LA-SEMIGROUPS CHARACTERIZED BY THE PROPERTIES OF INTERVAL VALUED (α, β)-FUZZY IDEALS

Abstract

The concept of interval-valued (${\alpha},{\beta}$)-fuzzy ideals, interval-valued (${\alpha},{\beta}$)-fuzzy generalized bi-ideals are introduced in LA-semigroups, using the ideas of belonging and quasi-coincidence of an interval-valued fuzzy point with an interval-valued fuzzy set and some related properties are investigated. We define the lower and upper parts of interval-valued fuzzy subsets of an LA-semigroup. Regular LA-semigroups are characterized by the properties of the lower part of interval-valued (${\in},{\in}{\vee}q$)-fuzzy left ideals, interval-valued (${\in},{\in}{\vee}q$)-fuzzy quasi-ideals and interval-valued (${\in},{\in}{\vee}q$)-fuzzy generalized bi-ideals. Main Facts.

1. Introduction

The concept of fuzzy sets was first introduced by Zadeh [16] and then the fuzzy sets have been used in the reconsideration of classical mathematics. Fuzzy set theory has been shown to be a useful tool to describe situations in which data is imprecise or vague. Fuzzy sets handle such situations by attributing a degree to which a certain object belongs to a set. The fuzzy algebraic structures play a prominent role in mathematics with wide applications in many other branches such as theoratical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, group theory, real analysis, measure theory etc. The notion of fuzzy subgroups was defined by Rosen ed [12]. A systematic exposition of fuzzy semigroup was given by Morde-son et. al. [7], and they have find theoratical results on fuzzy semigroups and their use in finite state machine, fuzzy languages and fuzzy coding. Using the notions "belong to" relation (∈) introduced by Pu and Liu [11], in [8] Morali proposed the concept of a fuzzy point belonging to a fuzzy subset under nat-ural equivalence on fuzzy subsets. Bhakat and Das introduced the concept of (α, β)-fuzzy subgroups by using the belong to relation (∈) and quasi-coincident with relation (q) between a fuzzy point and a fuzzy subgroup, and defined an (∈, ∈ ∨q)-fuzzy subgroup of a group [1]. Kazanci and Yamak [4] studied gener-alized types fuzzy bi-ideals of semigroups and defined -fuzzy bi-ideals of semigroups. In [14] Shabir et. al. characterized regular semigroups by the properties of (α, β)-fuzzy ideals, bi-ideals and quasi-ideals. In [15], Shabir and Yasir characterized regular semigroups by the properties of -fuzzy ideals, generalized bi-ideals and quasi-ideals of a semigroup.

Interval-valued fuzzy subsets were proposed about thirty years ago as a natu-ral extension of fuzzy sets by Zadeh [17]. Interval-valued fuzzy subsets have many applications in several areas such as the method of approximate inference. In [10] Al Narayanan and T. Manikantan introduced the notions of interval-valued fuzzy ideals generated by an interval-valued fuzzy subset in semigroups.

In this paper we introduced the concepts of interval-valued (∈, ∈ ∨q)-fuzzy ideals, interval-valued (∈,∈ ∨q)-fuzzy bi-ideals and interval-value (∈,∈ ∨ ∧q)-fuzzy quasi-ideals of an LA-semigroup, where α, β are any one of {∈, q, ∈ ∨q, ∈ ∧q} with α ≠ ∈ ∧q, by using belong to relation (∈) and quasi-coincidence with relation (q) between interval-valued fuzzy set and an interval-valued fuzzy point, and investigated related properties.

 

2. Preliminaries

We first recall some basic concepts. A groupoid (S, ∗) is called a left almost semigroup, abbreviated as an LA-semigroup, if it satisfies left invertive law:

A non-empty subset A of S is called a sub LA-semigroup of S if AA ⊆ A and is called left (resp. right) ideal of S if SA ⊆ A (AS ⊆ A). By a two-sided ideal or simply an ideal we mean a non-empty subset of S which is both a left and a right ideal of S. A non-empty subset B of S is called a generalized bi-ideal of S if (BS)B ⊆ B. A sub LA-semigroup B of S is called a bi-ideal of S if (BS)B ⊆ B. A non-empty subset Q of S is called a quasi-ideal of S if QS∩SQ ⊆ Q. Obviously every one-sided ideal of an LA-semigroup S is a quasi-ideal, every quasi-ideal is a bi-ideal and every bi-ideal is a generalized bi-ideal. An LA-semigroup S is called regular if for each element a of S, there exists an element x in S such that a = (ax)a. It is well known that for a regular LA-semigroup the concepts of generalized bi-ideal, bi-ideal and quasi-ideal coincide.

