Generalized Fuzzy Interior Ideals in Abel Grassmann's Groupoids

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 groupoids 𝑆 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.

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 𝑥_𝜆(0<𝜆<1) is said to be strongly belong to a fuzzy subset 𝐹, denoted by 𝑥_𝜆∈𝐹, if and only if 𝐹(𝑥)>𝜆. 𝜆-strong cut set 𝐹_𝜆 of 𝐹 is given by 𝐹_𝜆={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 𝑥_𝜆 is said to be strongly quasicoincident with 𝐹, denoted by 𝑥_𝜆𝑞𝐹, if and only if 𝜆+𝐹(𝑥)>1.

An Abel Grassmann's groupoid, abbreviated as 𝐴𝐺-groupoid, is a groupoid 𝑆 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 identity 𝑒. 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 (𝛼,𝛽)-fuzzy interior ideals of an AG-groupoid.

2. Preliminaries

For subsets 𝐴,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.

Let 𝑆 be an 𝐴𝐺-groupoid. By a fuzzy subset 𝐹 of 𝑆, we mean a mapping, 𝐹∶𝑆→[0,1].

For fuzzy subsets 𝐹₁ and 𝐹₂ of 𝑆, define

𝐹₁∘𝐹₂∶𝑆→[0,1],𝑎↦𝐹₁∘𝐹₂=(𝑎)_{𝑎=𝑦𝑧}min{𝐹₁(𝑦),𝐹₂(𝑧)},if𝑎=𝑦𝑧(∀𝑎,𝑥,𝑦∈𝑆),0,if𝑎≠𝑦𝑧. (2.1)

We denote by ℱ(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:

𝐹₁⊆𝐹₂iff𝐹₁(𝑥)≤𝐹₂(𝑥)∀𝑥∈𝑆,∀𝐹₁,𝐹₂∈ℱ(𝑆). (2.2)

For a nonempty family of fuzzy subsets {𝐹_𝑖}_𝑖∈𝐼, of an 𝐴𝐺-groupoid 𝑆, the fuzzy subsets ⋃_𝑖∈𝐼𝐹_𝑖 and ⋂_𝑖∈𝐼𝐹_𝑖 of 𝑆 are defined as follows:

_𝑖∈𝐼𝐹_𝑖∶𝐺→[0,1],𝑎↦(_𝑖∈𝐼𝐹_𝑖)(𝑎)∶=sup_𝑖∈𝐼{𝐹_𝑖(𝑎)},_𝑖∈𝐼𝐹_𝑖∶𝐺→[0,1],𝑎↦(_𝑖∈𝐼𝐹_𝑖)(𝑎)∶=inf_𝑖∈𝐼{𝐹_𝑖(𝑎)}. (2.3)

If 𝐼 is a finite set, say 𝐼={1,2,…,𝑛}, then clearly

_𝑖∈𝐼𝐹_𝑖(𝑎)=max{𝐹₁(𝑎),𝐹₂(𝑎),…,𝐹_𝑛(𝑎)},_𝑖∈𝐼𝐹_𝑖(𝑎)=min{𝐹₁(𝑎),𝐹₂(𝑎),…,𝐹_𝑛(𝑎)}. (2.4)

Definition 2.1 (cf. [21]). Let 𝑆 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)).
Let 𝐹 be a fuzzy subset of 𝑆 and ∅≠𝐴⊆𝑆, then the characteristic function 𝜒_𝐴 of 𝐴 is defined as 𝜒_𝐴∶𝑆→[0,1],𝑎↦𝜒_𝐴(𝑎):=1,if𝑎∈𝐴,0,if𝑎∉𝐴. (2.5)

Lemma 2.2 (cf. [21]). Let 𝑆 be an 𝐴𝐺-groupoid and 𝐹 a fuzzy subset of 𝑆. Then 𝐹 is a fuzzy interior ideal of 𝑆 if and only if 𝜒_𝐴 is a fuzzy interior ideal of 𝑆.
Let 𝑆 be an 𝐴𝐺-groupoid and 𝐹 a fuzzy subset of 𝑆. Then for every 𝜆∈(0,1] the set 𝑈(𝐹;𝜆):={𝑥∣𝑥∈𝑆,𝐹(𝑥)≥𝜆} (2.6) is called a level set of 𝐹.

