International Journal of Analysis and Applications Volume 17, Number 5 (2019), 752-770 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-17-2019-752 DIRECT PRODUCT OF FINITE FUZZY NORMAL SUBRINGS OVER NON-ASSOCIATIVE RINGS NASREEN KAUSAR1,∗, MUHAMMAD AZAM WAQAR2 1Department of Mathematics, University of Agriculture FSD, Pakistan 2Department of School of Business Mangment, NFC IEFR FSD, Pakistan ∗Corresponding author: kausar.nasreen57@gmail.com Abstract. In this paper, we define the concept of direct product of finite fuzzy normal subrings over non- associative and non-commutative rings (LA-ring) and investigate the some fundamental properties of direct product of fuzzy normal subrings. 1. Introduction A generalization of commutative semigroups has been established by Kazim et al [10]. In ternary com- mutative law: abc = cba, they introduced the braces on the left side of this law and explored a new pseudo associative law (ab)c = (cb)a. This law (ab)c = (cb)a is called the left invertive law. A groupoid S is left almost semigroup (abbreviated as LA-semigroup) if it satisfies the left invertive law: (ab)c = (cb)a. A groupoid S is medial (resp. paramedial) if (ab)(cd) = (ac)(bd) (resp. (ab)(cd) = (db)(ca)), in [5] (resp. [1]). In [10], 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 [19] and also satisfies a(bc) = b(ac) and (ab)(cd) = (dc)(ba). Received 2019-05-15; accepted 2019-06-24; published 2019-09-02. 2010 Mathematics Subject Classification. 03F55, 08A72, 20N25. Key words and phrases. Direct product of fuzzy sets, Direct product of fuzzy LA-subrings, Direct product of fuzzy normal LA-subrings. c©2019 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 752 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-752 Int. J. Anal. Appl. 17 (5) (2019) 753 S. Kamran [6], extended the concept of LA-semigroup to the left almost group (LA-group). An LA- semigroup S is left almost group, if there exists left identity e ∈ S such that ea = a for all a ∈ S and for every a ∈ S, there exists b ∈ S such that ba = e. Rehman et al [23], discussed the left almost ring (LA-ring) of finitely nonzero functions which is a gener- alization 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. 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 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. First time, the concept of fuzzy set introduced by Zadeh in his classical paper [26]. This concept has provided a useful mathematical tool for describing the behavior of systems that are too complex to admit precise mathematical analysis by classical methods and tools. Extensive applications of fuzzy set theory have been found in various fields such as artificial intelligence, computer science, management science, expert systems, finite state machines, Languages, robotics, coding theory and others. Liu [14], introduced the concept of fuzzy subrings and fuzzy ideals of a ring. Many authors have explored the theory of fuzzy rings (for example [2–4, 13, 15, 16, 25]). Gupta et al [4], gave the idea of intrinsic product of fuzzy subsets of a ring. Kuroki [13], characterized regular (intra-regular, both regular and intra-regular) rings in terms of fuzzy left (right, quasi, bi-) ideals. Sherwood [24], introduced the concept of product of fuzzy subgroups. After this, further study on this concept continued by Osman [17, 18] and Ray [20]. Zaid [27], gave the idea of normal fuzzy subgroups. Kausar et al [21] initiated the idea of intuitionistic fuzzy normal subrings over a non-associative ring and also characterized the non-associative rings by their intuitionistic fuzzy bi-ideals in [7]. Recently Kausar [9] explored the direct product of finite intuitionistic