Research Article | Open Access

# Generalized Fuzzy Interior Ideals in Abel Grassmann's Groupoids

**Academic Editor:**Peter Basarab-Horwath

#### 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 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 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 -fuzzy subgroups by using the “belongs to” relation () and “quasicoincident with” relation () between a fuzzy point and a fuzzy subgroup, and they introduced the concept of an (, )-fuzzy subgroup. In particular, ()-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 ()-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 ()-fuzzy subalgebra of a BCK/BCI-algebras. In [18], Luo introduced the concept of a strong neighborhood. According to him, a fuzzy point is said to be strongly belong to a fuzzy subset , denoted by , if and only if . -strong cut set of is given by where is a nonempty set. The idea of -neighborhood in fuzzy topology was introduced by Pu and Liu in [19]. According to them, a fuzzy point is said to be strongly quasicoincident with , denoted by , if and only if .

An *Abel Grassmann's groupoid*, abbreviated as -*groupoid*, is a *groupoid * whose elements satisfy the left invertive law: for all An -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 -groupoid the *medial law*, for all (see [20]) If there exists an element in an -groupoid such that for all then is called an -*groupoid with left identity *. If an -groupoid has the *right identity* then is a *commutative monoid*. If an -groupoid contains left identity then holds for all . Also holds for all

In this paper, we define ()-fuzzy interior ideals of an *AG*-groupoid and give some interesting characterizations of an -groupoids in terms of -fuzzy interior ideals. We also introduce the notion of ()-fuzzy interior ideals of an *AG*-groupoid.

#### 2. Preliminaries

For subsets of an *AG*-groupoid , we denote . A nonempty subset of an -groupoid is called an -*subgroupoid* of if . is called an *interior ideal* of if .

Let be an -groupoid. By a fuzzy subset of we mean a mapping, .

For fuzzy subsets and of , define

We denote by the set of all fuzzy subsets of . One can easily see that () becomes an -groupoid as shown in [21]. The order relation “” on is defined as follows:

For a nonempty family of fuzzy subsets , of an -groupoid , the fuzzy subsets and of are defined as follows:

If is a finite set, say , then clearly

*Definition 2.1 (cf. [21]). *Let be an -groupoid and a fuzzy subset of . Then is called a *fuzzy interior ideal* of , if it satisfies the following conditions. () ( min). () ().

Let be a fuzzy subset of and , then the *characteristic function * of is defined as

Lemma 2.2 (cf. [21]). *Let be an -groupoid and a fuzzy subset of . Then is a fuzzy interior ideal of if and only if is a fuzzy interior ideal of .**Let be an -groupoid and a fuzzy subset of . Then for every the set
**
is called a level set of *

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

Lemma 2.3. *Let be an -groupoid and a fuzzy subset of . Then is a fuzzy interior ideal of if and only if is an interior ideal of for every .*

#### 3. -Fuzzy Interior Ideal

In what follows let denote an -groupoid and let denote any one of

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

is called a *fuzzy point* with support and value and is denoted by . A fuzzy point is said to *belong to* (resp., *quasicoincident with*) a fuzzy set , written as (resp., ) if (resp., ). If or , then . The symbol means does not hold. A fuzzy point is said to be *strongly belong to* (resp., *strongly quasicoincident with*) a fuzzy set , written as (resp., if (resp., ). If or , then . The symbol means that does not hold.

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

Theorem 3.1. *For any fuzzy subset of . The conditions and of Definition 2.1, are equivalent to the following. ** ( (, ).** (.*

*Proof. *()(). Let and be such that and . Then and By we have
and so

()(). Let Since and . Then by , we have and so min

()(). Let and be such that . Then By we have
and so

()(). Let Since , by , we have and so

#### 4. -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 ()-fuzzy bi-ideal of an ordered semigroup and characterized ordered semigroups in terms of ()-fuzzy bi-ideals. In this section we define the notions of ()-fuzzy interior ideals of an Abel Grassmann's groupoid and investigate some of their properties in terms of ()-fuzzy interior ideals.