We now review some concepts of fuzzy subsets. A fuzzy subset λ of an LA-semigroup S is a mapping λ : S → [0, 1]. Throughout this paper S denotes an LA-semigroup.

Definition 1. A fuzzy subset λ of S is called a fuzzy sub LA-semigroup of S if for all x, y ∈ S

Definition 2. A fuzzy subset λ of S is called a fuzzy left (resp. right) ideal of S if for all x, y ∈ S

A fuzzy subset λ of S is called a fuzzy two-sided ideal or simply fuzzy ideal of S if it is both a fuzzy left and a fuzzy right ideal of S .

Definition 3. A fuzzy subsemigroup λ of S is called a fuzzy bi-ideal of S if for all x, y ∈ S

We will describe some results of an interval number. By an interval number on [0, 1], say ã is a closed subinterval of [0, 1], that is, ã = [a−, a+], where 0 ≤ a− ≤ a+ ≤ 1. Let D[0, 1] denote the family of all closed subintervals of Now we define ≤,=, ∧, ∨ in case of two elements in D[0, 1].

Consider two elements ã = [a−, a+], and in D[0, 1]. Then,

if and only if a− ≤ b− and a+ ≤ b+.

if and only if a− = b− and a+ = b+.

= [min{a−, b−}, min{a+, b+}].

= [max{a−, b−}, max{a+, b+}].

Let X be a set. A mapping : X → D[0, 1] is called an interval-valued fuzzy subset ( briey, an i-v fuzzy subset) of X, where for all x ∈ X, where λ−and λ+ are fuzzy subsets in X such that λ−(x) ≤ λ+(x) for all x ∈ X.

Let be two interval-valued fuzzy subsets of X. Define the relation ⊆ between λ and μ as follows:

if and only if for all x ∈ X, that is, λ−(x) ≤ μ−(x) and λ+(x) ≤ μ+(x). An interval-valued fuzzy subset in a universe X of the form

for all y ∈ X, is said to be an interval-valued fuzzy point with support x and interval value and is denoted by

J. Zhan et al [19] gave meaning to the symbol where α ∈ {∈, q, ∈ ∨q, ∈ ∧q}. An interval-valued fuzzy point is said to belongs to (resp. quasi-coincident with) an interval-valued fuzzy set written if and in this case means that To say that means that does not hold.

Let be two interval-valued fuzzy subsets of an LA-semigroup S. The product is defined by

 

3. Main results

Definition 4. An interval-valued fuzzy subset of an LA-semigroup S is called an interval-valued fuzzy sub LA-semigroup of S if for all x, y ∈ S

Definition 5. An interval-valued fuzzy subset of an LA-semigroup S is called an interval-valued fuzzy left (resp. right ) ideal of S if for all x, y ∈ S

Definition 6. An interval-valued fuzzy subset of an LA-semigroup S is called an interval-valued fuzzy generalized bi-ideal of S if for all x, y, z ∈ S

Definition 7. An interval-valued fuzzy sub LA-semigroup of S is called an interval-valued fuzzy bi-ideal of S if for all x, y, z ∈ S

Definition 8. An interval-valued fuzzy subset of an LA-semigroup S is called an interval-valued fuzzy quasi-ideal of S if

where : S → [1, 1].

Theorem 1 ([18]). Let S be an LA-semigroup with left identity e such that (xe)S = xS for all x ∈ S. Then, the following are equivalent

 

4. INTERVAL-VALUED (α, β)-FUZZY IDEALS OF LA-SEMIGROUPS

Let S be an LA-semigroup and α, β denote any one of ∈, q, ∈ ∨q, or ∈ ∧q unless otherwise specified.

Definition 9. An interval-valued fuzzy subsete an LA-semigroup S is called an interval-valued (α, β)-fuzzy sub LA-semigroup of S, where α ≠ ∈ ∧q, if implies that

Let be an interval-valued fuzzy subset of S such that, for all x ∈ S. Let x ∈ S and ∈ D[0, 1] such that, that is, It follows that This means that Therefore, the case α =∈ ∧q in above definition is omitted.