anti fuzzy normal subrings over non-associative rings. In this paper we explore the concept of [9, 21] in finite fuzzy normal subrings over non-associative and non- commutative rings. Recently Kausar et al [11] studied the fuzzy ideals in LA-rings and also Kausar et al [12], investigated a study on intuitionistic fuzzy ideals with thresholds (α, β] in LA-rings. In this paper, we give the concept of direct product of fuzzy normal LA-subrings. In the first section, we investigate the some basic properties of fuzzy normal LA-subrings of an LA-ring R. In the second section, we provide the some elementary properties of direct product of fuzzy normal LA-subrings of an LA-ring R1 × R2. In the third section, we define the direct product of fuzzy subsets µ1,µ2, ...,µn of LA-rings R1,R2, ...,Rn, Int. J. Anal. Appl. 17 (5) (2019) 754 respectively and examine the some fundamental properties of direct product of fuzzy normal LA-subrings of an LA-ring R1 ×R2 × ...×Rn. Specifically we will show that: (1) Let A and B be two LA-subrings of an LA-ring R. Then A∩B is an LA-subring of R if and only if the characteristic function χZ of Z = A∩B is a fuzzy normal LA-subring of R. (2) 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 R1 ×R2 if and only if the characteristic function χZ of Z = X ∩Y is a fuzzy normal LA-subring of R1 ×R2. (3) Let A = A1×A2×...×An and B = B1×B2×...×Bn be two LA-subrings of an LA-ring R1×R2×...×Rn. Then A∩B is an LA-subring of R1×R2×...×Rn if and only if the characteristic function χZ of Z = A∩B is a fuzzy normal LA-subring of R1 ×R2 × ...×Rn. 2. Fuzzy Normal LA-subrings In this section, we investigate the some basic properties of fuzzy normal LA-subrings of an LA-ring R. By a fuzzy subset µ of an LA-ring R, we mean a function µ : R → [0, 1] and the complement of µ is denoted by µ′, is a fuzzy subset of R defined by µ′(x) = 1 −µ(x) for all x ∈ R. A fuzzy subset µ of an LA-ring R is said to be a fuzzy LA-subring of R if µ(x−y) ≥ min{µ(x),µ(y)} and µ(xy) ≥ min{µ(x),µ(y)} for all x,y ∈ R. A fuzzy LA-subring of an LA-ring R is said to be a fuzzy normal LA-subring of R if µ(xy) = µ(yx) for all x,y ∈ R. Let A be a non-empty subset of an LA-ring R. The characteristic function of A is denoted by χA and defined by χA : R → [0, 1] | x → χA (x) =   1 if x ∈ A0 if x /∈ A Lemma 2.1. Let A be a non-empty subset of an LA-ring R. Then A is an LA-subring of R if and only if the characteristic function χA of A is a fuzzy normal LA-subring of R. Proof. Let A be an LA-subring of R and a,b ∈ R. If a,b ∈ A, then by definition of characteristic function χA(a) = 1 = χA(b). Since a− b,ab ∈ A, A being an LA-subring of R. This implies that χA(a− b) = 1 = 1 ∧ 1 = χA(a) ∧χA(b) and χA(ab) = 1 = 1 ∧ 1 = χA(a) ∧χA(b). Int. J. Anal. Appl. 17 (5) (2019) 755 Thus χA(a−b) ≥ min{χA(a),χA(b)} and χA(ab) ≥ min{χA(a),χA(b)}. Since ab and ba ∈ A, so χA(ab) = 1 = χA(ba), i.e., χA(ab) = χA(ba). Similarly we have χA(a− b) ≥ min{χA(a),χA(b)}, χA(ab) ≥ min{χA(a),χA(b)}, χA(ab) = χA(ba), when a,b /∈ A. Hence the characteristic function χA of A is a fuzzy normal LA-subring of R. Conversely, suppose that the characteristic function χA of A is a fuzzy normal LA-subring of R. Let a,b ∈ A, then by definition χA(a) = 1 = χA(b). By the supposition χA(a− b) ≥ χA(a) ∧χA(b) = 1 ∧ 1 = 1 and χA(ab) ≥ χA(a) ∧χA(b) = 1 ∧ 1 = 1. Thus χA(a− b) = 1 = χA(ab), i.e., a− b,ab ∈ A. Hence A is an LA-subring of R. � Lemma 2.2. If A and B are two LA-subrings of an LA-ring R, then their intersection A ∩ B is also an LA-subring of R. Proof. Straight forward. � Theorem 2.1. Let A and B be two LA-subrings of an LA-ring R. Then A ∩ B is an LA-subring of R if and only if the characteristic function χZ of