FUZZY IDEALS ON ORDERED AG-GROUPOIDS

In this paper, we define the concept of direct product of finite fuzzy normal subrings over nonassociative and non-commutative rings (LA-ring) and investigate the some fundamental properties of direct product of fuzzy normal subrings.


Introduction
In 1972, a generalization of commutative semigroups has been established by Kazim et al [12]. In ternary commutative law: abc = cba, they introduced the braces on the left side of this law and explored a new pseudo associative law, that is (ab)c = (cb)a. This law (ab)c = (cb)a is called the left invertive law. A groupoid S is said to be a left almost semigroup (abbreviated as LA-semigroup) if it satisfies the left invertive law : (ab)c = (cb)a. This structure is also known as Abel-Grassmann's groupoid (abbreviated as AG-groupoid) in [22]. An AG-groupoid is a midway structure between an abelian semigroup and a groupoid. Mushtaq et al [21], investigated the concept of ideals of AG-groupoids.
In [13], if (S, ·, ≤) is an ordered semigroup and ∅ = A ⊆ S, we define a subset of S as follows : (A] = {s ∈ S : s ≤ a for some a ∈ A}. A non-empty subset A of S is called a subsemigroup of S if A 2 ⊆ A. A is called a left (resp. right) ideal of S if following hold (1) SA ⊆ A (resp. AS ⊆ A). (2) If a ∈ A and b ∈ S such that b ≤ a implies b ∈ A. Equivalent definition: A is called a left (resp. right) ideal of S if (A] ⊆ A and SA ⊆ A (resp. AS ⊆ A).
In [13,15], an ordered semigroup S is said to be a regular if for every a ∈ S, there exists an element x ∈ S such that a ≤ axa. In [14,15], an ordered semigroup S is said to be an intra-regular if for every a ∈ S there exist elements x, y ∈ S such that a ≤ xa 2 y.
We will define the concept of fuzzy left (resp. right, interior, quasi-, bi-, generalized bi-) ideals of an ordered AG-groupoid S. We will establish a study by discussing the different properties of such ideals. We will also characterize regular (resp. intra-regular, both regular and intra-regular) ordered AG-groupoids by the properties of fuzzy left (right, quasi-, bi-, generalized bi-) ideals.

Fuzzy Ideals on ordered AG-groupoids
In [25] An ordered AG-groupoid S, is a partially ordered set, at the same time an AG-groupoid such that a ≤ b, implies ac ≤ bc and ca ≤ cb for all a, b, c ∈ S. Two conditions are equivalent to the one condition First time, Zadeh introduced the concept of fuzzy set in his classical paper [27] of 1965. 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.
Rosenfeld [23], was the first, who introduced the concept of fuzzy set in a group. The study of fuzzy set in semigroups was established by Kuroki [18,19]. He studied fuzzy ideals and fuzzy interior (resp. quasi-, bi-, generalized bi-, semiprime, semiprime quasi-) ideals of semigroups. A systematic exposition of fuzzy semigroups appeared by Mordeson et al [20], where one can find the theoretical results on fuzzy semigroups and their use in fuzzy finite state machines and languages. Fuzzy sets in ordered semigroups/ordered groupoids were first explored by Kehayopulu et al [16,17]. They also studied fuzzy ideals and fuzzy interior (resp. quasi-, bi-, generalized bi-) ideals in ordered semigroups.
By a fuzzy subset µ of an ordered AG-groupoid S, we mean a function µ : S → [0, 1], the complement of µ is denoted by µ , is a fuzzy subset of S given by µ ( for all x, y ∈ S. µ is a fuzzy ideal of S if µ is both a fuzzy left ideal and a fuzzy right ideal of S. Every fuzzy ideal (whether left, right, two-sided) is a fuzzy AG-subgroupoid of S but the converse is not true in general.
We denote by F (S), the set of all fuzzy subsets of S. We define an order relation "⊆" on F (S) such that µ ⊆ γ if and only if µ(x) ≤ γ(x) for all x ∈ S. Then (F (S), •, ⊆) is an ordered AG-groupoid.