Definition 10. An interval-valued fuzzy subset of an LA-semigroup S is called an interval-valued (α, β)-fuzzy left (resp. right ) ideal of S, where α ≠ ∈ ∧q, if it satisfies, and x ∈ S implies that for all x, y ∈ S.

An interval-valued fuzzy subset of an LA-semigroup S is called an interval-valued (α, β)-fuzzy ideal of S if it is both an interval-valued (α, β)-fuzzy left ideal and an interval-valued (α, β)-fuzzy right ideal of S.

Definition 11. An interval-valued fuzzy subset of an LA-semigroup S is called an interval-valued (α, β)-fuzzy generalized bi-ideal of S, where α ≠ ∈ ∧q, if it satisfies

for all x, y, z ∈ S and for alle where implies that

Definition 12. An interval-valued fuzzy subset of an LA-semigroup S is called an interval-valued (α, β)-fuzzy bi-ideal of S, where α ≠ ∈ ∧q, if it satisfies, the following two conditions.

(i) For all x, y ∈ S and for all where and implies that

(ii) For all x, y, z ∈ S and for all where and implies that

Lemma 1. An interval-valued fuzzy subset of an LA-semigroup S is an interval-valued fuzzy sub LA-semigroup of S if and only if it satisfies,

For all x, y ∈ S and for all such that implies that

Proof. Suppose that is an interval-valued fuzzy sub LA-semigroup of an LA-semigroup S. Let x, y ∈ S and where such that, Then Since is an interval-valued fuzzy sub LA-semigroup of S. So Hence,

Conversely, assume that satisfies the given condition. We show that On contrary, assume that there exist x, y ∈ S such that Let such that Then This contradicts our hypothesis. Thus,

Lemma 2. An interval-valued fuzzy subset of an LA-semigroup S is an interval-valued fuzzy left (resp. right) ideal of S if and only if it satisfies

for all x, y ∈ S and for all where such that implies that

Proof. Suppose that is an interval-valued fuzzy left ideal of an LA-semigroup S. If Since is an interval-valued fuzzy left ideal of S, so Hence,

Conversely, suppose that satisfies the given condition. We show that On contrary, assume that there exist x, y ∈ S such that Let be such that Then Which contradicts our hypothesis. Hence,

Remark 1. The above lemma shows that every fuzzy left (resp. right) ideal of S is an (∈, ∈)-fuzzy left (resp. right) ideal of S.

Lemma 3. An interval-valued fuzzy subset of an LA-semigroup S is an interval-valued fuzzy generalized bi-ideal of S if and only if it satisfies,

For all x, y, z ∈ S and for all where implies that

Proof. Suppose that is an interval-valued fuzzy generalized bi-ideal of S. Let x, y, z ∈ S and where such that Then, Since is an interval-valued fuzzy gener-alized bi-ideal of S. So Hence,

Conversely, suppose that satisfies the given condition. We show that Suppose contrary that Let be such that Then, Which contradicts our supposition. Hence,

Lemma 4. An interval-valued fuzzy subset of an LA-semigroup S is an interval-valued fuzzy bi-ideal of S if and only if it satisfy,

(1) for all x, y ∈ S and where implies that

(2) for all x, y, z ∈ S and for all where and implies that

Proof. Follows from Lemma 1 and Lemma 3.

Theorem 2. Let be a non-zero interval-valued (α, β)-fuzzy sub LA-semigroup of S. Then, the set is a sub LA-semigroup of S.

Proof. Let Thens, Let If α ∈ {∈, ∈ ∨q}, then but and So, for every β ∈ {∈, q, ∈ ∧q,∈ ∨q}, a contradiction. Hence, that is, Also, for every β ∈ {∈, q, ∈ ∨q, ∈ ∧q}. Hence, that is, Thus, is a sub LA-semigroup of S.

Theorem 3. Let be a non-zero interval-valued (α, β)-fuzzy left (resp. right) ideal of S. Then, the set is a left (resp. right) ideal of S.

Proof. Similar to the proof of Theorem 2.

Theorem 4. Let be a non-zero interval-valued (α, β)-fuzzy generalized bi-ideal of S. Then the set is a generalized bi-ideal of S.

Theorem 5. Let be a non-zero interval-valued (α, β)-fuzzy bi-ideal of S. Then the set is a bi-ideal of S.

Proof. Proof follows from Theorem 1 and Theorem 4.

