# Generalized Fuzzy Interior Ideals in Abel Grassmann's Groupoids

Authors: Asghar Khan; Young Bae Jun; Tahir Mahmood
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

Using the notion of a fuzzy point and its belongness to and quasicoincidence with a fuzzy subset, some new concepts of a fuzzy interior ideal
in Abel Grassmann's groupoidsS are introduced and their interrelations and related properties are invesitigated. We also introduce the notion of a strongly belongness and strongly quasicoincidence of a fuzzy point with a fuzzy subset and characterize fuzzy interior ideals of S in terms of these relations.


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## Body

## 1. Introduction

The idea of a quasicoincidence of a fuzzy point with a fuzzy set, which is mentioned in [1, 2], played a vital role to generate some different types of fuzzy subgroups. It is worth pointing out that Bhakat and Das [2] gave the concepts of (α,β)-fuzzy subgroups by using the  “belongs to” relation (∈) and “quasicoincident with” relation (q) between a fuzzy point and a fuzzy subgroup, and they introduced the concept of an (∈, ∈∨q)-fuzzy subgroup. In particular, (∈,∈∨q)-fuzzy subgroup is an important and useful generalization of Rosenfeld's fuzzy subgroup. It is now natural to investigate similar type of generalizations of the existing fuzzy subsystems of other algebraic structures. With this objective in view, Davvaz [3, 4] introduced the concept of (∈,∈∨q)-fuzzy sub-near-rings (R-subgroups, ideals) of a near-ring and investigated some of their interesting properties. Jun and Song [5] discussed general forms of fuzzy interior ideals in semigroups. Kazanci and Yamak introduced the concept of a generalized fuzzy bi-ideal [6]. Also Davvaz and many others used this concept in several other algebraic structures (see [7–16]). Jun [13, 17], gave the concept of (α,β)-fuzzy subalgebra of a BCK/BCI-algebras. In [18], Luo introduced the concept of a strong neighborhood. According to him, a fuzzy point xλ(0<λ<1) is said to be strongly belong to a fuzzy subset F, denoted by xλ∈̲F, if and only if F(x)>λ. λ-strong cut set Fλ̲ of F is given by Fλ̲={x∈X∣F(x)>λ}, where X is a nonempty set. The idea of Q-neighborhood in fuzzy topology was introduced by Pu and Liu in [19]. According to them, a fuzzy point xλ is said to be strongly quasicoincident with F, denoted by xλq̲F, if and only if λ+F(x)>1.AnAbel Grassmann's groupoid, abbreviated as AG-groupoid, is a groupoidS whose elements satisfy the left invertive law: (ab)c=(cb)a for all a,b,c∈S. An AG-groupoid is the midway structure between a commutative semigroup and a groupoid. It is a useful non-associative structure with wide applications in theory of flocks. In an AG-groupoid the medial law, (ab)(cd)=(ac)(bd) for all a,b,c∈S (see [20]). If there exists an element e in an AG-groupoid S such that ex=x for all x∈S then S is called an AG-groupoid with left identitye. If an AG-groupoid S has the right identity then S is a commutative monoid. If an AG-groupoid S contains left identity then (ab)(cd)=(dc)(ba) holds for all a,b,c∈S. Also a(bc)=b(ac) holds for all a,b,c∈S.In this paper, we define (α,β)-fuzzy interior ideals of an AG-groupoid and give some interesting characterizations of an AG -groupoids in terms of (α,β)-fuzzy interior ideals. We also introduce the notion of (α̲,β̲)-fuzzy interior ideals of an AG-groupoid.