By the symbols µ ∧ γ and µ ∨ γ, we mean the following fuzzy subsets: For ∅ = A ⊆ S, the characteristic function of A is denoted by χ A and defined by An ordered AG-groupoid S can be considered a fuzzy subset of itself and we write S = χ S , i.e., S(x) = χ S (x) = 1 for all x ∈ S. This implies that S(x) = 1 for all x ∈ S. Let x ∈ S, we define a set A x = {(y, z) ∈ S × S | x ≤ yz}. Let µ and γ be two fuzzy subsets of S, then product of µ and γ is denoted by µ • γ and defined by : Now we give the imperative properties of such ideals of an odered AG-groupoid S, which will be very helpful in later sections.
Lemma 2.1. Let S be an ordered AG-groupoid. Then the following properties hold.
Proof. Let µ, γ and β be fuzzy subsets of an ordered AG-groupoid S. We have to show that (µ Similarly, we can prove (2) .
Proposition 2.1. Let S be an ordered AG-groupoid with left identity e. Then the following assertions hold.
Theorem 2.1. Let A and B be two non-empty subsets of an ordered AG-groupoid S. Then the following assertions hold.
Theorem 2.2. Let A be a non-empty subset of an ordered AG-groupoid S. Then the following properties hold.
(1) A is an AG-subgroupoid of S if and only if χ A is a fuzzy AG-subgroupoid of S.
(2) A is a left (resp. right, two-sided) ideal of S if and only if χ A is a fuzzy left (resp. right, two-sided) ideal of S. Proof.
(1) Let A be an AG-subgroupoid of S and a, b ∈ S. If a, b ∈ A, then by definition χ A (a) = 1 = χ A (b). Since ab ∈ A, A being an AG-subgroupoid of S, this implies that χ A (ab) = 1. Thus Conversely, suppose that χ A is a fuzzy AG-subgroupoid of S and let a, b ∈ A. Since χ A (ab) ≥ χ A (a) ∧ χ A (b) = 1, χ A being a fuzzy AG-subgroupoid of S. Thus χ A (ab) = 1, i.e., ab ∈ A. Hence A is an AGsubgroupoid of S.
(2) Let A be a left ideal of S and a, b ∈ S. If a, b ∈ A, then by definition χ A (b) = 1. Since ab ∈ A, A being a left ideal of S, this means that χ A (ab) = 1. Thus when a, b / ∈ A. Therefore χ A is a fuzzy left ideal of S.
Conversely, assume that χ A is a fuzzy left ideal of S. Let a, b ∈ A and z ∈ S. Since χ A (zb) ≥ χ A (b) = 1, χ A being a fuzzy left ideal of S. Thus χ A (zb) = 1, i.e., zb ∈ A. Therefore A is a left ideal of S. Theorem 2.3. Let µ be a fuzzy subset of an ordered AG-groupoid S. Then the following assertions hold.
Hence µ is a fuzzy AG-subgroupoid of S.
(2) Suppose that µ is a fuzzy left ideal of S and x ∈ S.
Conversely, assume that S • µ ⊆ µ. Let y, z ∈ S such that x ≤ yz. Now Therefore µ is a fuzzy left ideal of S. Similarly, we can prove (3) .
Proof. Let µ and γ be two fuzzy AG-subgroupoids of S. We have to show that µ ∩ γ is also a fuzzy AG- Hence µ ∩ γ is a fuzzy AG-subgroupoids of S. Similarly, for ideals. Lemma 2.3. If µ and γ are two fuzzy AG-subgroupoids of an ordered AG-groupoid S, then µ • γ is also a fuzzy AG-subgroupoid of S.
Proof. Let µ and γ be two fuzzy AG-subgroupoids of S. We have to show that µ • γ is also a fuzzy AG- Proof. Let µ be a fuzzy right ideal of S and x, y ∈ S. Now µ (xy Thus µ is a fuzzy ideal of S. Lemma 2.6. If µ and γ are two fuzzy left (resp. right) ideals of an ordered AG-groupoid S with left identity e, then µ • γ is also a fuzzy left (resp. right) ideal of S.
Proof. Let µ and γ be two fuzzy left ideals of S. We have to show that µ • γ is also a fuzzy left ideal of S.
for right ideals.

Remark 2.2.
If µ is a fuzzy left (resp. right) ideal of an ordered AG-groupoid S with left identity e. Then µ • µ is a fuzzy ideal of S.
Proof. Let µ and γ be two fuzzy ideals of S and x ∈ S. (1) A is an interior ideal of S if and only if χ A is a fuzzy interior ideal of S. Proof.