Z = A∩B is a fuzzy normal LA-subring of R. Proof. Let Z = A ∩ B be an LA-subring of R and a,b ∈ R. If a,b ∈ Z = A ∩ B, then by definition of characteristic function χZ(a) = 1 = χZ(b). Since a − b,ab ∈ A,B, A and B being LA-subrings of R. This implies that χZ(a− b) = 1 = 1 ∧ 1 = χZ(a) ∧χZ(b) and χZ(ab) = 1 = 1 ∧ 1 = χZ(a) ∧χZ(b). Thus χZ(a− b) ≥ min{χZ(a),χZ(b)} and χZ(ab) ≥ min{χZ(a),χZ(b)}. As ab and ba ∈ Z, so χZ(ab) = 1 = χZ(ba), i.e., χZ(ab) = χZ(ba). Similarly we have χZ(a− b) ≥ min{χZ(a),χZ(b)}, χZ(ab) ≥ min{χZ(a),χZ(b)}, χZ(ab) = χZ(ba), when a,b /∈ Z. Hence the characteristic function χZ of Z is a fuzzy normal LA-subring of R. Int. J. Anal. Appl. 17 (5) (2019) 756 Conversely, assume that the characteristic function χZ of Z = A∩B is a fuzzy normal LA-subring of R. Let a,b ∈ Z = A∩B, then by definition of characteristic function χZ(a) = 1 = χZ(b). By our assumption χZ(a− b) ≥ χZ(a) ∧χZ(b) = 1 ∧ 1 = 1 and χZ(ab) ≥ χZ(a) ∧χZ(b) = 1 ∧ 1 = 1. Thus χZ(a− b) = 1 = χZ(ab), i.e., a− b and ab ∈ Z. Hence Z is an LA-subring of R. � Corollary 2.1. Let {Ai}i∈I be a family of LA-subrings of an LA-ring R. Then A = ∩Ai is an LA-subring of R if and only if the characteristic function χA of A = ∩Ai is a fuzzy normal LA-subring of R. Lemma 2.3. If µ and γ are two fuzzy normal LA-subrings of an LA-ring R, then their intersection µ∩γ is also a fuzzy normal LA-subring of R. Proof. Let µ and γ be two fuzzy normal LA-subrings of an LA-ring R. We have to show that β = µ∩γ is also a fuzzy normal LA-subring of R. Now β(z1 −z2) = (µ∩γ)(z1 −z2) = min{µ(z1 −z2),γ(z1 −z2)} ≥ {{µ(z1) ∧µ(z2)}∧{γ(z1) ∧γ(z2)}} = {µ(z1) ∧{µ(z2) ∧γ(z1)}∧γ(z2)} = {µ(z1) ∧{γ(z1) ∧µ(z2)}∧γ(z2)} = {{µ(z1) ∧γ(z1)}∧{µ(z2) ∧γ(z2)}} = min{(µ∩γ)(z1), (µ∩γ)(z2)} = min{β(z1),β(z2)}. ⇒ β(z1 −z2) ≥ min{β(z1),β(z2)}. Similarly, we have β(z1 ◦z2) ≥ min{β(z1),β(z2)}. Thus β is a fuzzy LA-subring of an LA-ring R. Now β(z1 ◦z2) = (µ∩γ)(z1 ◦z2) = min{µ(z1 ◦z2),γ(z1 ◦z2)} = min{µ(z2 ◦z1),γ(z2 ◦z1)} = (µ∩γ)(z2 ◦z1) = β(z2 ◦z1). Hence β = µ∩γ is a fuzzy normal LA-subring of R. � Int. J. Anal. Appl. 17 (5) (2019) 757 Corollary 2.2. If {µi}i∈I is a family of fuzzy normal LA-subrings of an LA-ring R, then µ = ∩µi is also a fuzzy normal LA-subring of R. 3. Direct Product of Fuzzy Normal LA-subrings In this section, we define the direct product of fuzzy subsets µ1,µ2 of LA-rings R1,R2, respectively and investigate the some elementary properties of direct product of fuzzy normal LA-subrings of an LA-ring R1 ×R2. Let µ1,µ2 be fuzzy subsets of LA-rings R1,R2, respectively. The direct product of fuzzy subsets µ1,µ2 is denoted by µ1 ×µ2 and defined as (µ1 ×µ2)(x1,x2) = min{µ1(x1),µ2(x2)}. A fuzzy subset µ1 ×µ2 of an LA-ring R1 ×R2 is said to be a fuzzy LA-subring of R1 ×R2 if (1) (µ1 ×µ2)(x−y) ≥ min{(µ1 ×µ2)(x), (µ1 ×µ2)(y)}, (2) (µ1 ×µ2)(xy) ≥ min{(µ1 ×µ2)(x), (µ1 ×µ2)(y)} for all x = (x1,x2) ,y = (y1,y2) ∈ R1 ×R2. A fuzzy LA-subring of an LA-ring R1 × R2 is said 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. Let A×B be a non-empty subset of an LA-ring R1 ×R2. The characteristic function of A×B is denoted by χA×B and defined as χA×B : R1 ×R2 → [0, 1] | x = (x1,x2) → χA×B (x) =   1 if x ∈ A×B0 if x /∈ A×B Lemma 3.1. [21, 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 3.1. 