The proof of the following lemma is easy and we omit it.

Lemma 2.3. Let 𝑆 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 let 𝑆 denote an 𝐴𝐺-groupoid and let 𝛼,𝛽 denote any one of ∈,𝑞,∈∨𝑞,∈⋀𝑞.

Let 𝑆 be an AG-groupoid and 𝐹 a fuzzy subset of 𝑆, then the set of the form

𝐹(𝑦):=𝜆(≠0),if𝑦=𝑥,0,if𝑦≠𝑥 (3.1) is called a fuzzy point with support 𝑥 and value 𝜆 and is denoted by 𝑥_𝜆. A fuzzy point 𝑥_𝜆 is said to belong to (resp., quasicoincident with) a fuzzy set 𝐹, written as 𝑥_𝜆∈𝐹 (resp., 𝑥_𝜆𝑞𝐹) if 𝐹(𝑥)≥𝜆 (resp., 𝐹(𝑥)+𝜆≥1). If 𝑥_𝜆∈𝐹 or 𝑥_𝜆𝑞𝐹, then 𝑥_𝜆∈∨𝑞𝐹. The symbol ∈∨𝑞 means ∈∨𝑞 does not hold. A fuzzy point 𝑥_𝜆 is said to be strongly belong to (resp., strongly quasicoincident with) a fuzzy set 𝐹, written as 𝑥_𝜆∈𝐹 (resp., 𝑥_𝜆𝑞𝐹) if 𝐹(𝑥)>𝜆 (resp., 𝜆+𝐹(𝑥)>1). If 𝑥_𝜆∈𝐹 or 𝑥_𝜆𝑞𝐹, then 𝑥_𝜆∈∨𝑞𝐹. The symbol ∈∨𝑞 means that ∈∨𝑞 does not hold.

Every fuzzy interior ideal of 𝑆 is an (∈,∈)-fuzzy interior ideal of 𝑆, as shown in the following theorem.

Theorem 3.1. For any fuzzy subset 𝐹 of 𝑆. The conditions (𝐵1) and (𝐵2) of Definition 2.1, are equivalent to the following. (𝐵3) (∀𝑥,𝑦∈𝑆)(∀𝜆₁,𝜆₂∈(0,1]) (𝑥_𝜆1∈𝐹, 𝑦_𝜆2∈𝐹→(𝑥𝑦)_{min{𝜆1,𝜆2}}∈𝐹).(𝐵4) (∀𝑥,𝑦,𝑎∈𝑆)(∀𝜆∈(0,1])(𝑎_𝜆∈𝐹→((𝑥𝑎)𝑦)_𝜆∈𝐹).

Proof. (𝐵1)→(𝐵3). Let 𝑥,𝑦∈𝑆 and 𝜆₁,𝜆₂∈(0,1] be such that 𝑥_𝜆1∈𝐹 and 𝑦_𝜆2∈𝐹. Then 𝐹(𝑥)≥𝜆₁ and 𝐹(𝑦)≥𝜆₂. By (𝐵1) we have 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦)}≥min{𝜆₁,𝜆₂}, (3.2) and so (𝑥𝑦)_{min{𝜆1,𝜆2}}∈𝐹.
(𝐵3)→(𝐵1). Let 𝑥,𝑦∈𝑆. Since 𝑥_𝐹(𝑥)∈𝐹 and 𝑦_𝐹(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 𝑎_𝜆∈𝐹. Then 𝐹(𝑎)≥𝜆. By (𝐵2) we have 𝐹((𝑥𝑎)𝑦)≥𝐹(𝑎)≥𝜆, (3.3) and so ((𝑥𝑎)𝑦)_𝜆∈𝐹.
(𝐵4)→(𝐵2). Let 𝑥,𝑦∈𝑆. Since 𝑎_𝐹(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.

Let 𝐹 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 𝑥_𝜆∈⋀q𝐹. Then 𝑥_𝜆∈𝐹 and 𝑥_𝜆𝑞𝐹 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∣𝑥_𝜆∈⋀𝑞𝐹}=∅.