(1) Let A be an interior ideal of S and x, y, a ∈ S. If a ∈ A, then by definition χ A (a) = 1. Since (xa)y ∈ A, A being an interior ideal of S, this means that χ A ((xa)y) = 1. Thus Similarly, we have χ A ((xa)y) ≥ χ A (a), when a / ∈ A. Hence χ A is a fuzzy interior ideal of S.
Conversely, suppose that χ A is a fuzzy interior ideal of S. Let x, y ∈ S and a ∈ A, so χ A (a) = 1. Since is an interior ideal of S.
Therefore χ A is a fuzzy quasi-ideal of S.
Conversely, assume that χ A is a fuzzy quasi-ideal of S. Let x be an element of AS ∩ SA. Now This implies that x ∈ A, i.e., AS ∩ SA ⊆ A. Therefore A is a quasi-ideal of S.
(3) Let A be a bi-ideal of S, this implies that χ A is a fuzzy AG-subgroupoid of S by the Theorem 2.2. Let Conversely, suppose that χ A is a fuzzy bi-ideal of S, this means that A is an AG-subgroupoid of S by the Proof. Suppose that µ is a fuzzy interior ideal of S and x ∈ S.
otherwise there exist a, b, c, d ∈ S such that x ≤ ab and a ≤ cd. Since µ is a fuzzy interior ideal of S, this Conversely, assume that (S • µ) • S ⊆ µ and let y, z ∈ S such that a ≤ (xy)z. Now Therefore µ is a fuzzy interior ideal of S.
Theorem 2.6. Let µ be a fuzzy AG-subgroupoid of an ordered AG-groupoid S. Then µ is a fuzzy bi-ideal of Proof. Same as Theorem 2.5.
Theorem 2.7. Let µ be a fuzzy subset of an ordered AG-groupoid S. Then µ is a fuzzy generalized bi-ideal Proof. Same as Theorem 2.5.
Proof. Let µ and γ be two fuzzy bi-ideals of S. This implies that µ and γ be two fuzzy AG-subgroupoids of S, Hence µ ∩ γ is a fuzzy bi-ideal of S.
Lemma 2.10. If µ and γ are two fuzzy bi-(resp. generalized bi-, interior) ideals of an ordered AG-groupoid S with left identity e, then µ • γ is also a fuzzy bi-(resp. generalized bi-, interior) ideal of S.
Proof. Let µ and γ be two fuzzy bi-ideals of S. We have to show that µ • γ is also a fuzzy bi-ideal of S. Since µ and γ are fuzzy AG-subgroupoids of S, then µ • γ is also a fuzzy AG-subgroupoid of S by the Lemma 2.3.
Therefore µ • γ is a fuzzy bi-ideal of S.
Lemma 2.11. Every fuzzy ideal of an ordered AG-groupoid S is a fuzzy interior ideal of S. The converse is not true in general.
Proof. Straight forward. Proof. Let µ be a fuzzy interior ideal of S and x, y ∈ S. Now µ(xy) = µ((ex)y) ≥ µ(x), thus µ is a fuzzy right ideal of S. Hence µ is a fuzzy ideal of S by the Lemma 2.5. Converse is true by the Lemma 2.11.
Lemma 2.12. Every fuzzy left (right, two-sided) ideal of an ordered AG-groupoid S is a fuzzy bi-ideal of S.
The converse is not true in general.
Lemma 2.13. Every fuzzy bi-ideal of an ordered AG-groupoid S is a fuzzy generalized bi-ideal of S. The converse is not true in general.
Lemma 2.14. Every fuzzy left (right, two-sided) ideal of an ordered AG-groupoid S is a fuzzy quasi-ideal of S. The converse is not true in general.
Proposition 2.3. Every fuzzy quasi-ideal of an ordered AG-groupoid S is a fuzzy AG-subgroupoid of S.
Proof. Let µ be a fuzzy quasi-ideal of S.
So µ is a fuzzy AG-subgroupoid of S.
Proposition 2.4. Let µ be a fuzzy right ideal and γ be a fuzzy left ideal of an ordered AG-groupoid S, respectively. Then µ ∩ γ is a fuzzy quasi-ideal of S. Proof. Let µ be a fuzzy quasi-ideal of S. Since µ • µ ⊆ µ by the Proposition 2.3. Now Hence µ is a fuzzy bi-ideal of S.
Proposition 2.5. If µ and γ are two fuzzy quasi-ideals of an ordered AG-groupoid S with left identity e, such that (xe)S = xS for all x ∈ S. Then µ • γ is a fuzzy bi-ideal of S.