 

5. INTERVAL-VALUED (∈, ∈ ∨q)-FUZZY IDEALS

Theorem 6. Let A be a sub LA-semigroup of an LA-semigroup S and let be an interval-valued fuzzy subset in S such that

Then,

(1) is an interval-valued (q, ∈ ∨q)-fuzzy sub LA-semigroup of S.

(2) is an interval-valued (∈, ∈ ∨q)-fuzzy sub LA-semigroup of S.

Proof. (1) Let x, y ∈ S and where such that, Then, So, x ∈ A. Therefore, xy ∈ A. Thus, if Then, If min So, Therefore, Then is an interval-valued (q, ∈ ∨q)-fuzzy sub LA-semigroup of S.

(2) Let x, y ∈ S and where such that, Then, So, x ∈ A and y ∈ A. Therefore, xy ∈ A. If So, If So,

Therefore, Hence, is an interval-valued (∈,∈ ∨q)-fuzzy sub LA-semigroup of S.

Theorem 7. Let L be a left (resp. right) ideal of S and let be an interval-valued fuzzy subset in S such that

Then,

Proof. Similar to the proof of Theorem 6.

Theorem 8. Let B be a generalized bi-ideal of S and let be an interval-valued fuzzy subset in S such that

Then,

Proof. Similar to the proof of Theorem 6.

Theorem 9. Let B be a bi-ideal of S and let be an interval-valued fuzzy subset in S such that

Then,

(1) is an interval-valued (q, ∈ ∨q)-fuzzy bi-ideal of S.

(2) is an interval-valued (∈, ∈ ∨q)-fuzzy bi-ideal of S.

Proof. Follows from Theorem 6 and Theorem 8.

Lemma 5. Let be an interval-valued fuzzy subset of an LA-semigroup S. Then is an interval-valued (∈, ∈ ∨q)-fuzzy left (right) ideal of S if and only if

Proof. Let be an interval-valued (∈, ∈ ∨q)-fuzzy left (right) ideal of S. On contrary, suppose that Choose, such that, Then, Which is contradiction. Hence, Conversely, assume that Let Now If So, So, So, Therefore,

Corollary 1. Let be an interval-valued fuzzy subset of an LA-semigroup S. Then, is an interval-valued (∈, ∈ ∨q)-fuzzy two-sided ideal of S if and only if

Lemma 6. Intersection of interval-valued (∈,∈ ∨q)-fuzzy left ideals of an LA-semigroup S is an (∈, ∈ ∨q)-fuzzy left ideal of S.

Proof. Let be a family of interval-valued (∈,∈ ∨q)-fuzzy left ideals of S. Let x, y ∈ S. Then,

Since each is an interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S. So, Thus,

Hence, is an interval-valued (∈, ∈ ∨q) fuzzy left ideal of S.

Similarly, we can prove that intersection of interval-valued (∈, ∈ ∨q)-fuzzy right ideals of an LA-semigroup S is an interval-valued (∈,∈ ∨q)-fuzzy right ideal of S. Thus, the intersection of interval-valued (∈,∈ ∨q)-fuzzy two-sided ideals of an LA-semigroup S is an interval-valued (∈, ∈ ∨q)-fuzzy two-sided ideal of S.

Now, we show that If and are interval-valued (∈, ∈ ∨q)-fuzzy ideals of an LA-semigroup S, then

Example 1. Consider an LA-semigroup S = {1, 2, 3, 4},

Let be interval-valued fuzzy subsets of S such that (1) = [0.8, 0.9], (2) = [0.7, 0.75], (3) = [0.6, 0.65], (4) = [0.5, 0.55], (1) = [0.2, 0.3], (2) = [0.6, 0.65], (3) = [0.5, 0.55], (4) = [0.7, 0.75], Then, are interval-valued (∈, ∈ ∨q)-fuzzy ideals of S. Now,

Hence, in general.

Theorem 10. An interval-valued fuzzy subset of an LA-semigroup S is an interval-valued (∈, ∈ ∨q)-fuzzy sub LA-semigroup of S if and only if for all x,y ∈ S.

Proof. Suppose that is an interval-valued (∈,∈ ∨q)-fuzzy sub LA-semigroup of S. On contrary, suppose that there exist x, y ∈ S such that, Choose, where such that, Then, which is contradiction. Thus, for all x, y ∈ S. Conversely, assume that for all x, y ∈ S. Let Then, So, Now if Then, If So, Therefore, Hence, is an interval-valued (∈, ∈ ∨q)-fuzzy sub LA-semigroup of S.