Definition 4.1. A fuzzy subset 𝐹 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]) (𝑥_𝜆1𝛼𝐹,𝑦_𝜆2𝛼𝐹→(𝑥𝑦)_{min{𝜆1,𝜆2}}𝛽𝐹).(𝐵6) (∀𝑥,𝑦,𝑎∈𝑆)(∀𝜆∈(0,1]) (𝑎_𝜆𝛼𝐹→((𝑥𝑎)𝑦)_𝜆𝛽𝐹).

Proposition 4.2. Let 𝐹 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 subset 𝐹 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 Proposition 4.2, we have the following characterization of (∈,∈∨𝑞)-fuzzy interior ideals of an 𝐴𝐺-groupoid.

Lemma 4.4. Let 𝑆 be an 𝐴𝐺-groupoid and ∅≠𝐼⊆𝑆. Then 𝐼 is an interior ideal of 𝑆 if and only if the characteristic function 𝜒_𝐼 of 𝐼 is an (∈,∈∨q )-fuzzy interior ideal of 𝑆.

The converse of Theorem 3.1 is not true in general, as shown in the following example.

Example 4.5. Let 𝑆={𝑎,𝑏,𝑐,𝑑,𝑒} be an AG-groupoid with the following multiplication: ⋅𝑎𝑏𝑐𝑑𝑒𝑎𝑎𝑎𝑎𝑎𝑎𝑏𝑎𝑎𝑎𝑎𝑎𝑐𝑎𝑎𝑒𝑐𝑑𝑑𝑎𝑎𝑑𝑒𝑐𝑒𝑎𝑎𝑐𝑑𝑒 (4.1)
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 𝐹(𝑎)=0.8,𝐹(𝑐)=0.6,𝐹(𝑑)=0.4,𝐹(𝑒)=0.2,𝐹(𝑏)=0.1. (4.2) Then 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]. (4.3) 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 𝑑_0.38∈𝐹 but (𝑑𝑑)_{min{0.38,0.38}}=𝑒_0.38∈𝐹. (4.4)(ii)𝐹 is not an (∈,𝑞)-fuzzy interior ideal of 𝑆, since 𝑑_0.36∈𝐹 but (𝑑𝑑)_{min{0.36,0.36}}=𝑒_0.36𝑞𝐹. (4.5)(iii)𝐹 is not a (𝑞,∈)-fuzzy interior ideal of 𝑆, since 𝑐_0.52𝑞𝐹 and 𝑒_0.82𝑞𝐹 but (𝑐𝑒)_{min{0.52,0.82}}=𝑑_0.52∈𝐹. (4.6)(iv)𝐹 is not a (𝑞,∈∨𝑞)-fuzzy bi-ideal of 𝑆, since 𝑐_0.52𝑞𝐹 and 𝑒_0.82𝑞𝐹 but (𝑐𝑒)_{min{0.52,0.82}}=𝑑_0.52∈∨𝑞𝐹. (4.7)(v)𝐹 is not an (∈∨𝑞,∈⋀𝑞)-fuzzy interior ideal of 𝑆, since 𝑑_0.38∈∨𝑞𝐹 but (𝑑𝑑)_{min{0.38,0.38}}=𝑒_0.38∈𝑞𝐹. (4.8)(vi)𝐹 is not an (∈∨𝑞,𝑞)-fuzzy interior ideal of 𝑆, since 𝑐_0.56∈∨𝑞𝐹 and 𝑒_0.18∈∨q𝐹 but (𝑐𝑒)_{min{0.56,0.18}}=𝑑_0.18𝑞𝐹. (4.9)(vii)𝐹 is not an (∈∨𝑞,∈)-fuzzy interior ideal of 𝑆, since 𝑑_0.38∈∨𝑞𝐹, but (𝑑𝑑)_{min{0.38,0.38}}=𝑑_0.38∈𝐹. (4.10)(viii)𝐹 is not (∈⋀𝑞,∈)-fuzzy interior ideal of 𝑆, 𝑑_0.38∈⋀𝑞𝐹, but (𝑑𝑑)_{min{0.38,0.38}}=𝑑_0.38∈𝐹. (4.11)(ix)𝐹 is not a (𝑞,𝑞)-fuzzy interior ideal of 𝑆, since 𝑐_0.52𝑞𝐹 and 𝑒_0.82𝑞𝐹 but (𝑐𝑒)_{min{0.52,0.82}}=𝑑_0.52𝑞𝐹. (4.12)(x)𝐹 is not an (∈,∈∨𝑞)-fuzzy interior ideal of 𝑆, since 𝑐_0.52∈𝐹 and 𝑒_0.82∈𝐹 but (𝑐𝑒)_{min{0.52,0.82}}=𝑑_0.52∈∨𝑞𝐹. (4.13)(xi)𝐹 is not an (∈∨𝑞,∈∨𝑞)-fuzzy interior ideal of 𝑆, 𝑐_0.58∈𝐹 and 𝑒_0.86∈𝐹 but (𝑐𝑒)_{min{0.58,0.86}}=𝑑_0.58∈∨𝑞𝐹. (4.14)