## 2. Preliminaries

For subsetsA,B of an AG-groupoid S, we denote  AB={ab∈S∣a∈A,b∈B}. A nonempty subset A of an AG-groupoid S is called an AG-subgroupoid of S if A2⊆A. A is called an interior ideal of S if (SA)S⊆A.LetS be an AG-groupoid. By a fuzzy subset F of S, we mean a mapping, F:S→[0,1].For fuzzy subsetsF1 and F2 of S, define(2.1)F1∘F2:S→[0,1],a↦F1∘F2(a)={⋁a=yzmin{F1(y),F2(z)},ifa=yz(∀a,x,y∈S),0,ifa≠yz.We denote byℱ(S) the set of all fuzzy subsets of S. One can easily see that (ℱ(S),∘) becomes an AG-groupoid as shown in [21]. The order relation “⊆” on ℱ(S) is defined as follows:(2.2)F1⊆F2iffF1(x)≤F2(x)∀x∈S,∀F1,F2∈ℱ(S).For a nonempty family of fuzzy subsets{Fi}i∈I, of an AG-groupoid S, the fuzzy subsets ⋃i∈IFi and ⋂i∈IFi of S are defined as follows:(2.3)⋃i∈IFi:G→[0,1],a↦(⋃i∈IFi)(a):=supi∈I{Fi(a)},⋂i∈IFi:G→[0,1],a↦(⋂i∈IFi)(a):=infi∈I{Fi(a)}.IfI is a finite set, say I={1,2,…,n}, then clearly(2.4)⋃i∈IFi(a)=max{F1(a),F2(a),…,Fn(a)},⋂i∈IFi(a)=min{F1(a),F2(a),…,Fn(a)}.Definition 2.1 (cf. [21]).
LetS be an AG-groupoid and F a fuzzy subset of S. Then F is called a fuzzy interior ideal of S, if it satisfies the following conditions.(B1)
(forallx,y∈S) (F(xy)≥ min{F(x),F(y)}).(B2)
(forallx,y,a∈S) (F((xa)y)≥F(a)).
LetF be a fuzzy subset of S and ∅≠A⊆S, then the characteristic functionχA of A is defined as
(2.5)χA:S→[0,1],a↦χA(a):={1,ifa∈A,0,ifa∉A.Lemma 2.2 (cf. [21]).
LetS be an  AG-groupoid and  F a fuzzy subset of  S. Then  F is a fuzzy interior ideal of  S if and only if  χA is a fuzzy interior ideal of  S.
LetS be an AG-groupoid and F a fuzzy subset of S. Then for every λ∈(0,1] the set
(2.6)U(F;λ):={x∣x∈S,F(x)≥λ}
is called a level set  of F.The proof of the following lemma is easy and we omit it.Lemma 2.3.
LetS be an  AG-groupoid and  F a fuzzy subset of  S. Then  F is a fuzzy interior ideal of  S if and only if  U(F;λ)(≠∅) is an interior ideal of  S for every  λ∈(0,1].


## 3.(α,β)-Fuzzy Interior Ideal

In what follows letS denote an AG-groupoid and let α,β denote any one of ∈,q,∈∨q,∈⋀q.LetS be an AG-groupoid and F a fuzzy subset of S, then the set of the form(3.1)F(y):={λ(≠0),ify=x,0,ify≠x
is called a fuzzy point with support x and value λ and is denoted by xλ. A fuzzy point xλ is said to belong to (resp., quasicoincident with) a fuzzy set F, written as xλ∈F (resp., xλqF) if F(x)≥λ (resp., F(x)+λ≥1). If xλ∈F or xλqF, then xλ∈∨qF. The symbol ∈∨q¯ means ∈∨q does not hold. A fuzzy point xλ is said to be strongly belong  to (resp., strongly quasicoincident with) a fuzzy set F, written as xλ∈̲F (resp., xλq̲F) if F(x)>λ (resp., λ+F(x)>1). If xλ∈̲F or xλq̲F, then xλ∈̲∨q̲F. The symbol ∈̲∨q̲¯ means that ∈̲∨q̲ does not hold.Every fuzzy interior ideal ofS is an (∈,∈)-fuzzy interior ideal of S, as shown in the following theorem.Theorem 3.1.
For any fuzzy subsetF of  S. The conditions  (B1) and  (B2) of Definition 2.1, are equivalent to the following. (B3)
(∀x,y∈S)(∀λ1,λ2∈(0,1]) (xλ1∈F, yλ2∈F→(xy)min{λ1,λ2}∈F).(B4)
(∀x,y,a∈S)(∀λ∈(0,1])(aλ∈F→((xa)y)λ∈F).Proof.