Proof. Let µ and γ be two fuzzy quasi-ideals of S, this implies that µ and γ be two fuzzy bi-ideals of S, by the Lemma 2.15. Then µ • γ is also a fuzzy bi-ideal of S by the Lemma 2.10.

Regular Ordered AG-groupoids
An ordered AG-groupoid S will be called a regular if for every x ∈ S, there exists an element a ∈ S such that x ≤ (xa)x. Equivalent definitions are as follows: (1) A ⊆ ((AS)A] for every A ⊆ S.
In this section, we characterize regular ordered AG-groupoids by the properties of fuzzy (left, right, quasi-, bi-, generalized bi-) ideals.
Lemma 3.1. Every fuzzy right ideal of a regular ordered AG-groupoid S is a fuzzy ideal of S.
Lemma 3.2. Every fuzzy ideal of a regular ordered AG-groupoid S is a fuzzy idempotent.
Proof. Assume that µ is a fuzzy ideal of S and µ • µ ⊆ µ. We have to show that µ ⊆ µ • µ. Let x ∈ S, this means that there exists an element a ∈ S such that x ≤ (xa)x. Thus Therefore µ = µ • µ.
Remark 3.1. Every fuzzy right ideal of a regular ordered AG-groupoid S is a fuzzy idempotent.  Proof. Assume that µ is a fuzzy right ideal of S. Then (µ • S) ∩ (S • µ) ⊆ µ, because every fuzzy right ideal of S is a fuzzy quasi-ideal of S by the Lemma 2.14. Let x ∈ S, this implies that there exists an element a ∈ S, such that x ≤ (xa)x. Thus Similarly, we have µ ⊆ S • µ, i.e., µ ⊆ (µ • S) ∩ (S • µ). Therefore (µ • S) ∩ (S • µ) = µ. Proof. Since µ • γ ⊆ µ ∩ γ for every fuzzy right ideal µ and every fuzzy left ideal γ of S by the Lemma 2.8.
Let x ∈ S, this means that there exists an element a ∈ S such that x ≤ (xa)x. Thus Hence µ • γ = µ ∩ γ.  Thus (aS ∪ Sa)S ⊆ aS ∪ Sa and also (aS ∪ Sa] ⊆ aS ∪ Sa. Therefore aS ∪ Sa is a right ideal of S.
Since a ∈ Sa, i.e., a ∈ aS ∪ Sa. Let I be another right ideal of S containing a. Now aS ∈ IS ⊆ I and  Then the following conditions are equivalent.
(1) S is a regular.
Proof. Suppose that (1) holds and β be a fuzzy quasi-ideal of S. Then (β • S) • β ⊆ β, because every fuzzy quasi-ideal of S is a fuzzy bi-ideal of S by the Lemma 2.15. Let x ∈ S, this implies that there exists an element a of S such that x ≤ (xa)x. Thus Therefore β = (β • S) • β, i.e., (1) implies (3) . Assume that (3) holds. Let µ be a fuzzy right ideal and γ be a fuzzy left ideal of S. This means that µ and γ be fuzzy quasi-ideals of S by the Lemma 2.14, so µ ∩ γ be also a fuzzy quasi-ideal of S. Then by our assumption, This means that a ∈ ((aS)a], i.e., a is regular. Hence S is a regular, i.e., (2) ⇒ (1) .
Theorem 3.2. Let S be an ordered AG-groupoid with left identity e, such that (xe)S = xS for all x ∈ S.
Then the following conditions are equivalent.
(1) S is a regular.
(1) S is a regular.
Proof. Suppose that (1) holds. Let δ be a fuzzy generalized bi-ideal and γ be a fuzzy ideal of S. Now means that there exists an element a ∈ S such that x ≤ (xa)x. Now xa ≤ ((xa)x)a = (ax)(xa) = x((ax)a).
Thus Then the following conditions are equivalent.
(1) S is a regular.
Theorem 3.5. Let S be an ordered AG-groupoid with left identity e, such that (xe)S = xS for all x ∈ S.
Then the following conditions are equivalent.
(1) S is a regular.

Intra-regular Ordered AG-groupoids
An ordered AG-groupoid S will be called an intra-regular ordered AG-groupoid if for every x ∈ S there exist elements a, b ∈ S such that x ≤ (ax 2 )b. Equivalent definitions are as follows: (1) A ⊆ ((SA 2 )S] for every A ⊆ S.