Remark 4.6. By Remark 4.3, 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 the 𝐴𝐺-groupoid given in Example 4.5, and define a fuzzy subset 𝐹∶𝑆→[0,1] by 𝐹(𝑎)=0.8,𝐹(𝑐)=0.6,𝐹(𝑑)=0.4,𝐹(𝑒)=0.2,𝐹(𝑏)=0.1. (4.15)
Clearly 𝐹 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. Let 𝐹 be an (∈∨𝑞,∈∨𝑞)-fuzzy interior ideal of 𝑆. Let 𝑥,𝑦∈𝑆 and 𝜆₁,𝜆₂∈(0,1] be such that 𝑥_𝜆1,𝑦_𝜆2∈𝐹. Then 𝑥_𝜆1,𝑦_𝜆2∈∨𝑞𝐹, which implies that (𝑥𝑦)_{min{𝜆1,𝜆2}}∈∨𝑞𝐹. Let 𝑥,𝑦,𝑎∈𝑆 and 𝜆∈(0,1] be such that 𝑎_𝜆∈𝐹. Then 𝑎_𝜆∈∨𝑞𝐹, and we have ((𝑥𝑎)𝑦)_𝜆∈∨𝑞𝐹.

Theorem 4.10. Let 𝐹 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. Let 𝑥,𝑦∈𝐹₀. Then 𝐹(𝑥)>0 and 𝐹(𝑦)>0. Assume that 𝐹(𝑥𝑦)=0. If 𝛼∈{∈,∈∨𝑞}, then 𝑥_𝐹(𝑥)𝛼𝐹 and 𝑦_𝐹(𝑦)𝛼𝐹 but (𝑥𝑦)_{min{𝐹(𝑥),𝐹(𝑦)}}𝛽𝐹 for every 𝛽∈{∈,𝑞,∈∨𝑞,∈⋀𝑞}, a contradiction. Note that 𝑥₁𝑞𝐹 and 𝑦₁𝑞𝐹 but (𝑥𝑦)_min{1,1}=(𝑥𝑦)₁𝛽𝐹 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, 𝑎_𝐹(𝑎)𝛼𝐹 but ((𝑥𝑎)𝑦)_𝐹(𝑎)𝛽𝐹 for every 𝛽∈{∈,𝑞,∈∨𝑞,∈⋀𝑞}, a contradiction. Note that 𝑎₁q𝐹 but ((𝑥𝑎)𝑦)_min{1,1}=((𝑥𝑎)𝑦)₁𝛽𝐹 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. Let 𝐼 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. Let 𝑥,𝑦∈𝑆 and 𝜆₁,𝜆₂∈(0,1] be such that 𝑥_𝜆1𝑞𝐹 and 𝑦_𝜆2𝑞𝐹. 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 𝐹(𝑥𝑦)+min{𝜆₁,𝜆₂}>0.5+0.5=1, (4.16) 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 𝑎_𝜆1𝑞𝐹. 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 𝐹((𝑥𝑎)𝑦)+𝜆≥0.5+0.5=1, (4.17) and so ((𝑥𝑎)𝑦)_𝜆𝑞𝐹. Therefore ((𝑥𝑎)𝑦)_𝜆∈∨𝑞𝐹. Therefore 𝐹 is a (𝑞,∈∨𝑞)-fuzzy interior ideal of 𝑆.
Let 𝑥,𝑦∈𝑆 and 𝜆₁,𝜆₂∈(0,1] be such that 𝑥_𝜆1∈𝐹 and 𝑦_𝜆2∈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 𝐹(𝑥𝑦)+min{𝜆₁,𝜆₂}>0.5+0.5=1, (4.18) 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 𝑎_𝜆∈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 𝐹((𝑥𝑎)𝑦)+𝜆>0.5+0.5=1, (4.19) and so ((𝑥𝑎)𝑦)_𝜆𝑞𝐹. Therefore ((𝑥𝑎)𝑦)_𝜆∈∨𝑞𝐹, (4.20) and so 𝐹 is an (∈,∈∨𝑞)-fuzzy interior ideal of 𝑆.