(B1)→(B3). Let x,y∈S and λ1,λ2∈(0,1] be such that xλ1∈F and yλ2∈F. Then F(x)≥λ1 and F(y)≥λ2. By (B1) we have
(3.2)F(xy)≥min{F(x),F(y)}≥min{λ1,λ2},
and so (xy)min{λ1,λ2}∈F.
(B3)→(B1). Let x,y∈S. Since xF(x)∈F and yF(y)∈F. Then by (B3), we have (xy)min{F(x),F(y)}∈F and so F(xy)≥ min{F(x),F(y)}.
(B2)→(B4). Let x,y,a∈S and λ∈(0,1] be such that aλ∈F. Then F(a)≥λ. By (B2) we have
(3.3)F((xa)y)≥F(a)≥λ,
and so ((xa)y)λ∈F.
(B4)→(B2). Let x,y∈S. Since aF(a)∈F, by (B4), we have ((xa)y)F(a)∈F and so F((xa)y)≥F(a).


## 4.(∈,∈∨q)-Fuzzy Interior Ideals

In [5], Jun and Song introduced the concept of a generalized fuzzy interior ideal of a semigroup. In [12], Jun et al. introduced the concept an (α,β)-fuzzy bi-ideal of an ordered semigroup and characterized ordered semigroups in terms of (α,β)-fuzzy bi-ideals. In this section we define the notions of (∈,∈∨q)-fuzzy interior ideals of an Abel Grassmann's groupoid and investigate some of their properties in terms of (∈,∈∨q)-fuzzy interior ideals.LetF be a fuzzy subset of S and F(x)≤0.5 for all x∈S. Let x∈S and λ∈(0,1] be such that xλ∈⋀qF. Then xλ∈F and xλqF and so F(x)≥λ and F(x)+λ≥1. It follows that 1<F(x)+λ≤F(x)+F(x)=2F(x), and so F(x)>0.5, which is a contradiction. This means that {x∈S∣xλ∈⋀qF}=∅.Definition 4.1.
A fuzzy subsetF of S is called an(α,β)-fuzzy interior ideal of S, where α≠∈⋀q, if it satisfies the following conditions:(B5)
(∀x,y∈S)(∀λ1,λ2∈(0,1]) (xλ1αF,yλ2αF→(xy)min{λ1,λ2}βF).(B6)
(∀x,y,a∈S)(∀λ∈(0,1]) (aλαF→((xa)y)λβF).Proposition 4.2.
LetF be a fuzzy subset of  S. If  α=∈ and  β=∈∨q in Definition 4.1. Then  (B5), and  (B6), respectively, of Definition 4.1, are equivalent to the following conditions: (B7)
(∀x,y∈S)(F(xy)≥min{F(x),F(y),0.5}).(B8)
(∀x,y,a∈S)(F((xa)y)≥min{F(a),0.5}).Remark 4.3.
A fuzzy subsetF of an AG-groupoid S is an (∈,∈∨q)-fuzzy interior ideal of S if and only if it satisfies conditions (B7), and (B8) of the above proposition.Using Proposition4.2, we have the following characterization of (∈,∈∨q)-fuzzy interior ideals of an AG-groupoid.Lemma 4.4.
LetS be an  AG-groupoid and  ∅≠I⊆S. Then  I is an interior ideal of  S if and only if the characteristic function  χI of  I is an  (∈,∈∨q )-fuzzy interior ideal of  S.The converse of Theorem3.1 is not true in general, as shown in the following example.Example 4.5.
LetS={a,b,c,d,e} be an AG-groupoid with the following multiplication:
(4.1)·abcdeaaaaaabaaaaacaaecddaadeceaacde
The(S,·) is an AG-groupoid. The interior ideals of S are {a} and {a,c,d,e}. Define a fuzzy subset F:S→[0,1] by
(4.2)F(a)=0.8,F(c)=0.6,F(d)=0.4,F(e)=0.2,F(b)=0.1.