Let be a fuzzy subset of and for all . Let and be such that q. Then and and so and It follows that and so , which is a contradiction. This means that

*Definition 4.1. *A fuzzy subset of is called an*-fuzzy interior ideal* of where , if it satisfies the following conditions:() ()() ().() ()() ().

Proposition 4.2. *Let be a fuzzy subset of . If and q in Definition 4.1. Then , and , respectively, of Definition 4.1, are equivalent to the following conditions: *()*).*()*.*

*Remark 4.3. *A fuzzy subset of an -groupoid is an ()-fuzzy interior ideal of if and only if it satisfies conditions , and of the above proposition.

Using Proposition 4.2, we have the following characterization of ()-fuzzy interior ideals of an -groupoid.

Lemma 4.4. *Let be an -groupoid and . Then is an interior ideal of if and only if the characteristic function of is an q -fuzzy interior ideal of .*

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

*Example 4.5. *Let be an *AG*-groupoid with the following multiplication:

The is an *AG*-groupoid. The interior ideals of are and . Define a fuzzy subset by
Then
Obviously, is an ()-fuzzy interior ideal of by Lemma 4.4. But we have the following.(i) is not an ()-fuzzy interior ideal of , since but
(ii) is not an (,)-fuzzy interior ideal of , since but (iii) is not a -fuzzy interior ideal of , since and but (iv) is not a ()-fuzzy bi-ideal of , since and but (v) is not an (-fuzzy interior ideal of , since but (vi) is not an (-fuzzy interior ideal of , since and q but (vii) is not an (-fuzzy interior ideal of , since , but (viii) is not (-fuzzy interior ideal of , , but (ix) is not a ()-fuzzy interior ideal of , since and but (x) is not an ()-fuzzy interior ideal of , since and but (xi) is not an ()-fuzzy interior ideal of , and but

*Remark 4.6. *By Remark 4.3, every fuzzy interior ideal of an -groupoid is an -fuzzy interior ideal of . However, the converse is not true, in general.

*Example 4.7. *Consider the -groupoid given in Example 4.5, and define a fuzzy subset by

Clearly is an -fuzzy interior ideal of . But is not an -fuzzy interior ideal of as shown in Example 4.5.

Theorem 4.8. *Every -fuzzy interior ideal of is an ()-fuzzy interior ideal of .*

*Proof. *It is straightforward.

Theorem 4.9. *Every qq-fuzzy interior ideal of is -fuzzy interior ideal of .*

*Proof. *Let be an ()-fuzzy interior ideal of Let and be such that Then which implies that Let and be such that . Then and we have

Theorem 4.10. *Let be a nonzero -fuzzy interior ideal of . Then the set is an interior ideal of .*

*Proof. *Let Then and . Assume that . If , then and but for every , a contradiction. Note that and but for every ,, a contradiction. Hence , that is, . Let and . Then Assume that If then, but for every a contradiction. Note that q but for every , a contradiction. Hence , that is, . Consequently, is an interior ideal of .

Theorem 4.11. *Let be an interior ideal and a fuzzy subset of such that *(1)*,*(2)*.**Then *(a)* is a (q, q -fuzzy interior ideal of ,*(b)* is an ( q)-fuzzy interior ideal of .*

*Proof. *Let and be such that and . Then and we have . If min, then min and hence . If min, then
and so q. Therefore q. Let and be such that . Then and we have (. If , then and hence . If , then
and so . Therefore . Therefore is a (-fuzzy interior ideal of .

Let and be such that and . Then and we have . If min, then min and hence . If min, then
and so . Therefore . Now let and be such that . Then and we have . If , then and hence . If , then
and so . Therefore
and so is an (,)-fuzzy interior ideal of .

From Example 4.5, we see that an ()-fuzzy interior ideal is not a ()-fuzzy interior ideal (Example 4.5, iv).

In the following theorem we give a condition for an ()-fuzzy interior ideal to be an ()-fuzzy interior ideal of .