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 characteristic function χZ of Z = A × B is a fuzzy normal LA-subring of an LA-ring R1 ×R2. Proof. Let Z = A×B be an LA-subring of R1×R2 and a = (a1,a2),b = (b1,b2) ∈ R1×R2. If a,b ∈ Z = A×B, then by definition of characteristic function χZ(a) = 1 = χZ(b). Since a − b and ab ∈ Z, Z being an LA- subring of an LA-ring R1 ×R2. This implies that χZ(a− b) = 1 = 1 ∧ 1 = χZ(a) ∧χZ(b) and χZ(ab) = 1 = 1 ∧ 1 = χZ(a) ∧χZ(b). Int. J. Anal. Appl. 17 (5) (2019) 758 Thus χZ(a−b) ≥ min{χZ(a),χZ(b)} and χZ(ab) ≥ min{χZ(a),χZ(b)}. Since ab and ba ∈ Z, so µχZ (ab) = 1 = µχZ (ba), i.e., χZ(ab) = χZ(ba). Similarly we have χZ(a− b) ≥ min{χZ(a),χZ(b)}, χZ(ab) ≥ min{χZ(a),χZ(b)}, χZ(ab) = χZ(ba), when a,b /∈ Z. Hence the characteristic function χZ of Z = A×B is a fuzzy normal LA-subring of R1×R2. Conversely, suppose that the characteristic function χZ of Z = A × B is a fuzzy normal LA-subring of R1 ×R2. We have to show that Z = A×B is an LA-subring of R1 ×R2. Let a,b ∈ Z, where a = (a1,a2) and b = (b1,b2) , a1,b1 ∈ A, a2,b2 ∈ B. By definition, we have χZ(a) = 1 = χZ(b). By our supposition χZ(a− b) ≥ χZ(a) ∧χZ(b) = 1 ∧ 1 = 1 and χZ(ab) ≥ χZ(a) ∧χZ(b) = 1 ∧ 1 = 1. Thus χZ(a−b) = 1 = χZ(ab), i.e., a−b and ab ∈ Z. Hence Z = A×B is an LA-subring of R1 ×R2. � Lemma 3.2. 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 R1 ×R2. Proof. Straight forward. � Theorem 3.1. 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 R1 × R2 if and only if the characteristic function χZ of Z = X ∩ Y is a fuzzy normal LA-subring of 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 characteristic function χZ(a) = 1 = χZ(b). Since a−b and ab ∈ Z, Z being an LA-subring of R1 ×R2. This implies that χZ(a− b) = 1 = 1 ∧ 1 = χZ(a) ∧χZ(b) and χZ(ab) = 1 = 1 ∧ 1 = χZ(a) ∧χZ(b). Thus χZ(a − b) ≥ min{χZ(a),χZ(b)} and χZ(ab) ≥ min{χZ(a),χZ(b)}. Since ab and ba ∈ Z, then by definition χZ(ab) = 1 = χZ(ba), i.e., χZ(ab) = χZ(ba). Similarly we have χZ(a− b) ≥ min{χZ(a),χZ(b)}, χZ(ab) ≥ min{χZ(a),χZ(b)}, χZ(ab) = χZ(ba), Int. J. Anal. Appl. 17 (5) (2019) 759 when a,b /∈ Z. Hence the characteristic function χZ of Z is a fuzzy normal LA-subring of R1 ×R2. Conversely, assume that the characteristic function χZ of Z = X ∩Y is a fuzzy normal LA-subring of an LA-ring R1 ×R2. Let a,b ∈ Z = X ∩Y, then by definition χZ(a) = 1 = χZ(b). By our assumption χZ(a− b) ≥ χZ(a) ∧χZ(b) = 1 ∧ 1 = 1 and χZ(ab) ≥ χZ(a) ∧χZ(b) = 1 ∧ 1 = 1. Thus χZ(a−b) = 1 = χZ(ab), i.e., a−b and ab ∈ Z. Hence Z is an LA-subring of an LA-ring R1×R2. � Corollary 3.1. 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 R1 ×R2 if and only if the characteristic function χC of C = ∩Ci is a fuzzy normal LA-subring of R1 ×R2. Lemma 3.3. If µ and γ are fuzzy normal LA-subrings of LA-rings R1 and R2, respectively, then µ×γ is a fuzzy normal LA-subring of an LA-ring R1 ×R2. Proof. Let µ and γ be fuzzy normal LA-subrings of LA-ring R1 and R2, respectively. We have to show that β = µ×γ is a fuzzy normal LA-subring of an LA-ring R1 ×R2. Now β((a,b) − (c,d)) = (µ×γ)(a− c,b−d) = min{µ(a− c),γ(b−d)} = µ(a− c) ∧γ(b−d) ≥ {µ(a) ∧µ(c)}∧{γ(b) ∧γ(d)} = µ(a) ∧{µ(c) ∧γ(b)}∧γ(d) = µ(a) ∧{γ(b) ∧µ(c)}∧γ(d) = {µ(a) ∧γ(b)}∧{µ(c) ∧γ(d)} = min{(µ×γ)(a,b), (µ×γ)(c,d)} = min{β(a,b),β(c,d)}. ⇒ β((a,b) − (c,d)) ≥ min{β(a,b),β(c,d)}. Int. J. Anal. Appl. 17 (5) (2019) 760 Similarly, we have β((a,b) ◦ (c,d)) ≥ min{β(a,b),β(c,d)}. Thus µ×γ is a fuzzy LA-subring of R1 ×R2. Now β((a,b) ◦ (c,d)) = (µ×γ)(ac,bd) = min{µ(ac),γ(bd)} = min{µ(ca),γ(db)} = (µ×γ)(ca,db) = β((c,d) ◦ (a,b)). Hence µ×γ is a fuzzy normal LA-subring of R1 ×R2. � Proposition 3.2. If µ = µ1×µ2 and γ = γ1×γ2 are two fuzzy normal LA-subrings of an LA-ring R1×R2, then their intersection β = µ∩γ is also a fuzzy normal LA-subring of R1 ×R2. Proof. Let µ = µ1 ×µ2 and γ = γ1 ×γ2 be two fuzzy normal LA-subrings of an LA-ring R1 ×R2. We have to show that β = µ∩γ is also a fuzzy normal LA-subring of R1 ×R2. Now β((z1,z2) − (z3,z4)) = (µ∩γ)((z1,z2) − (z3,z4)) = min{µ((z1,z2) − (z3,z4)),γ((z1,z2) − (z3,z4))} ≥ {{µ(z1,z2) ∧µ(z3,z4)}∧{γ(z1,z2) ∧γ(z3,z4)}} = {µ(z1,z2) ∧{µ(z3,z4) ∧γ(z1,z2)}∧γ(z3,z4)} = {µ(z1,z2) ∧{γ(z1,z2) ∧µ(z3,z4)}∧γ(z3,z4)} = {{µ(z1,z2) ∧γ(z1,z2)}∧{µ(z3,z4) ∧γ(z3,z4)}} = min{(µ∩γ)(z1,z2), (µ∩γ)(z3,z4)} = min{β(z1,z2),β(z3,z4)}. ⇒ β((z1,z2) − (z3,z4)) ≥ min{β(z1,z2),β(z3,z4)}. Similarly, we have β((z1,z2) ◦ (z3,z4)) ≥ min{β(z1,z2),β(z3,z4)}. Thus β = µ∩γ is a fuzzy LA-subring of an LA-ring R1 ×R2. Now β((z1,z2) ◦ (z3,z4)) = (µ∩γ)((z1,z2) ◦ (z3,z4)) = min{µ((z1,z2) ◦ (z3,z4)),γ((z1,z2) ◦ (z3,z4))} = min{µ((z3,z4) ◦ (z1,z2)),γ((z3,z4) ◦ (z1,z2))} = (µ∩γ)((z3,z4) ◦ (z1,z2)) = β((z3,z4) ◦ (z1,z2)). Int. J. Anal. Appl. 17 (5) (2019) 761 Hence β = µ∩γ is a fuzzy normal LA-subring of an LA-ring R1 ×R2. � Corollary 3.2. If {βi}i∈I = {µi ×γi}i∈I is a family of fuzzy normal LA-subrings of an LA-ring R1 ×R2, then β = ∩βi is also a fuzzy normal LA-subring of R1 ×R2. Theorem 3.2. If µ = µ1 ×µ2 and γ = γ1 ×γ2 are fuzzy normal LA-subrings of LA-rings R′ = R1 ×R2 and R′′ = R3 × R4, respectively, then β = µ × γ is a fuzzy normal LA-subring of an LA-ring R′ × R′′ = (R1 ×R2) × (R3 ×R4). Proof. Let µ = µ1 × µ2 and γ = γ1 × γ2 be fuzzy normal LA-subrings of LA-rings R′ = R1 × R2 and R′′ = R3 ×R4, respectively. We have to show that β = µ×γ is a fuzzy normal LA-subring of an LA-ring R′ ×R′′. Now β(((z1,z2), (z3,z4)) − ((z5,z6), (z7,z8))) = µ×γ(((z1,z2), (z3,z4)) − ((z5,z6), (z7,z8))) = µ×γ(((z1,z2) − (z5,z6)), ((z3,z4) − (z7,z8))) = min{µ((z1,z2) − (z5,z6)),γ((z3,z4) − (z7,z8))} ≥ min{(µ(z1,z2) ∧µ(z5,z6)), (γ(z3,z4) ∧γ(z7,z8))} = ((µ(z1,z2) ∧µ(z5,z6)) ∧ (γ(z3,z4) ∧γ(z7,z8))) = ((µ(z1,z2) ∧γ(z3,z4)) ∧ (µ(z5,z6) ∧γ(z7,z8))) = min{(µ(z1,z2) ∧γ(z3,z4)), (µ(z5,z6) ∧γ(z7,z8))} = min{µ×γ((z1,z2), (z3,z4)),µ×γ((z5,z6), (z7,z8))} = min{β((z1,z2), (z3,z4)),β((z5,z6), (z7,z8))}, . Similarly, we have β(((z1,z2), (z3,z4)) ◦ ((z5,z6), (z7,z8))) ≥ min{β((z1,z2), (z3,z4)),β((z5,z6), (z7,z8))}. Int. J. Anal. Appl. 17 (5) (2019) 762 Thus β = µ∩γ is a fuzzy LA-subring of an LA-ring R′×R′′. Now β(((z1,z2), (z3,z4)) ◦ ((z5,z6), (z7,z8))) = µ×γ(((z1,z2), (z3,z4)) ◦ ((z5,z6), (z7,z8))) = µ×γ(((z1,z2) ◦ (z5,z6)), ((z3,z4) ◦ (z7,z8))) = min{µ((z1,z2) ◦ (z5,z6)),γ((z3,z4) ◦ (z7,z8))} = min{µ((z5,z6) ◦ (z1,z2)),γ((z7,z8) ◦ (z3,z4))} = µ×γ(((z5,z6) ◦ (z1,z2)), ((z7,z8) ◦ (z3,z4))) = µ×γ(((z5,z6), (z7,z8)) ◦ ((z1,z2), (z3,z4))) = β(((z5,z6), (z7,z8)) ◦ ((z1,z2), (z3,z4))). Hence β = µ∩γ is a fuzzy normal LA-subring of an LA-ring R′×R′′. � Lemma 3.4. Let µ and γ be fuzzy subsets of LA-rings R1 and R2 with left identities e1 and e2, respectively. If µ×γ is a fuzzy LA-subring of an LA-ring R1 ×R2, then at least one of the following two statements must hold. (1) µ (x) ≤ γ (e2) , for all x ∈ R1. (2) µ (x) ≤ γ (e1) , for all x ∈ R2. Proof. Let µ×γ be a fuzzy LA-subring of R1 ×R2. By contraposition, suppose that none of the statements (1) and (2) holds. Then we can find a and b in R1 and R2, respectively such that µ (a) ≥ γ (e2) and µ (b) ≥ γ (e1) . Thus, we have (µ×γ)(a,b) = min{µ(a),γ(b)} ≥ min{µ(e1),γ(e2)} = (µ×γ)(e1,e2). So µ×γ is not a fuzzy LA-subring of R1×R2. Hence either µ (x) ≤ γ (e2) , for all x ∈ R1 or µ (x) ≤ γ(e1) for all x ∈ R2. � Lemma 3.5. Let µ and γ be fuzzy subsets of LA-rings R1 and R2 with left identities e1 and e2, respectively and µ×γ is a fuzzy normal LA-subring of an LA-ring R1 ×R2, then the following conditions are true. (1) If µ (x) ≤ γ(e2), for all x ∈ R1, then µ is a fuzzy normal LA-subring of R1. (2) If µ (x) ≤ γ(e1), for all x ∈ R2, then γ is a fuzzy normal LA-subring of R2. Int. J. Anal. Appl. 17 (5) (2019) 763 Proof. (1) Let µ (x) ≤ γ (e2) for all x ∈ R1, and y ∈ R1. We have to show that µ is a fuzzy normal LA-subring of R1. Now µ(x−y) = µ(x + (−y)) = min{µ(x + (−y)),γ(e2 + (−e2))} = (µ×γ)(x + (−y),e2 + (−e2)) = (µ×γ)((x,e2) + (−y,−e2)) = (µ×γ)((x,e2) − (y,e2)) ≥ (µ×γ)(x,e2) ∧ (µ×γ)(y,e2) = min{min{µ(x),γ(e2)},min{µ(y),γ(e2)}} = µ(x) ∧µ(y). and µ(xy) = min{µ(xy),γ(e2e2)} = (µ×γ)(xy,e2e2) = (µ×γ)((x,e2) ◦ (y,e2)) ≥ (µ×γ)(x,e2) ∧ (µ×γ)(y,e2) = min{min{µ(x),γ(e2)},min{µ(y),γ(e2)}} = µ(x) ∧µ(y). Thus µ is a fuzzy LA-subring of R1. Now µ(xy) = min{µ(xy),γ(e2e2)} = (µ×γ) (xy,e2e2) = (µ×γ) ((x,e2) ◦ (y,e2)) = (µ×γ) ((y,e2) ◦ (x,e2)) = (µ×γ)(yx,e2e2) = min{µ(yx),γ(e2e2)} = µ(yx). Hence µ is a fuzzy normal LA-subring of R1. (2) is same as (1) . � Int. J. Anal. Appl. 17 (5) (2019) 764 4. Direct Product of Finite Fuzzy Normal LA-subrings In this section, we define the direct product of fuzzy subsets µ1,µ2, ...,µn, of LA-rings R1,R2, ...,Rn, respectively and examine the some fundamental properties of direct product of 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 said 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 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. Let A1 ×A2 × ...×An be a non-empty subset of an LA-ring R = R1 ×R2 × ...×Rn. The characteristic function of A = A1 ×A2 × ...×An is denoted by χA and defined as χA : R → [0, 1] | x = (x1,x2, ...,xn) → χA (x) =   1 if x ∈ A0 if x /∈ A Lemma 4.1. 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 [21]. Proof. Straight forward. � Proposition 4.1. 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 characteristic function χA of A = A1 ×A2 × ...×An is a fuzzy normal LA-subring of R1 ×R2 × ...×Rn. Proof. Let A = A1 × A2 × ... × An be an LA-subring of 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 characteristic function χA(a) = 1 = χA(b). Since a − b and ab ∈ A, A being an LA-subring of R1 × R2 × ... × Rn. This implies that χA(a− b) = 1 = 1 ∧ 1 = χA(a) ∧χA(b) and χA(ab) = 1 = 1 ∧ 1 = χA(a) ∧χA(b). Int. J. Anal. Appl. 17 (5) (2019) 765 Thus χA(a − b) ≥ min{χA(a),χA(b)} and χA(ab) ≥ min{χA(a),χA(b)}. Since ab and ba ∈ A, then by definition χA(ab) = 1 = χA(ba), i.e., µχA (ab) = µχA (ba). Similarly we have χA(a− b) ≥ min{χA(a),χA(b)}, χA(ab) ≥ min{χA(a),χA(b)}, χA(ab) = χA(ba), when a,b /∈ A. Hence the characteristic function χA of A = A1×A2×...×An is a fuzzy normal LA-subring of R1 ×R2 × ...×Rn. Conversely, suppose that the characteristic function χA of A = A1 ×A2 × ...×An is a fuzzy normal LA- subring of R1×R2×...×Rn. We have to show that A = A1×A2×...×An is an LA-subring of R1×R2×...×Rn. Let a,b ∈ A, where a = (a1,a, ...,an) and b = (b1,b2, ...,bn) , then by definition χA(a) = 1 = χA(b). By our supposition χA(a− b) ≥ χA(a) ∧χA(b) = 1 ∧ 1 = 1 and χA(ab) ≥ χA(a) ∧χA(b) = 1 ∧ 1 = 1. Thus χA(a− b) = 1 = χA(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 4.2. 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 R1 ×R2 × ...×Rn. Proof. Straight forward. � Theorem 4.1. Let A = A1 ×A2 × ...×An and B = B1 ×B2 × ...×Bn be two LA-subrings of an LA-ring R1 × R2 × ... × Rn. Then A ∩ B is an LA-subring of R1 × R2 × ... × Rn if and only if the characteristic function χZ of Z = A∩B is a fuzzy normal LA-subring of R1 ×R2 × ...×Rn. Proof. Let Z = A∩B be an LA-subring of 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 characteristic function χZ(a) = 1 = χZ(b). Since a− b and ab ∈ Z, Z being an LA-subring. This implies that χZ(a− b) = 1 = 1 ∧ 1 = χZ(a) ∧χZ(b) and χZ(ab) = 1 = 1 ∧ 1 = χZ(a) ∧χZ(b). Int. J. Anal. Appl. 17 (5) (2019) 766 Thus χZ(a − b) ≥ min{χZ(a),χZ(b)} and χZ(ab) ≥ min{χZ(a),χZ(b)}. As ab and ba ∈ Z, then by definition χZ(ab) = 1 = χZ(ba), i.e., χZ(ab) = χZ(ba). Similarly, we have χZ(a− b) ≥ min{χZ(a),χZ(b)}, χZ(ab) ≥ min{χZ(a),χZ(b)}, χZ(ab) = χZ(ba), when a,b /∈ Z. Hence the characteristic function χZ of Z is a fuzzy normal LA-subring of R1×R2×...