Then
(4.3)U(F;λ):={S,ifλ∈(0,0.1],{a,c,d,e},ifλ∈(0.1,0.2],{a},ifλ∈(0.6,1],∅,ifλ∈(0.8,1].
Obviously, F is an (∈,∈∨q)-fuzzy interior ideal of S by Lemma 4.4. But we have the following.(i)
F is not an (∈,∈)-fuzzy interior ideal of S, since d0.38∈F but
(4.4)(dd)min{0.38,0.38}=e0.38∈¯F.(ii)
F is not an (∈,q)-fuzzy interior ideal of S, since d0.36∈F but(4.5)(dd)min{0.36,0.36}=e0.36q¯F.(iii)
F is not a (q,∈)-fuzzy interior ideal of S, since c0.52qF and e0.82qF but(4.6)(ce)min{0.52,0.82}=d0.52∈¯F.(iv)
F is not a (q,∈∨q)-fuzzy bi-ideal of S, since c0.52qF and e0.82qF but(4.7)(ce)min{0.52,0.82}=d0.52∈∨q¯F.(v)
F is not an (∈∨q,∈⋀q)-fuzzy interior ideal of S, since d0.38∈∨qF but(4.8)(dd)min{0.38,0.38}=e0.38∈⋀q¯F.(vi)
F is not an (∈∨q,q)-fuzzy interior ideal of S, since c0.56∈∨qF and e0.18∈∨qF but(4.9)(ce)min{0.56,0.18}=d0.18q¯F.(vii)
F is not an (∈∨q,∈)-fuzzy interior ideal of S, since d0.38∈∨qF, but(4.10)(dd)min{0.38,0.38}=d0.38∈¯F.(viii)
F is not (∈⋀q,∈)-fuzzy interior ideal of S, d0.38∈⋀qF, but(4.11)(dd)min{0.38,0.38}=d0.38∈¯F.(ix)
F is not a (q,q)-fuzzy interior ideal of S, since c0.52qF and e0.82qF but(4.12)(ce)min{0.52,0.82}=d0.52q¯F.(x)
F is not an (∈,∈∨q)-fuzzy interior ideal of S, since c0.52∈F and e0.82∈F but(4.13)(ce)min{0.52,0.82}=d0.52∈∨q¯F.(xi)
F is not an (∈∨q,∈∨q)-fuzzy interior ideal of S, c0.58∈F and e0.86∈F but(4.14)(ce)min{0.58,0.86}=d0.58∈∨q¯F.Remark 4.6.
By Remark4.3, every fuzzy interior ideal of an AG-groupoid S is an (∈,∈∨q)-fuzzy interior ideal of  S. However, the converse is not true, in general.Example 4.7.
Consider theAG-groupoid given in Example 4.5, and define a fuzzy subset F:S→[0,1] by
(4.15)F(a)=0.8,F(c)=0.6,F(d)=0.4,F(e)=0.2,F(b)=0.1.
ClearlyF is an (∈,∈∨q)-fuzzy interior ideal of S. But F is not an (α,β)-fuzzy interior ideal of S as shown in Example 4.5.Theorem 4.8.
Every(∈,∈)-fuzzy interior ideal of S is an (∈,∈∨q)-fuzzy interior ideal of S.Proof.
It is straightforward.Theorem 4.9.
Every(∈∨q,∈∨q)-fuzzy interior ideal of S is (∈,∈∨q)-fuzzy interior ideal of S.Proof.
LetF be an (∈∨q,∈∨q)-fuzzy interior ideal of S. Let x,y∈S and λ1,λ2∈(0,1] be such that xλ1,yλ2∈F. Then xλ1,yλ2∈∨qF, which implies that (xy)min{λ1,λ2}∈∨qF. Let x,y,a∈S and λ∈(0,1] be such that aλ∈F. Then aλ∈∨qF, and we have ((xa)y)λ∈∨qF.Theorem 4.10.
LetF be a nonzero  (α,β)-fuzzy interior ideal of  S. Then the set  F0:={x∈S∣F(x)>0} is an interior ideal of  S.Proof.