Theorem 11. An interval-valued fuzzy subset of an LA-semigroup S is an interval-valued (∈,∈ ∨q)-fuzzy generalized bi-ideal of S if and only if for all x, y ∈ S.

Proof. Straightforward.

Theorem 12. An interval-valued fuzzy subset of an LA-semigroup S is an interval-valued (∈,∈ ∨q)-fuzzy bi-ideal of S if and only if it satisfy the following conditions

Proof. Follows from Theorem 10 and Theorem 11.

Definition 13. An LA-semigroup S is called a left regular if for each element a of S, there exists an element x in S such that a = (aa)x.

Lemma 7. Every interval-valued (∈, ∈ ∨q)-fuzzy generalized bi-ideal of a left regular LA-semigroup S, with left identity e, is an interval-valued (∈, ∈ ∨q)-fuzzy bi-ideal of S.

Proof. Let be any interval-valued (∈, ∈ ∨q)-fuzzy generalized bi-ideal of a left regular LA-semigroup S, with left identity e. Let a, b ∈ S. Then, there exists an element x ∈ S such that a = (aa)x. Thus, we have

Thus, we have This shows that is an interval-valued (∈, ∈ ∨q)-fuzzy sub LA-semigroup of S and so is an interval-valued (∈, ∈ ∨q)-fuzzy bi-ideal of S.

Definition 14. An interval-valued fuzzy subset of an LA-semigroup S is called an interval-valued (∈,∈ ∨q)-fuzzy quasi-ideal of S, if it satisfies where is an interval-valued fuzzy subset of S mapping every element of S on

Theorem 13. Let be an interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal of an LA-semigroup S. Then the set is a quasi-ideal of S.

Proof. In order to show that is a quasi-ideal of S, we have to show that This implies that So, a = sx and a = yr for some s, r ∈ S and and Now,

Since

Similarly,

Thus,

Thus, Hence, is a quasi-ideal of S.

Remark 2. Every interval-valued fuzzy quasi-ideal of S is an interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal of S.

Definition 15. Let S be a non-empty LA-semigroup. Let A be any subset of S. Then, an interval-valued characteristic function of A, is the function of S into D[0, 1] defined by

Lemma 8. A non-empty subset Q of an LA-semigroup S is a quasi-ideal of S if and only if interval-valued characteristic function e is an interval-valued(∈, ∈ ∨q)-fuzzy quasi-ideal of S.

Proof. Suppose that Q is a quasi-ideal of S. Let be an interval-valued characteristic function of Q. Let x ∈ S. If x ∉ Q, then x ∉ SQ. If x ∉ SQ, then Hence, is an interval-valued (∈,∈ ∨q)-fuzzy quasi-ideal of S.

Conversely, assume that is an interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal of S. Let a ∈ QS ∩ SQ. Then, there exist b, c ∈ S and x, y ∈ Q such that, a = xb and a = cy. Then,

So, Similarly,

So, Hence,

Thus, which implies that a ∈ Q. Hence, SQ∩ QS ⊆ Q, that is, Q is a quasi-ideal of S.

Lemma 9. An interval-valued characteristic function e is an interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S if and only if L is a left ideal of S.

Proof. Suppose that L is a left ideal of S and x, y ∈ S. If y ∈ L, then xy ∈ L. So,

If y ∉ L, then So,

Thus, is an interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S.

Conversely, assume that is an interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S. Let y ∈ L and x ∈ S. As

This implies that that is, xy ∈ L. So, L is a left ideal of S.

Similarly, an interval-valued characteristic function e is an interval-valued (∈, ∈ ∨q)-fuzzy right ideal of S if and only if R is a right ideal of S . Hence, it follows that interval-valued characteristic function e is an interval-valued (∈, ∈ ∨q)-fuzzy two-sided ideal of S if and only if I is two-sided ideal of S .

Theorem 14. Every (∈, ∈ ∨q)-fuzzy left ideal of S is an (∈,∈ ∨q)-fuzzy quasi-ideal of S.

Proof. Let x ∈ S. Then

This implies that

Thus, Hence, Thus, is an interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal of S. Similarly, we have this result for interval-valued (∈, ∈ ∨q)-fuzzy right ideals.

Similarly, we can show that every interval-valued (∈, ∈ ∨q)-fuzzy right ideal of S is an interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal of S.

Lemma 10. Let S be an LA-semigroup with left identity e, such that, (xe)S = xS. Then every interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal of S is an interval-valued (∈, ∈ ∨q)-fuzzy bi-ideal of S.