In this section, we characterize intra-regular ordered AG-groupoids by the properties of fuzzy (left, right, quasi-, bi-, generalized bi-) ideals.
Lemma 4.1. Every fuzzy left (right) ideal of an intra-regular ordered AG-groupoid S is a fuzzy ideal of S.
So µ is a fuzzy right ideal of S, hence µ is a fuzzy ideal of S by the Lemma 4.1. Converse is true by the Lemma 2.11.  Proof. Let µ be a fuzzy left ideal and γ be a fuzzy right ideal of S. Let x ∈ S, this implies that there exist Then the following conditions are equivalent.
Theorem 4.2. Let S be an ordered AG-groupoid with left identity e, such that (xe)S = xS for all x ∈ S.
Then the following conditions are equivalent.
Proof. Suppose that (1) holds. Let δ be a fuzzy generalized bi-ideal and γ be a fuzzy ideal of S. Now Thus Hence δ ∩ γ = (δ • γ) • δ, i.e., (1) ⇒ (4) . It is clear that (4) ⇒ (3) and (3) ⇒ (2) . Assume that (2) is true. Let µ be a fuzzy right ideal and γ be a fuzzy two-sided ideal of S. Since every fuzzy right ideal of S is a fuzzy quasi-ideal of S by the Lemma 2.14, so µ is a fuzzy quasi-ideal of S. By our assumption Then the following conditions are equivalent.
Proof. Assume that (1) holds. Let δ be a fuzzy generalized bi-ideal and γ be a fuzzy left ideal of S. Let x ∈ S, this means that there exist elements a, b ∈ S such that x ≤ (ax 2 )b. Now x ≤ (a(xx))b = (x(ax))b = (b(ax))x. Thus Hence (1) implies (4) . It is clear that (4) ⇒ (3) and (3) ⇒ (2) . Suppose that (2) holds. Let µ be a fuzzy right ideal and γ be a fuzzy left ideal of S. Since every fuzzy right ideal of S is a fuzzy quasi-ideal of S, this implies that µ is a fuzzy quasi-ideal of S. By our supposition, µ ∩ γ ⊆ γ • µ. Thus S is an intra-regular by the Theorem 4.1, i.e., (2) ⇒ (1) .
Theorem 4.4. Let S be an ordered AG-groupoid with left identity e, such that (xe)S = xS for all x ∈ S.
Then the following conditions are equivalent.
• λ for every fuzzy generalized bi-ideal δ, every fuzzy left ideal γ and every fuzzy right ideal λ of S.

Regular and Intra-regular Ordered AG-groupoids
In this section, we characterize both regular and intra-regular ordered AG-groupoid by the properties of fuzzy (left, right, quasi-, bi-, generalized bi-) ideals.
Theorem 5.1. Let S be an odered AG-groupoid with left identity e, such that (xe)S = xS for all x ∈ S.
Then the following conditions are equivalent.
(1) S is both a regular and an intra-regular.
Proof. Suppose that (1) holds. Let µ be a fuzzy bi-ideal of S and µ • µ ⊆ µ. Let x ∈ S, this implies that there exists an element a ∈ S such that x ≤ (xa)x, also there exist elements a, b ∈ S such that x ≤ (ax 2 )b. ⇒ µ ⊆ µ • µ.
Theorem 5.2. Let S be an ordered AG-groupoid with left identity e, such that (xe)S = xS for all x ∈ S.
Then the following conditions are equivalent.
(1) S is both a regular and an intra-regular.
(2) Every fuzzy quasi-ideal of S is a fuzzy idempotent.
Proof. Suppose that S is both a regular and an intra-regular. Let µ be a fuzzy quasi-ideal of S. Then µ be a fuzzy bi-ideal of S and µ • µ ⊆ µ. Let x ∈ S, this means that there exists an element a ∈ S such that x ≤ (xa)x, and also there exist elements a, b ∈ S such that x ≤ (ax 2 )b. Since x ≤ (xa)x = ((xw)x)x by the ⇒ µ ⊆ µ • µ.
Hence µ = µ • µ. Conversely, assume that every fuzzy quasi-ideal of S is a fuzzy idempotent. Let a ∈ S, then Sa is a left ideal of S containing a by the Lemma 3.5.This implies that Sa is a quasi-ideal of S, so