×Rn. Conversely, assume that the characteristic function χZ of Z = A ∩ B is a fuzzy normal LA-subring of R1 ×R2 × ...×Rn. Let a,b ∈ Z = A∩B, then by definition χZ(a) = 1 = χZ(b). By our assumption χZ(a− b) ≥ χZ(a) ∧χZ(b) = 1 ∧ 1 = 1 and χZ(ab) ≥ χZ(a) ∧χZ(b) = 1 ∧ 1 = 1. Thus χZ(a−b) = 1 = χZ(ab), i.e., a−b and ab ∈ Z. Hence Z is an LA-subring of R1 ×R2 × ...×Rn. � Corollary 4.1. Let {Ai}i∈I = {Ai1 ×Ai2 × ...×Ain}i∈I be a family of LA-subrings of an LA-ring R1 × R2 × ...×Rn, then A = ∩Ai is an LA-subring of R1 ×R2 × ...×Rn if and only if the characteristic function χA of A = ∩Ai is a fuzzy normal LA-subring of R1 ×R2 × ...×Rn. Theorem 4.2. If µ = µ1 × µ2 × ... × µn and γ = γ1 × γ2 × ... × γn are two fuzzy normal LA-subrings of an LA-ring R1 × R2 × ... × Rn, then their intersection β = µ ∩ γ is also a fuzzy normal LA-subring of R1 ×R2 × ...×Rn. Proof. Let µ = µ1 ×µ2 × ...×µn and γ = γ1 ×γ2 × ...×γn be two fuzzy normal LA-subrings of an LA-ring R1 ×R2 × ...×Rn. We have to show that β = µ∩γ is also a fuzzy normal LA-subring of R1 ×R2 × ...×Rn. Let z = (z1,z2, ...,zn) and w = (w1,w2, ...,wn) ∈ R1 ×R2 × ...×Rn. Now β(z −w) = (µ∩γ)(z −w) = min{µ(z −w),γ(z −w)} ≥ {{µ(z) ∧µ(w)}∧{γ(z) ∧γ(w)}} = {µ(z) ∧{µ(w) ∧γ(z)}∧γ(w)} = {µ(z) ∧{γ(z) ∧µ(w)}∧γ(w)} = {{µ(z) ∧γ(z)}∧{µ(w) ∧γ(w)}} = min{(µ∩γ)(z), (µ∩γ)(w)} = min{β(z),β(w)}. Thus β((z1,z2, ...,zn) − (w1,w2, ...wn)) ≥ min{β(z1,z2, ...,zn),β(w1,w2, ...wn)}. Int. J. Anal. Appl. 17 (5) (2019) 767 Similarly, we have β((z1,z2, ...,zn) ◦ (w1,w2, ...wn)) ≥ min{β(z1,z2, ...,zn),β(w1,w2, ...wn)}. Therefore β = µ∩γ is a fuzzy LA-subring of an LA-ring R1 ×R2 × ...×Rn. Now β((z1,z2, ...,zn) ◦ (w1,w2, ...,wn)) = (µ∩γ)(z1w1,z2w2, ...,znwn) = min{µ(z1w1,z2w2, ...,znwn),γ(z1w1,z2w2, ...,znwn)} = min{µ(w1z1,w2z2, ...,wnzn),γ(w1z1,w2z2, ...,wnzn)} = (µ∩γ)(w1z1,w2z2, ...,wnzn) = β((w1,w2, ...,wn) ◦ (z1,z2, ...,zn)). Hence β = µ∩γ is a fuzzy normal LA-subring of an LA-ring R1 ×R2 × ...×Rn. � Corollary 4.2. If {µi}i∈I = {µi1 ×µi2 × ...×µin}i∈I is a family of fuzzy normal LA-subrings of an LA-ring R1 ×R2 × ...×Rn, then µ = ∩µi is also a fuzzy normal LA-subring of R1 ×R2 × ...×Rn. Proposition 4.2. Let µ = µ1 × µ2 × ... × µn and γ = γ1 × γ2 × ... × γn be fuzzy subsets 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 µ×γ is a fuzzy LA-subring of an LA-ring R×R′. Then at least one of the following two statements must hold. (1) µ (x) ≤ γ (e′) , for all x ∈ R. (2) µ (x) ≤ γ (e) , for all x ∈ R′. Proof. Let µ×γ be a fuzzy LA-subring of R×R′. By contraposition, suppose that none of the statements (1) and (2) holds. Then we can find a and b in R and R′, respectively such that µ (a) ≥ γ (e′) and µ (b) ≥ γ (e) . Thus, we have µ×γ(a,b) = min{µ(a),γ(b)} ≥ min{µ(e),γ(e′)} = (µ×γ)(e,e′). Therefore µ × γ is not a fuzzy LA-subring of R × R′. Hence either µ (x) ≤ γ (e′) , for all x ∈ R or µ (x) ≤ γ(e) for all x ∈ R′. � Int. J. Anal. Appl. 17 (5) (2019) 768 Proposition 4.3. Let µ = µ1 × µ2 × ... × µn and γ = γ1 × γ2 × ... × γn be fuzzy subsets 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 µ×γ is a fuzzy normal LA-subring of an LA-ring R×R′. Then the following conditions are true. (1) If µ (x) ≤ γ (e′) , for all x ∈ R, then µ is a fuzzy normal LA-subring of R. (2) If µ (x) ≤ γ(e), for all x ∈ R′, then γ is a fuzzy normal LA-subring of R′. Proof. (1) Let µ (x) ≤ γ (e′) , for all x ∈ R, and y ∈ R. We have to show that µ is a fuzzy normal LA-subring of R. Now µ(x−y) = µ(x + (−y)) = min{µ(x + (−y)),γ(e′ + (−e′))} = (µ×γ)(x + (−y),e′ + (−e′)) = (µ×γ)((x,e′) + (−y,−e′)) = (µ×γ)((x,e′) − (y,e′)) ≥ (µ×γ)(x,e′) ∧ (µ×γ)(y,e′) = min{min{µ(x),γ(e′)},min{µ(y),γ(e′)}} = µ(x) ∧µ(y). and µ(xy) = min{µ(xy),γ(e′e′)} = (µ×γ)(xy,e′e′) = (µ×γ)((x,e′) ◦ (y,e′)) ≥ (µ×γ)(x,e′) ∧ (µ×γ)(y,e′) = min{min{µ(x),γ(e′)},min{µ(y),γ(e′)}} = µ(x) ∧µ(y). Int. J. Anal. Appl. 17 (5) (2019) 769 Thus µ is a fuzzy LA-subring of R. Now µ(xy) = min{µ(xy),γ(e′e′)} = (µ×γ) (xy,e′e′) = (µ×γ) ((x,e′) ◦ (y,e′)) = (µ×γ) ((y,e′) ◦ (x,e′)) = (µ×γ)(yx,e′e′) = min{µ(yx),γ(e′e′)} = µ(yx). Hence µ is a fuzzy normal LA-subring of R. (2) is same as (1) . � References [1] R. J. Cho, J. Jezek and T. Kepka, Paramedial groupoids, Czechoslovak Math. J., 49 (1999) 277-290. [2] K. A. Dib, N. Galhum and A. A. M. Hassan, Fuzzy rings and fuzzy ideals, Fuzzy Math., 4 (1996) 245-261. [3] V. N. Dixit, R. Kumar and N. Ajmal, Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy Set Syst., 44 (1991) 127-138. [4] K. C. Gupta and M. K. Kantroo, The intrinsic product of fuzzy subsets of a ring, Fuzzy Set Syst., 57 (1993) 103-110. [5] J. Jezek and T. Kepka, Medial groupoids, Rozpravy CSAV Rada Mat. a Prir. Ved., 93/2, 1983, 93 pp. [6] M. S. Kamran, Conditions for LA-semigroups to resemble associative structures, Ph.D. Thesis, Quaid-i-Azam University, Islamabad, 1993. [7] N. Kausar, M. Waqar, Characterizations of non-associative rings by their intuitionistic fuzzy bi-ideals, Eur. J. Pure Appl. Math. 12 (2019), 226-250. [8] N. Kausar, Characterizations of non-associative ordered semigroups by the properties of their fuzzy ideals with thresholds (α,β], Prikl. Diskr. Mat. 43 (2019), 37-59. [9] N. Kausar, Direct product of finite intuitionistic fuzzy normal subrings over non-associative rings, Eur. J. Pure Appl. Math., 12 (2019), 622-648. [10] M. A. Kazim and M. Naseeruddin, On almost semigroups, Alig. Bull. Math., 2 (1972), 1-7. [11] N. Kausar, B. Islam, M. Javaid, S, Amjad, U. Ijaz, Characterizations of non-associative rings by the properties of their fuzzy ideals, J. Taibah Univ. Sci. 13 (2019), 820-833. [12] N. Kausar, B. Islam, S. Amjad, M. Waqar, Intuitionistics fuzzy ideals with thresholds(,] in LA-rings, Eur. J. Pure Appli. Math. 12 (2019) 906-943. [13] N. Kuroki, Regular fuzzy duo rings, Inform. Sci., 94 (1996), 119-139. [14] W. J. Liu, Fuzzy invariant subgroups and ideals, Fuzzy Sets Syst., 8 (1982), 133-139. [15] T. K. Mukherjee and M. K. Sen, On fuzzy ideals of a ring 1, Fuzzy Sets Syst., 21 (1987), 99-104. [16] T. K. Mukherjee and M. K. Sen, Prime fuzzy ideals in rings, Fuzzy Sets Syst., 32 (1989), 337-341. [17] M. T. A. Osman, On the direct product of fuzzy subgroups, Fuzzy Sets Syst., 12 (1984), 87-91. [18] M. T. A. Osman, On some product of fuzzy subgroups, Fuzzy Sets Syst., 24 (1987), 79-86. Int. J. Anal. Appl. 17 (5) (2019) 770 [19] P. V. Protic and N. Stevanovic, AG-test and some general properties of Abel-Grassmann’s groupoids, Pure Math. Appl., 6 (1995) 371-383. [20] A. K. Ray, Product of fuzzy subgroups, Fuzzy Sets Syst., 105 (1999), 181-183. [21] T. Shah, N. Kausar and I. Rehman, Intuitionistic fuzzy normal subrings over a non-associative ring, An. St. Univ. Ovidius Constanta, 1 (2012) 369-386. [22] T. Shah, N. Kausar, Characterizations of non-associative ordered semigroups by their fuzzy bi-ideals, Theor. Comput. Sci. 529 (2014), 96-110. [23] T. Shah and I. Rehman, On LA-rings of finitely non-zero functions, Int. J. Contemp. Math. Sci., 5 (2010) 209-222. [24] H. Sherwood, Product of fuzzy subgroups, Fuzzy Sets Syst., 11 (1983) 65-77. [25] U. M. Swamy and K. L. N. Swamy, Fuzzy prime ideals of rings, J. Math. Anal. Appl., 134 (1988) 94-103. [26] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965) 338-353. [27] S. A. Zaid, On normal fuzzy subgroups, J. Fac. Educ. Ain Shams Univ. Cairo, 13 (1988), 115-125. 1. Introduction 2. Fuzzy Normal LA-subrings 3. Direct Product of Fuzzy Normal LA-subrings 4. Direct Product of Finite Fuzzy Normal LA-subrings References