Letx,y∈F0. Then F(x)>0 and F(y)>0. Assume that F(xy)=0. If α∈{∈,∈∨q}, then xF(x)αF and yF(y)αF but (xy)min{F(x),F(y)}β¯F for every β∈{∈,q,∈∨q,∈⋀q}, a contradiction. Note that x1qF and y1qF but (xy)min{1,1}=(xy)1β¯F for every β∈{∈,q,∈∨q,∈⋀q}, a contradiction. Hence F(xy)>0, that is, xy∈F0. Let a∈F0 and x,y∈S. Then F(a)>0. Assume that F((xa)y)=0. If α∈{∈,∈∨q} then, aF(a)αF but ((xa)y)F(a)β¯F for every β∈{∈,q,∈∨q,∈⋀q}, a contradiction. Note that a1qF but ((xa)y)min{1,1}=((xa)y)1β¯F for every β∈{∈,q,∈∨q,∈⋀q}, a contradiction. Hence F((xa)y)>0, that is, (xa)y∈F0. Consequently, F0 is an interior ideal of S.Theorem 4.11.
LetI be an interior ideal and  F a fuzzy subset of  S such that (1)
(∀x∈S∖I)(F(x)=0),(2)
(∀x∈I)(F(x)≥0.5).
Then(a)
F is a (q, ∈∨q )-fuzzy interior ideal of  S,(b)
F is an ( ∈,∈∨q)-fuzzy interior ideal of  S.Proof.
Letx,y∈S and λ1,λ2∈(0,1] be such that xλ1qF and yλ2qF. Then x,y∈I and we have xy∈I. If min{λ1,λ2}≤0.5, then F(xy)≥0.5≥min{λ1,λ2} and hence (xy)min{λ1,λ2}∈F. If min{λ1,λ2}>0.5, then
(4.16)F(xy)+min{λ1,λ2}>0.5+0.5=1,
and so (xy)min{λ1,λ2}qF. Therefore (xy)min{λ1,λ2}∈∨qF. Let x,y,a∈S and λ∈(0,1] be such that aλ1qF. Then a∈I and we have (xa)y∈(SI)S⊆I. If λ1≤0.5, then F((xa)y)≥0.5≥λ and hence ((xa)y)λ∈F. If λ1>0.5, then
(4.17)F((xa)y)+λ≥0.5+0.5=1,
and so ((xa)y)λqF. Therefore ((xa)y)λ∈∨qF. Therefore F is a (q,∈∨q)-fuzzy interior ideal of S.
Letx,y∈S and λ1,λ2∈(0,1] be such that xλ1∈F and yλ2∈F. Then x,y∈I and we have xy∈I. If min{λ1,λ2}≤0.5, then F(xy)≥0.5≥min{λ1,λ2} and hence (xy)min{λ1,λ2}∈F. If min{λ1,λ2}>0.5, then
(4.18)F(xy)+min{λ1,λ2}>0.5+0.5=1,
and so (xy)min{λ1,λ2}qF. Therefore (xy)min{λ1,λ2}∈∨qF. Now let x,y,a∈S and λ∈(0,1] be such that aλ∈F. Then a∈I and we have (xa)y∈I. If λ≤0.5, then F((xa)y)≥0.5≥λ and hence ((xa)y)λ∈F. If λ>0.5, then
(4.19)F((xa)y)+λ>0.5+0.5=1,
and so ((xa)y)λqF. Therefore
(4.20)((xa)y)λ∈∨qF,
and so F is an (∈,∈∨q)-fuzzy interior ideal of S.From Example4.5, we see that an (∈,∈∨q)-fuzzy interior ideal is not a (q,∈∨q)-fuzzy interior ideal (Example 4.5, iv).In the following theorem we give a condition for an (∈,∈∨q)-fuzzy interior ideal to be an (∈,∈)-fuzzy interior ideal of S.Theorem 4.12.
LetF be an ( ∈,∈∨q)-fuzzy interior ideal of  S such that  F(x)<0.5 for all  x∈S. Then  F is an ( ∈,∈)-fuzzy interior ideal of  S.Proof.