Theorem 4.12. *Let be an ( )-fuzzy interior ideal of such that for all . Then is an ( )-fuzzy interior ideal of .*

*Proof. *Let and be such that Then and and so min min min and hence Now, let and be such that . Then and we have
consequently, Therefore is an ( )-fuzzy interior ideal of .

For any fuzzy subset of an *AG*-groupoid and , we denote

Obviously,

We call an -level interior ideal of and a q-level interior ideal of .

We have given a characterization of ()-fuzzy interior ideals by using level subsets (see Proposition 4.2). Now we provide another characterization of ()-fuzzy interior ideals by using the set

Theorem 4.13. *Let be an AG-groupoid and a fuzzy subset of . Then is an -fuzzy interior ideal of if and only if is an interior ideal of for all .*

*Proof. *Let be an ()-fuzzy interior ideal of Let for Then and , that is, or , and or . Since is an -fuzzy interior ideal of , we have

We discuss the following cases.*Case 1. *Let and . If , then
and hence If Then
and so . Hence *Case 2. *Let and . If , then
and we discuss the following cases.

If , then
that is, and thus . If , then
and so . Hence .*Case 3. *Let and . If , then
that is, and hence . If , then
and so . Hence .*Case 4. *Let and If , then
that is, and thus . If then
and so . Thus in any case, we have . Therefore . Now, let for . Then , that is, or . Since is an ()-fuzzy interior ideal of , we have
*Case 1. *Let . If , then
and hence If , then
and so . Hence *Case 2. *Let and . If , then
If , then
that is, and thus . If , then
and so . Hence .Thus in any case, we have . Therefore (.

Conversely, let be a fuzzy subset of and let be such that min for some Then , it implies that . Hence or , a contradiction. Hence for all . Now let for some . Choose such that Then It follows that . This implies that or , a contradiction. Hence for all . By Proposition 4.2, it follows that is an ()-fuzzy interior ideal of .

and are interior ideals of for all , but is not an interior ideal of for all 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 by

Then for all Since and but hence is not an interior ideal of for all

Proposition 4.15. *If is a family of ()-fuzzy bi-ideals of an -groupoid , then is an ()-fuzzy bi-ideal of .*

*Proof. *Let be a family of ()-fuzzy bi-ideals of . Let . Then
Let . Then
Thus is an ()-fuzzy interior ideal of .

*Definition 4.16. *Let be an *AG*-groupoid and a fuzzy subset of . Then is called a *strongly fuzzy interior ideal* of , if it satisfies the following conditions.()Every fuzzy interior ideal of an *AG*-groupoid is strongly fuzzy interior ideal of .

Theorem 4.17. *For any fuzzy subset of . The conditions and of Definition 4.16 are equivalent to the following. ** (.**.*

*Proof. *()(). Let be a fuzzy subset of . Let and be such that , . Then and Using (B9)
and so

()(). Let Since and . Then by , we have and so min

()(). Let and be such that . Then By we have
and so

()(). Let Since , by , we have and so

#### 5. -Fuzzy Interior Ideals

In this section we define the notions of )-fuzzy interior ideals of an Abel Grassmann's groupoid and investigate some of their properties in terms of )-fuzzy interior ideals.

Let be a fuzzy subset of and for all . Let and be such that . Then and and so and It follows that and so , which is a contradiction. This means that

*Definition 5.1. *A fuzzy subset of is called an* ( **)-fuzzy interior ideal* of where , if it satisfies the following conditions...

Proposition 5.2. *Let be a fuzzy subset of . If and in Definition 5.1. Then , and , respectively, of Definition 5.1, are equivalent to the following conditions. **.**.*

*Remark 5.3. *A fuzzy subset of an -groupoid is an ( )-fuzzy interior ideal of if and only if it satisfies conditions and of the above proposition.

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

Lemma 5.4. *Let be an -groupoid and . Then is an interior ideal of if and only if the characteristic function of is an -fuzzy interior ideal of .*

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Copyright © 2010 Asghar Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.