Proof. Suppose that is an interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal of an LA-semigroup S. Let x, y ∈ S. Then

So,

Also, if x, y, z ∈ S then

Now,

So,

Also,

Since (xy)z = (xy)(ez) = (xe)(yz) ∈ xe)S = xS. So, (xy)z = xr for some r ∈ S. Then,

Thus,

Hence, is an interval-valued (∈, ∈ ∨q)-fuzzy bi-ideal of S.

 

6. LOWER AND UPPER PARTS OF INTERVAL VALUED (∈, ∈ ∨q)-FUZZY IDEALS

Definition 16. Let λ be an interval-valued fuzzy subset of an LA-semigroup S. We define upper and lower parts as follows.

Lemma 11. Let be interval-valued fuzzy subsets of an LA-semigroup S. Then, the following holds.

Proof. (i) For all a ∈ S

Thus,

Thus,

(iii) If α is not expressible as α = bc for some b, c ∈ S, then Thus,

Since α is not expressible as α = bc, so Thus, in this case

If α is expressible as α = xy for some x, y ∈ S, then

Thus,

Lemma 12. Let be an interval-valued fuzzy subsets of S. Then, the following hold.

If every element x of S is expressible as x = bc, then

Proof. (i) For all a ∈ S

Thus,

Thus,

(iii) If α is not expressible as α = bc for some b, c ∈ S, then Thus,

But If α is expressible as α = xy for some x, y ∈ S, then

Thus,

Definition 17. Let A be a non-empty subset of an LA-semigroup S. Then the lower and upper parts of an interval-valued characteristic function is

Lemma 13. Let A and B be non-empty subsets of an LA-semigroup S. Then the following properties hold

Proof. The proof is obvious.

Lemma 14. The lower part of an interval-valued characteristic function is an interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S if and only if L is a left ideal of S.

Proof. The proof is obvious.

Similarly, we can prove that the lower part of an interval-valued characteristic function is an interval-valued (∈,∈ ∨q)-fuzzy right ideal of S if and only if R is a right ideal of S. Thus, lower part of an interval-valued characteristic function is an interval-valued (∈,∈ ∨q)-fuzzy two-sided ideal of S if and only if I is a two-sided ideal of S.

Lemma 15. Let Q be a non-empty subset of an LA-semigroup S. Then, Q is a quasi-ideal of S if and only if lower part of an interval-valued characteristic function is an interval-valued (∈,∈ ∨q)-fuzzy quasi-ideal of S.

Proof. Suppose that Q is a quasi-ideal of S. Let be lower part of an interval-valued characteristic function of Q. Let x ∈ S. If x ≠ Q then x ≠ SQ or x ≠ QS. If x ≠ SQ, then and so,

If x ∈ Q, then

Hence, is an interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal of S. Conversely, assume that is an interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal of S. Let α ∈ SQ∩QS then there exist b, c ∈ S and x, y ∈ Q such that α = xb and α = cy. Then,

So,

Similarly,

So,

Hence,

Thus, This implies that α ∈ Q. Hence, QS ∩ SQ ⊆ Q, that is, a quasi-ideal of S.

We have shown in Lemma 1 that every fuzzy left (right) ideal of an LA-semigroup S is an (∈, ∈)-fuzzy left (right) ideal of S. Obviously, every (∈, ∈)-fuzzy left (right) ideal of S is an interval-valued (∈, ∈ ∨q)-fuzzy left (right) ideal of S. But interval-valued (∈, ∈ ∨q)-fuzzy left (right) ideal of S need not to be interval-valued fuzzy left (right) ideal of S.

Example 2. Consider the LA-semigroup given in Example 1. Then, the interval-valued fuzzy subset of S defined by is an (∈, ∈∨q)-fuzzy left ideal of S but is not an interval-valued fuzzy left ideal of S, because

Theorem 15. For an LA-semigroup S the following conditions are equivalent.

(1) S is regular.

(2) for every interval-valued (∈, ∈ ∨q)-fuzzy right ideal and every interval-valued (∈,∈ ∨q)-fuzzy left ideal of S.