Letx,y∈S and λ1,λ2∈(0,1] be such that xλ1,yλ2∈F. Then F(x)≥λ1 and F(y)≥λ2 and so F(xy)≥ min{F(x),F(y),0.5}≥ min{λ1,λ2,0.5}= min{λ1,λ2} and hence (xy)min{λ1,λ2}∈F. Now, let x,y,a∈S and λ∈(0,1] be such that aλ∈F. Then F(a)≥λ and we have
(4.21)F((xa)y)≥F(a)≥λ;
consequently, ((xa)y)λ∈F. Therefore F is an (∈,∈ )-fuzzy interior ideal of S.For any fuzzy subsetF of an AG-groupoid S and λ∈(0,1], we denote(4.22)Q(F;λ):={x∈S∣xλqF},[F]λ:={x∈S∣xλ∈∨qF}.Obviously,[F]λ=U(F;λ)∪Q(F;λ).We call[F]λ an (∈∨q)-level interior ideal of F and Q(F;λ) a q-level interior ideal of F.We have given a characterization of (∈,∈∨q)-fuzzy interior ideals by using level subsets (see Proposition 4.2). Now we provide another characterization of (∈,∈∨q)-fuzzy interior ideals by using the set [F]λ.Theorem 4.13.
LetS be an AG-groupoid and  F a fuzzy subset of  S. Then  A is an  (∈,∈∨q)-fuzzy interior ideal of  S if and only if  [F]λ is an interior ideal of  S for all  λ∈(0,1].Proof.
LetF be an (∈,∈∨q)-fuzzy interior ideal of S. Let x,y∈[F]λ for λ∈(0,1]. Then xλ∈∨qF and yλ∈∨qF, that is, F(x)≥t or F(x)+t≥1, and F(y)≥t or F(y)+t≥1. Since F is an (∈,∈∨q)-fuzzy interior ideal of S, we have
(4.23)F(xy)≥min{F(x),F(y),0.5}.
We discuss the following cases.
Case 1.
LetF(x)≥λ and F(y)≥λ. If λ>0.5, then
(4.24)F(xy)≥min{F(x),F(y),0.5}=0.5,
and hence (xy)λ∈F. If λ≤0.5. Then
(4.25)F(xy)≥min{F(x),F(y),0.5}≥λ,
and so (xy)λ∈F. Hence (xy)λ∈∨qF.Case 2.
LetF(x)≥λ and F(y)+λ≥1. If λ>0.5, then
(4.26)F(xy)≥min{F(x),F(y),0.5},
and we discuss the following cases.
Ifλ>0.5, then
(4.27)F(xy)≥min{F(x),F(y),0.5}=min{F(y),0.5}≥min{1-λ,0.5}=1-λ,
that is, F(xy)+t≥1 and thus (xy)tqF. If λ≤0.5, then
(4.28)F(xy)≥min{F(x),F(y),0.5}≥min{λ,1-λ,0.5}=λ,
and so (xy)λ∈F. Hence (xy)λ∈∨qF.Case 3.
LetF(x)+λ≥1 and F(y)≥λ. If λ<0.5, then
(4.29)F(xy)≥min{F(x),F(y),0.5}≥min{F(x),0.5}≥min{1-λ,0.5}=1-λ,
that is, F(xy)+λ≥1 and hence (xy)λqF. If λ<0.5, then
(4.30)F(xy)≥min{F(x),F(y),0.5}≥min{1-λ,λ,0.5}=λ,
and so (xy)λ∈F. Hence (xy)λ∈∨qF.Case 4.
LetF(x)+λ≥1 and F(y)+λ≥1. If λ>0.5, then
(4.31)F(xy)≥min{F(x),F(y),0.5}>min{1-λ,0.5}=1-λ,
that is, F(xy)+λ≥1 and thus (xy)λqF. If λ≤0.5, then
(4.32)F(xy)≥min{F(x),F(y),0.5}≥min{1-λ,0.5}=0.5≥λ,
and so (xy)λ∈F. Thus in any case, we have (xy)λ∈∨qF. Therefore xy∈[F]λ. Now, let a∈[F]λ for λ∈(0,1]. Then aλ∈∨qF, that is, F(x)≥λ or F(x)+λ≥1. Since F is an (∈,∈∨q)-fuzzy interior ideal of S, we have
(4.33)F((xa)y)≥F(a).Case 1.
