)di = (ai(xx))di = (x(aix))di = (di(aix))x. Thus ((µL ◦βα µC) ◦βα µD)(x) = {((µL ◦ µC) ◦ µD)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 piqi {∧ni=1 {(µL ◦ µC) (pi) ∧ µD (qi)}} ) ∧ β} ∨ α ≥ {{(µL ◦ µC) (bi(aix)) ∧ µD (x)} ∧ β} ∨ α = ((µL ◦ µC) (bi(aix)) ∨ α) ∧ (µD (x) ∨ α) ∧ (β ∨ α) ≥ ((µL ◦ µC) (bi(aix)) ∨ α) ∧ µD(x) ∧ β = ( ( ∨bi(aix)=∑n i=1mini {∧ni=1 {µL (mi) ∧ µC (ni)}} ) ∨ α) ∧ µD(x) ∧ β ≥ ({µL(di(aix)) ∧ µC(x)} ∨ α) ∧ µD(x) ∧ β = (µL(di(aix)) ∨ α) ∧ (µC(x) ∨ α) ∧ µD(x) ∧ β ≥ (µL(x) ∧ β) ∧ µC(x) ∧ µD(x) ∧ β = µL(x) ∧ µC(x) ∧ µD(x) ∧ β = (µL ∧ µC ∧ µD)(x) ∧ β = {(µC ∧ µL ∧ µD)(x) ∧ β} ∨ α = (µC ∧βα µL ∧βα µD)(x). ⇒ µC ∧βα µL ∧βα µD ⊆ (µL ◦βα µC) ◦βα µD. Similarly, we have µC∨βαµL∨βαµD ⊇ (µL◦βαµC)◦βαµD. Hence C∧βαL∧βαD ⊆ (L◦βαC)◦βαD, i.e., (1) implies (4) . Since (4) ⇒ (3) and (3) ⇒ (2) . Suppose that (2) holds. Then A∧βα R∧βαD ⊆ (R ◦βα A) ◦βαD, where A is an intuitionistic fuzzy left ideal with thresholds (α, β] of R, i.e., A ∧βα D ⊆ A ◦βα D. Therefore R is an intra-regular, i.e., (2)⇒ (1) . 5. Regular and Intra-regular LA-rings In this section, we characterize both regular and intra-regular LA-rings in terms of intuitionistic fuzzy left (right, quasi-, bi-, generalized bi-) ideals with thresholds (α, β]. Theorem 17. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is both a regular and an intra-regular. (2) Every intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R is an intuitionistic fuzzy idempotent with thresholds (α, β]. Proof. Suppose that R is both a regular and an intra-regular. Let A = (µA, γA) be an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R.Then A be an intuitionistic fuzzy bi-ideal with thresholds (α, β] of R and A ◦βα A ⊆ Aβα. Let x ∈ R, this implies K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 939 that there exists a ∈ R such that x = (xa)x, and also there exist ai, bi ∈ R such that x = ∑n i=1(aix 2)bi. Now x = (xa)x xa = ((aix 2)bi)a = (abi)(aix 2) = ci(ai(xx)) = ci(x(aix)) = x(ci(aix)) = x((eci)(aix)) = x((xai)(cie)) = x((xai)di) = x((diai)x) = x(lix) = li(xx) = (eli)(xx) = (xx)(lie) = (xx)mi = (mix)x. mix = mi((aix 2)bi) = (aix 2)(mibi) = (ai(xx))ni = (x(aix))ni = (x(aix))(eni) = (xe)((aix)ni) = (xe)((aix)(eni)) = (xe)((aie)(xni)) = (xe)(x((aie)ni)) = (xe)(xui) = x((xe)ui) = xwi. ⇒ xa = (mix)x = (xwi)x. Thus (µA ◦βα µA)(x) = {(µA ◦ µA)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 piqi {∧ni=1 {µA (pi) ∧ µA (qi)}} ) ∧ β} ∨ α ≥ {{µA ((xwi)x) ∧ µA (x)} ∧ β} ∨ α = (µA ((xwi)x) ∨ α) ∧ (µA (x) ∨ α) ∧ (β ∨ α) ≥ (µA (x) ∧ µA (x) ∧ β) ∧ µA (x) ∧ β = µA (x) ∧ β = (µA (x) ∧ β) ∨ α = (µA)βα(x). ⇒ (µA)βα ⊆ µA ◦βα µA. Similarly, we have (γA)βα ⊇ γA ◦βα γA. Hence Aβα = A ◦βα A. Conversely, assume that every intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R is an intuitionistic fuzzy idempotent with thresholds (α, β]. Let a ∈ R, then Ra is a left ideal of R containing a by the Lemma 19. This implies that Ra is a quasi-ideal of R, so χRa is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R by the Theorem 4. By our assumption (χRa) β α = χRa ◦βα χRa = (χ(Ra)(Ra)) β α, i.e., Ra = (Ra)(Ra). Since a ∈ Ra, i.e., a ∈ (Ra)(Ra). Thus a is both a regular and an intra-regular by the Theorems 8 and 13, respectively. Hence R is both a regular and an intra-regular, i.e., (2)⇒ (1) . Theorem 18. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is both a regular and an intra-regular. (2) A ∧βα B ⊆ A ◦βα B for all intuitionistic fuzzy quasi-ideals A and B with thresholds (α, β] of R. K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 940 (3) A∧βαB ⊆ A◦βαB for every intuitionistic fuzzy quasi-ideal A with thresholds (α, β] and every intuitionistic fuzzy bi-ideal B with thresholds (α, β] of R. (4) A∧βαB ⊆ A ◦βαB for every intuitionistic fuzzy bi-ideal A with thresholds (α, β] and every intuitionistic fuzzy quasi-ideal B with thresholds (α, β] of R. (5) A ∧βα B ⊆ A ◦βα B for all intuitionistic fuzzy bi-ideals A and B with thresholds (α, β] of R. (6) A∧βαB ⊆ A◦βαB for every intuitionistic fuzzy bi-ideal A with thresholds (α, β] and every intuitionistic fuzzy generalized bi-ideal B with thresholds (α, β] of R. (7) A∧βαB ⊆ A◦βαB for every intuitionistic fuzzy generalized bi-ideal A with thresholds (α, β] and every intuitionistic fuzzy quasi-ideal B with thresholds (α, β] of R. (8) A∧βαB ⊆ A◦βαB for every intuitionistic fuzzy generalized bi-ideal A with thresholds (α, β] and every intuitionistic fuzzy bi-ideal B with thresholds (α, β] of R. (9) A ∧βα B ⊆ A ◦βα B for all intuitionistic fuzzy generalized bi-ideals A and B with thresholds (α, β] of R. Proof. Assume that (1) holds. Let A = (µA, γA) and B = (µB, γB) be two intuitionistic fuzzy generalized bi-ideals with thresholds (α, β] of R. Let x ∈ R, then means that there exists an element a ∈ R such that x = (xa)x, and also there exist elements ai, bi ∈ R such that x = ∑n i=1(aix 2)bi. Since x = (xa)x = ((xwi)x)x by the Theorem 17. Thus (µA ◦βα µB)(x) = {(µA ◦ µB)(x) ∧ β} ∨ α = { ( ∨x=∑n i=1 piqi {∧ni=1 {µA (pi) ∧ µB (qi)}} ) ∧ β} ∨ α ≥ {{µA ((xwi)x) ∧ µB (x)} ∧ β} ∨ α = (µA ((xwi)x) ∨ α) ∧ (µB (x) ∨ α) ∧ (β ∨ α) ≥ (µA (x) ∧ µA (x) ∧ β) ∧ µB (x) ∧ β = µA (x) ∧ µB (x) ∧ β = (µA ∧ µB) (x) ∧ β = {(µA ∧ µB) (x) ∧ β} ∨ α = (µA ∧βα µB)(x). ⇒ µA ∧βα µB ⊆ µA ◦βα µB. Similarly, we have γA ∨βα γB ⊇ γA ◦βα γB. Therefore A ∧βα B ⊆ A ◦βα B, i.e., (1) implies (9) . It is clear that (9) ⇒ (8) ⇒ (7) ⇒ (4) ⇒ (2) and (9) ⇒ (6) ⇒ (5) ⇒ (3) . Suppose that (2) holds. Let A be an intuitionistic fuzzy right ideal with thresholds (α, β] and B be an intuitionistic fuzzy left ideal with thresholds (α, β] of R. Since every intuitionistic fuzzy right ideal with thresholds (α, β] and intuitionistic fuzzy left ideal with thresholds (α, β] of R is an intuitionistic fuzzy quasi-ideal with thresholds (α, β] of R by the Lemma 14. By our supposition, A ∧βα B ⊆ A ◦βα B. Since A ◦βα B ⊆ A ∧βα B, so A ∧βα B = A ◦βα B, i.e., R is a regular. Again by our supposition, A ∧βα B = B ∧βα A ⊆ B ◦βα A, i.e., R is an intra-regular. Therefore R is both a regular and an intra-regular, i.e., (2) ⇒ (1) . In similar way, we can prove that (3)⇒ (1) . K. Nasreen et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 906-943 941 Theorem 19. Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent. (1) R is both a regular and an intra-regular. (2) A ∧βα B ⊆ (A ◦βα B) ∧ (B ◦βα A) for every intuitionistic fuzzy right ideal A with thresholds (α, β] and every intuitionistic fuzzy left ideal B with thresholds (α, β] of R. (3) A ∧βα B ⊆ (A ◦βα B) ∧ (B ◦βα A) for every intuitionistic fuzzy right ideal A with thresholds (α, β] and every intuitionistic fuzzy quasi-ideal B with thresholds (α, β] of R. (4) A ∧βα B ⊆ (A ◦βα B) ∧ (B ◦βα A) for every intuitionistic fuzzy right ideal A with thresholds (α, β] and every intuitionistic fuzzy bi-ideal B with thresholds (α, β] of R. (5) A ∧βα B ⊆ (A ◦βα B) ∧ (B ◦βα A) for every intuitionistic fuzzy right ideal A with thresholds (α, β] and every intuitionistic fuzzy generalized bi-ideal B with thresholds (α, β] of R. (6) A∧βαB ⊆ (A◦βαB)∧(B◦βαA) for every intuitionistic fuzzy left ideal A with thresholds (α, β] and every intuitionistic fuzzy quasi-ideal B with thresholds (α, β] of R. (7) A∧βαB ⊆ (A◦βαB)∧(B◦βαA) for every intuitionistic fuzzy left ideal A with thresholds (α, β] and every intuitionistic fuzzy bi-ideal B with thresholds (α, β] of R. (8) A∧βαB ⊆ (A◦βαB)∧(B◦βαA) for every intuitionistic fuzzy left ideal A with thresholds (α, β] and every intuitionistic fuzzy generalized bi-ideal B with thresholds (α, β] of R. (9) A∧βαB ⊆ (A ◦βαB)∧ (B ◦βαA) for all intuitionistic fuzzy quasi-ideals A and B with thresholds (α, β] of R. (10) A ∧βα B ⊆ (A ◦βα B) ∧ (B ◦βα A) for every intuitionistic fuzzy quasi-ideal A with thresholds (α, β] and every intuitionistic fuzzy bi-ideal B with thresholds (α, β] of R. (11) A ∧βα B ⊆ (A ◦βα B) ∧ (B ◦βα A) for every intuitionistic fuzzy quasi-ideal A with thresholds (α, β] and every intuitionistic fuzzy generalized bi-ideal B with thresholds (α, β] of R. (12) A∧βαB ⊆ (A◦βαB)∧(B ◦βαA) for all intuitionistic fuzzy bi-ideals A with thresholds (α, β] and B with thresholds (α, β] of R. (13) A∧βαB ⊆ (A◦βαB)∧(B◦βαA) for every intuitionistic fuzzy bi-ideal A with thresholds (α, β] and every intuitionistic fuzzy generalized bi-ideal B with thresholds (α, β] of R. (14) A ∧βα B ⊆ (A ◦βα B) ∧ (B ◦βα A) for all intuitionistic fuzzy generalized bi-ideals A and B with thresholds (α, β] of R. Proof. Consider that (1) holds. Since A ∧βα B ⊆ A ◦βα B and A ∧βα B ⊆ B ◦βα A for all intuitionistic fuzzy generalized bi-ideals A and B with thresholds (α, β] of R by the Theorem 18. Hence A∧βαB ⊆ (A ◦βαB)∧ (B ◦βαA), i.e., (1)⇒ (14) . It is clear that (14)⇒ (13)⇒ (12)⇒ (9)⇒ (6)⇒ (2) , (14)⇒ (11)⇒ (10)⇒ (9) , (14)⇒ (8)⇒ (7)⇒ (6) and (14) ⇒ (5) ⇒ (4) ⇒ (3) ⇒ (2) . Suppose that (2) holds. Let A be an intuitionistic fuzzy right ideal with thresholds (α, β] and B be an intuitionistic fuzzy left ideal with thresholds (α, β] of R. By our supposition A ∧βα B ⊆ (A ◦βα B) ∧ (B ◦βα A) ⊆ B ◦βα A, i.e., R is an intra-regular. Again A ∧βα B ⊆ (A ◦βα B) ∧ (B ◦βα A) ⊆ A ◦βα B. 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