From Example 4.5, 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. Let 𝐹 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. Let 𝑥,𝑦∈𝑆 and 𝜆₁,𝜆₂∈(0,1] be such that 𝑥_𝜆1,𝑦_𝜆2∈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 𝑎_𝜆∈𝐹. Then 𝐹(𝑎)≥𝜆 and we have 𝐹((𝑥𝑎)𝑦)≥𝐹(𝑎)≥𝜆; (4.21) consequently, ((𝑥𝑎)𝑦)_𝜆∈𝐹. Therefore 𝐹 is an (∈,∈ )-fuzzy interior ideal of 𝑆.

For any fuzzy subset 𝐹 of an AG-groupoid 𝑆 and 𝜆∈(0,1], we denote

𝑄(𝐹;𝜆):={𝑥∈𝑆∣𝑥_𝜆𝑞𝐹},[𝐹]_𝜆∶={𝑥∈𝑆∣𝑥_𝜆∈∨𝑞𝐹}. (4.22)

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. Let 𝑆 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. Let 𝐹 be an (∈,∈∨𝑞)-fuzzy interior ideal of 𝑆. Let 𝑥,𝑦∈[𝐹]_𝜆 for 𝜆∈(0,1]. Then 𝑥_𝜆∈∨𝑞𝐹 and 𝑦_𝜆∈∨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 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦),0.5}. (4.23)
We discuss the following cases.
Case 1. Let 𝐹(𝑥)≥𝜆 and 𝐹(𝑦)≥𝜆. If 𝜆>0.5, then 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦),0.5}=0.5, (4.24) and hence (𝑥𝑦)_𝜆∈𝐹. If 𝜆≤0.5. Then 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦),0.5}≥𝜆, (4.25) and so (𝑥𝑦)_𝜆∈𝐹. Hence (𝑥𝑦)_𝜆∈∨𝑞𝐹.Case 2. Let 𝐹(𝑥)≥𝜆 and 𝐹(𝑦)+𝜆≥1. If 𝜆>0.5, then 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦),0.5}, (4.26) and we discuss the following cases.
If 𝜆>0.5, then 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦),0.5}=min{𝐹(𝑦),0.5}≥min{1−𝜆,0.5}=1−𝜆, (4.27) that is, 𝐹(𝑥𝑦)+𝑡≥1 and thus (𝑥𝑦)_𝑡𝑞𝐹. If 𝜆≤0.5, then 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦),0.5}≥min{𝜆,1−𝜆,0.5}=𝜆, (4.28) and so (𝑥𝑦)_𝜆∈𝐹. Hence (𝑥𝑦)_𝜆∈∨𝑞𝐹.Case 3. Let 𝐹(𝑥)+𝜆≥1 and 𝐹(𝑦)≥𝜆. If 𝜆<0.5, then 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦),0.5}≥min{𝐹(𝑥),0.5}≥min{1−𝜆,0.5}=1−𝜆, (4.29) that is, 𝐹(𝑥𝑦)+𝜆≥1 and hence (𝑥𝑦)_𝜆𝑞𝐹. If 𝜆<0.5, then 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦),0.5}≥min{1−𝜆,𝜆,0.5}=𝜆, (4.30) and so (𝑥𝑦)_𝜆∈𝐹. Hence (𝑥𝑦)_𝜆∈∨𝑞𝐹.Case 4. Let 𝐹(𝑥)+𝜆≥1 and 𝐹(𝑦)+𝜆≥1. If 𝜆>0.5, then 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦),0.5}>min{1−𝜆,0.5}=1−𝜆, (4.31) that is, 𝐹(𝑥𝑦)+𝜆≥1 and thus (𝑥𝑦)_𝜆𝑞𝐹. If 𝜆≤0.5, then 𝐹(𝑥𝑦)≥min{𝐹(𝑥),𝐹(𝑦),0.5}≥min{1−𝜆,0.5}=0.5≥𝜆, (4.32) 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 𝑎_𝜆∈∨𝑞𝐹, that is, 𝐹(𝑥)≥𝜆 or 𝐹(𝑥)+𝜆≥1. Since 𝐹 is an (∈,∈∨𝑞)-fuzzy interior ideal of 𝑆, we have 𝐹((𝑥𝑎)𝑦)≥𝐹(𝑎). (4.33)Case 1. Let 𝐹(𝑎)≥𝜆. If 𝜆≥0.5, then 𝐹((𝑥𝑎)𝑦)≥𝐹(𝑎)≥0.5 (4.34) and hence ((𝑥𝑎)𝑦)_𝜆𝑞𝐹. If 𝜆<0.5, then 𝐹((𝑥𝑎)𝑦)≥𝐹(𝑎)≥𝜆, (4.35) and so ((𝑥𝑎)𝑦)_𝜆∈𝐹. Hence ((𝑥𝑎)𝑦)_𝜆∈∨𝑞𝐹.Case 2. Let 𝐹(𝑎)≥𝜆 and 𝐹(𝑎)+𝜆≥1. If 𝜆≥0.5, then 𝐹((𝑥𝑎)𝑦)≥𝐹(𝑎)𝜆≥0.5. (4.36) If 𝜆<0.5, then 𝐹((𝑥𝑎)𝑦)≥𝐹(𝑎)≥min{1−𝜆,0.5}=1−𝜆, (4.37) that is, 𝐹((𝑥𝑎)𝑦)+𝜆≥1 and thus ((𝑥𝑎)𝑦)_𝜆𝑞𝐹. If 𝜆≤0.5, then 𝐹((𝑥𝑎)𝑦)≥𝐹(𝑎)≥min{𝜆,1−𝜆,0.5}=𝜆, (4.38) 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, let 𝐹 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 the AG-groupoid as given in Example 4.5. Define a fuzzy subset 𝐹 by 𝐹(𝑎)=0.8,𝐹(𝑐)=0.6,𝐹(𝑑)=0.4,𝐹(𝑒)=0.2,𝐹(𝑏)=0.1. (4.39)