LetF(a)≥λ. If λ≥0.5, then
(4.34)F((xa)y)≥F(a)≥0.5
and hence ((xa)y)λqF. If λ<0.5, then
(4.35)F((xa)y)≥F(a)≥λ,
and so ((xa)y)λ∈F. Hence ((xa)y)λ∈∨qF.Case 2.
LetF(a)≥λ and F(a)+λ≥1. If λ≥0.5, then
(4.36)F((xa)y)≥F(a)λ≥0.5.
If λ<0.5, then
(4.37)F((xa)y)≥F(a)≥min{1-λ,0.5}=1-λ,
that is, F((xa)y)+λ≥1 and thus ((xa)y)λqF. If λ≤0.5, then
(4.38)F((xa)y)≥F(a)≥min{λ,1-λ,0.5}=λ,
and so ((xa)y)λ∈F. Hence ((xa)y)λ∈∨qA.Thus in any case, we have ((xa)y)λ∈∨qF. Therefore (xa)y∈[F]λ.
Conversely, letF be a fuzzy subset of S and let x,y,a∈S be such that F(xy)<λ< min {F(x),F(y),0.5} for some λ∈(0,0.5]. Then x,y∈U(F;λ)⊆[F]λ, it implies that xy∈[F]λ. Hence F(xy)≥λ or F(xy)+λ≥1, a contradiction. Hence F(xy)≥min{F(x),F(y),0.5} for all x,y∈S. Now let F((xa)y)<F(a) for some x,y,a∈S. Choose λ such that F((xa)y)<λ<F(a). Then a∈U(F;λ)⊆[F]λ. It follows that (xa)y∈[F]λ. This implies that F((xa)y)≥λ or F((xa)y)+λ>1, a contradiction. Hence F((xa)y)≥min{F(a),0.5} for all x,y,a∈S. By Proposition 4.2, it follows that F is an (∈,∈∨q)-fuzzy interior ideal of S.U(F;λ) and [F]λ are interior ideals of S for all λ∈(0,1], but Q(F;λ) is not an interior ideal of G for all λ∈(0,1], in general. As shown in the following example.Example 4.14.
Consider theAG-groupoid as given in Example 4.5. Define a fuzzy subset F by
(4.39)F(a)=0.8,F(c)=0.6,F(d)=0.4,F(e)=0.2,F(b)=0.1.ThenQ(F;λ)={a,c,d} for all 0.2<λ≤0.4. Since c0.56∈F and e0.18∈F but (ce)min{0.56,0.18}=d0.18q¯F, hence Q(F;λ) is not an interior ideal of S for all λ∈(0.2,0.4].Proposition 4.15.
If{Fi}i∈I is a family of (∈,∈∨q)-fuzzy bi-ideals of an AG-groupoid S, then ⋂i∈IFi is an (∈,∈∨q)-fuzzy bi-ideal of S.Proof.
Let{Fi}i∈I be a family of (∈,∈∨q)-fuzzy bi-ideals of S. Let x,y∈S. Then
(4.40)(⋂i∈IFi)(xy)=⋀i∈IFi(xy)≥⋀i∈I(Fi(x)⋀Fi(y))=(⋀i∈IFi(x)⋀⋀i∈IFi(y))=(⋂i∈IFi)(x)⋀(⋂i∈IFi)(y).
Let x,y,a∈G. Then
(4.41)(⋂i∈IFi)((xa)y)=⋀i∈IFi((xa)y)≥⋀i∈I(Fi(a))=(⋂i∈IFi)(a).
Thus ⋂i∈IFi is an (∈,∈∨q)-fuzzy interior ideal of S.Definition 4.16.