Proof. First we assume that (1) holds. Let be an interval-valued (∈, ∈ ∨q)-fuzzy right ideal and an interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S. Let, α ∈ S then

So,

Since S is regular, so there exists an element x ∈ S such that, α = (αx)α. So,

So,

Thus,

Conversely, assume that (2) holds. Let R and L be right ideal and left ideal of S respectively. In order to see that R ∩ L = RL holds. Let α be any element of R∩L. Then, by Lemma 14, the lower part of an interval-valued characteristic functions of R and L are interval-valued (∈, ∈ ∨q)-fuzzy right ideal and interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S, respectively. Thus, we have

Thus, R ∩ L = RL. Hence, by Theorem 1, S is regular and so (2)⇒(1).

Theorem 16. Let S be an LA-semigroup with left identity e such that (xe)S = xS for all x ∈ S. Then, the following conditions are equivalent.

(1) S is regular.

(2) for every interval-valued (∈, ∈ ∨q)- fuzzy right ideal every interval-valued (∈, ∈ ∨q)-fuzzy generalized bi-ideal and every interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S.

(3) for every interval-valued (∈, ∈ ∨q)-fuzzy right ideal every interval-valued (∈, ∈ ∨q)-fuzzy bi-ideal and every interval-valued (∈,∈ ∨q)-fuzzy left ideal of S.

(4) for every interval-valued (∈, ∈ ∨q)-fuzzy right ideal every interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal and every interval-valued (∈,∈ ∨q)-fuzzy left ideal of S.

Proof. (1)⇒(2)

Let and be any interval-valued (∈,∈ ∨q)-fuzzy right ideal, interval-valued (∈, ∈ ∨q)-fuzzy generalized bi-ideal and for any interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S, respectively. Let α be any element of S. Since S is regular, so there exists an element x ∈ S such that α = (αx)α. Hence, we have

Now, α = (αx)α = (αx)(eα) = (α) (xα) = α(xaα) because (xe)S = xS for all x ∈ S.

Thus,

Hence, (1)⇒ (2).

(2) ⇒(3) is straight forward because every interval-valued (∈, ∈ ∨q)-fuzzy bi-ideal is an interval-valued (∈,∈ ∨q)-fuzzy generalized bi-ideal of S.

(3)⇒(4) is also straight forward because every interval-valued (∈,∈ ∨q)-fuzzy quasi-ideal is an interval-valued (∈, ∈ ∨q)-fuzzy bi-ideal of S.

(4)⇒(1) Let and be any interval-valued (∈, ∈ ∨q)-fuzzy right ideal and any interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S respectively. Since is an (∈, ∈ ∨q)-fuzzy quasi-ideal of S, by the assumption, we have

Thus, it follows that for every interval-valued (∈, ∈ ∨q)-fuzzy right ideal and every interval-valued (∈,∈ ∨q)-fuzzy left ideal of S. But always. So, Hence, it follows from Theorem 15 that S is regular.

Theorem 17. Let S be an LA-semigroup with left identity e such that, (xe)S = xS, for all x ∈ S. Then, the following conditions are equivalent.

(1) S is regular.

(2) for every interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal and every interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S.

(3) for every interval-valued (∈, ∈ ∨q)-fuzzy bi-ideal and every interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S.

(4) for every interval-valued (∈, ∈ ∨q)-fuzzy generalized bi-ideal and every interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S.

Proof. (1)⇒(4)

Let be any interval-valued (∈, ∈ ∨q)-fuzzy generalized bi-ideal and any interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S respectively. Let a be any element of S. Then, there exist an element x ∈ S such that α = (αx)α. Thus, we have

Now, α = (αx)α = (αx)(eα) = (αe)(xα) = α(xα) because (xe)S = xS for all x ∈ S. So,

Hence,

(4)⇒(3) is obvious because every interval-valued (∈,∈ ∨q)-fuzzy bi-ideal is an interval-valued (∈,∈ ∨q)-fuzzy generalized bi-ideal of S.

(3)⇒(2) is obvious because every interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideal is an interval-valued (∈, ∈ ∨q)-fuzzy bi-ideal of S.

(2)⇒(1)

Let be an interval-valued (∈,∈ ∨q)-fuzzy right ideal and be an interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S. Since every interval-valued (∈, ∈ ∨q)-fuzzy right ideal of S is an interval-valued (∈,∈ ∨q)-fuzzy quasi-ideal of S. So, Now,

So, Hence, for every interval-valued (∈, ∈ ∨q)-fuzzy right ideal of S and every interval-valued (∈, ∈ ∨q)-fuzzy left ideal of S. Thus, by Theorem 15, S is regular.