Then 𝑄(𝐹;𝜆)={𝑎,𝑐,𝑑} for all 0.2<𝜆≤0.4. Since 𝑐_0.56∈𝐹 and 𝑒_0.18∈𝐹 but (𝑐𝑒)_{min{0.56,0.18}}=𝑑_0.18𝑞𝐹, hence 𝑄(𝐹;𝜆) is not an interior ideal of 𝑆 for all 𝜆∈(0.2,0.4].

Proposition 4.15. If {𝐹_𝑖}_𝑖∈𝐼 is a family of (∈,∈∨𝑞)-fuzzy bi-ideals of an 𝐴𝐺-groupoid 𝑆, then ⋂_𝑖∈𝐼𝐹_𝑖 is an (∈,∈∨𝑞)-fuzzy bi-ideal of 𝑆.

Proof. Let {𝐹_𝑖}_𝑖∈𝐼 be a family of (∈,∈∨𝑞)-fuzzy bi-ideals of 𝑆. Let 𝑥,𝑦∈𝑆. Then (_𝑖∈𝐼𝐹_𝑖)(𝑥𝑦)=_𝑖∈𝐼𝐹_𝑖(𝑥𝑦)≥_𝑖∈𝐼(𝐹_𝑖(𝑥)𝐹_𝑖(𝑦))=(_𝑖∈𝐼𝐹_𝑖(𝑥)_𝑖∈𝐼𝐹_𝑖(𝑦))=(_𝑖∈𝐼𝐹_𝑖)(𝑥)(_𝑖∈𝐼𝐹_𝑖)(𝑦). (4.40) Let 𝑥,𝑦,𝑎∈𝐺. Then (_𝑖∈𝐼𝐹_𝑖)((𝑥𝑎)𝑦)=_𝑖∈𝐼𝐹_𝑖((𝑥𝑎)𝑦)≥_𝑖∈𝐼(𝐹_𝑖(𝑎))=(_𝑖∈𝐼𝐹_𝑖)(𝑎). (4.41) Thus ⋂_𝑖∈𝐼𝐹_𝑖 is an (∈,∈∨𝑞)-fuzzy interior ideal of 𝑆.