LetS be an AG-groupoid and F a fuzzy subset of S. Then F is called a strongly fuzzy interior ideal of S, if it satisfies the following conditions.(B9)
(∀x,y∈S)(F(xy)>min{F(x),F(y)}).(B10)
(∀x,y,a∈S)((F(xa)y)>F(a)).Every fuzzy interior ideal of an AG-groupoid S is strongly fuzzy interior ideal of S.Theorem 4.17.
For any fuzzy subsetF of  S. The conditions  (B9) and  (B10) of Definition 4.16 are equivalent to the following. (B11)
(∀x,y∈S)(∀λ1,λ2∈(0,1])(xλ1∈̲F,yλ2∈̲F→(xy)min{λ1,λ2}∈̲F).(B12)
(∀x,y,a∈S)(∀λ∈(0,1])(aλ∈̲F→((xa)y)λ∈̲F).Proof.
(B9)→(B11). Let F be a fuzzy subset of S. Let x,y∈S and λ1,λ2∈(0,1] be such that xλ1∈̲F, yλ2∈̲F. Then F(x)>λ1 and F(y)>λ2. Using (B9)
(4.42)F(xy)>min{F(x),F(y)}>min{λ1,λ2},
and so (xy)min{λ1,λ2}∈F.
(B11)→(B9). Let x,y∈S. Since xF(x)∈̲F and yF(y)∈̲F. Then by (B9), we have (xy)min{F(x),F(y)}∈̲F and so F(xy)>min{F(x),F(y)}.
(B10)→(B11). Let x,y,a∈S and λ∈(0,1] be such that aλ∈̲F. Then F(a)>λ. By (B10) we have
(4.43)F((xa)y)>F(a)>λ,
and so ((xa)y)λ∈̲F.
(B11)→(B10). Let x,y∈S. Since aF(a)∈̲F, by (B11), we have ((xa)y)F(a)∈̲F and so F((xa)y)>F(a).


## 5.(∈̲,∈̲∨q̲)-Fuzzy Interior Ideals

In this section we define the notions of(∈̲,∈̲∨q̲)-fuzzy interior ideals of an Abel Grassmann's groupoid and investigate some of their properties in terms of (∈̲,∈̲∨q̲)-fuzzy interior ideals.LetF be a fuzzy subset of S and F(x)<0.5 for all x∈S. Let x∈S and λ∈(0,1] be such that xλ∈̲⋀q̲F. Then xλ∈̲F and xλq̲F and so F(x)>λ and F(x)+λ>1. It follows that 1<F(x)+λ<F(x)+F(x)=2F(x), and so F(x)>0.5, which is a contradiction. This means that {x∈S∣xλ∈̲⋀q̲F}=∅.Definition 5.1.
A fuzzy subsetF of S is called an(α̲,β̲)-fuzzy interior ideal of S, where α≠∈̲⋀q̲, if it satisfies the following conditions.(B13)
(∀x,y∈S)(∀λ1,λ2∈(0,1])(xλ1α̲F,yλ2α̲F→(xy)min{λ1,λ2}β̲F).(B14)
(∀x,y,a∈S)(∀λ∈(0,1])(aλα̲F→((xa)y)λβ̲F).Proposition 5.2.
LetF be a fuzzy subset of  S. If  α̲=∈̲ and  β̲=∈̲∨q̲ in  Definition 5.1. Then  (B13), and  (B14), respectively, of Definition 5.1, are equivalent to the following conditions. (B15)
(∀x,y∈S)(F(xy)≥min{F(x),F(y),0.5}).(B16)
(∀x,y,a∈S)(F((xa)y)≥min{F(a),0.5}).Remark 5.3.
A fuzzy subsetF of an AG-groupoid S is an (∈̲,∈̲∨q̲ )-fuzzy interior ideal of S if and only if it satisfies conditions (B15) and (B16) of the above proposition.Using Proposition5.2, we have the following characterization of (∈̲,∈̲∨q̲)-fuzzy interior ideals of an AG-groupoid.Lemma 5.4.
LetS be an  AG-groupoid and  ∅≠I⊆S. Then  I is an interior ideal of  S if and only if the characteristic function  χI of  I is an  (∈̲,∈̲∨q̲)-fuzzy interior ideal of  S.



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