Definition 4.16. Let 𝑆 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 subset 𝐹 of 𝑆. The conditions (𝐵9) and (𝐵10) of Definition 4.16 are equivalent to the following. (𝐵11) (∀𝑥,𝑦∈𝑆)(∀𝜆₁,𝜆₂∈(0,1])(𝑥_𝜆1∈𝐹,𝑦_𝜆2∈𝐹→(𝑥𝑦)_{min{𝜆1,𝜆2}}∈𝐹).(𝐵12)(∀𝑥,𝑦,𝑎∈𝑆)(∀𝜆∈(0,1])(𝑎_𝜆∈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 𝑥_𝜆1∈𝐹, 𝑦_𝜆2∈𝐹. Then 𝐹(𝑥)>𝜆₁ and 𝐹(𝑦)>𝜆₂. Using (B9) 𝐹(𝑥𝑦)>min{𝐹(𝑥),𝐹(𝑦)}>min{𝜆₁,𝜆₂}, (4.42) and so (𝑥𝑦)_{min{𝜆1,𝜆2}}∈𝐹.
(𝐵11)→(𝐵9). Let 𝑥,𝑦∈𝑆. Since 𝑥_𝐹(𝑥)∈𝐹 and 𝑦_𝐹(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 𝑎_𝜆∈𝐹. Then 𝐹(𝑎)>𝜆. By (𝐵10) we have 𝐹((𝑥𝑎)𝑦)>𝐹(𝑎)>𝜆, (4.43) and so ((𝑥𝑎)𝑦)_𝜆∈𝐹.
(𝐵11)→(𝐵10). Let 𝑥,𝑦∈𝑆. Since 𝑎_𝐹(𝑎)∈𝐹, by (𝐵11), we have ((𝑥𝑎)𝑦)_𝐹(𝑎)∈𝐹 and so 𝐹((𝑥𝑎)𝑦)>𝐹(𝑎).

5. (∈,∈∨𝑞)-Fuzzy Interior 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.

Let 𝐹 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 𝑥_𝜆∈⋀𝑞𝐹. Then 𝑥_𝜆∈𝐹 and 𝑥_𝜆𝑞𝐹 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∣𝑥_𝜆∈⋀𝑞𝐹}=∅.

Definition 5.1. A fuzzy subset 𝐹 of 𝑆 is called an ( 𝛼,𝛽)-fuzzy interior ideal of 𝑆, where 𝛼≠∈⋀𝑞, if it satisfies the following conditions.(𝐵13)(∀𝑥,𝑦∈𝑆)(∀𝜆₁,𝜆₂∈(0,1])(𝑥_𝜆1𝛼𝐹,𝑦_𝜆2𝛼𝐹→(𝑥𝑦)_{min{𝜆1,𝜆2}}𝛽𝐹).(𝐵14)(∀𝑥,𝑦,𝑎∈𝑆)(∀𝜆∈(0,1])(𝑎_𝜆𝛼𝐹→((𝑥𝑎)𝑦)_𝜆𝛽𝐹).

Proposition 5.2. Let 𝐹 be a fuzzy subset of 𝑆. If 𝛼=∈ and 𝛽=∈∨𝑞 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 subset 𝐹 of an 𝐴𝐺-groupoid S is an (∈,∈∨𝑞 )-fuzzy interior ideal of 𝑆 if and only if it satisfies conditions (B15) and (B16) of the above proposition.

Using Proposition 5.2, we have the following characterization of (∈,∈∨𝑞)-fuzzy interior ideals of an 𝐴𝐺-groupoid.

Lemma 5.4. Let 𝑆 be an 𝐴𝐺-groupoid and ∅≠𝐼⊆𝑆. Then 𝐼 is an interior ideal of 𝑆 if and only if the characteristic function 𝜒_𝐼 of 𝐼 is an (∈,∈∨𝑞)-fuzzy interior ideal of 𝑆.