INTERVAL VALUED (α, β)-INTUITIONISTIC FUZZY BI-IDEALS OF SEMIGROUPS

Abstract

The concept of quasi-coincidence of interval valued intuitionistic fuzzy point with an interval valued intuitionistic fuzzy set is considered. By using this idea, the notion of interval valued (α, β)-intuitionistic fuzzy bi-ideals, (1,2)ideals in a semigroup introduced and consequently, a generalization of interval valued intuitionistic fuzzy bi-ideals and intuitionistic fuzzy bi-ideals is defined. In this paper, we study the related properties of the interval valued (α, β)-intuitionistic fuzzy bi-ideals, (1,2) ideals and in particular, an interval valued (Є, Є ∨q)-fuzzy bi-ideals and (1,2) ideals in semigroups will be investigated.

1. Introduction

The concept of a fuzzy set was first initiated by Zadeh [1]. Fuzzy set theory has been shown to be a useful tool to describe situations in which the data are 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 theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, group theory, real analysis, measure theory etc. After the introduction of the concept of fuzzy sets by Zadeh, several researches conducted the researches on the generalizations of the notion of fuzzy set with huge applications in computer, logics, automata and many branches of pure and applied mathematics. The notion of i-v fuzzy sets was first introduced by Zadeh [1] as an extension of fuzzy sets in which the values of the membership degrees are intervals of numbers instead of the numbers. Thus, i-v fuzzy sets provide a more adequate description of uncertainty than the traditional fuzzy sets. I-v fuzzy set theory has been shown to be a useful tool to describe situations in which the data are imprecise or vague. Rosenfeld studied fuzzy subgroups of a group [2]. The study of fuzzy semigroups was studied by Kuroki in his classical paper [3] and Kuroki initiated fuzzy ideals, bi-ideals, semi-prime ideals, quasi-ideals of semigroups [4,5,6,7,8,9,10]. A systematic exposition of fuzzy semigroup was given by Mordeson et.al. [11], and they have find theoretical results on fuzzy semigroups and their use in fuzzy finite state machines, fuzzy languages and fuzzy coding. Mordeson and Malik studied monograph in [12] deals with the application of fuzzy approch to the concepts of formal languages and automata. In 2008, Shabir and Khan introduced the concept of i-v fuzzy ideals generated by i-v fuzzy subset in ordered semigroup [43]. Using the notions ”belong to” relation (∈) introduced by Pu and Lia [19]. In [20], Morali proposed the concept of a fuzzy point belonging to a fuzzy subset under natural equivalence on fuzzy subset. Bhakat and Das introduced the concepts 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 [21]. Kazanci and Yamak [22] studied generalized types fuzzy bi-ideals of semigroups and defined bi-ideals of semigroups. Jun and Song [23] studied generalized fuzzy interior ideals of semigroups. In [24], Shabir et. al. characterized regular semigroups by the properties of (α, β)-fuzzy ideals, bi-ideals and quasi-ideals. In [25], Shabir and Yasir characterized regular semigroups by the properties of ideals, generalized bi-ideals and quasi-ideals of a semigroup. M. Shabir et. al. defined some types of (∈, ∈ ∨qk)-fuzzy ideals of semigroups and characterized regular semigroups by these ideals [26]. In [27], M. Shabir and T. Mehmood studied (∈, ∈ ∨qk)-fuzzy h-ideals of hemirings and characterized different classes of hemirings by the properties of (∈, ∈ ∨qk)-fuzzy h-ideals. Recently, M. Aslam et al. [28] initiated the concept of (α,β)-fuzzy Γ-ideals of Γ-LA-semigroups and given some characterization of Γ-LA-semigroups by (α,β)-fuzzy Γ-ideals.

In fuzzy sets theory, there is no means to incorporate the hesitation or uncertainty in the membership degrees. In 1986, Atanassov [29] premised the concept of an intuitionistic fuzzy set (IFS) and more operations defined in [30]. An Atanassov intuitionistic fuzzy set is deliberated as a generalization of fuzzy set [1] and has been found to be useful to deal with vagueness. In the sense of an IFS is characterized by a pair of functions valued in [0, 1]: the membership function and the non-membership function. The evaluation degrees of membership and non-membership are independent. Thus, an Atanassov intuitionistic fuzzy set is more material and concise to describe the essence of fuzziness, and Atanassov intuitionistic fuzzy set theory may be more suitable than fuzzy set theory for dealing with imperfect knowledge in many problems. The concept has been applied to various algebraic structures. Atanassov and Gargov [31] initiated the notion of i-v intuitionistic fuzzy sets which is a generalization of both intuitionistic fuzzy sets and interval valued fuzzy sets. In [?], Akram and Dudek have defined interval valued intuitionistic fuzzy Lie ideals of Lie algebras and some interesting results are abtained. Biswas [32] introduced the notion of intuitionistic fuzzy subgroup of a group by using the notion of intuitionistic fuzzy sets. In [33], Kim and Jun defined intuitionistic fuzzy ideals of semigroups. Kim and Lee [34] studied intuitionistic fuzzy bi-ideals of semigroups. Kim and Jun initiated the concept intuitionistic fuzzy interior ideals of semigroups [35]. Coker and Demirci introduced the notion of intuitionistic fuzzy point [36] of 1995. Jun [37] introduced the notion of (Φ, Ψ)-intuitionistic fuzzy subgroups, where Φ,Ψ are any two of {∈, q, ∈ ∨q, ∈ ∧q} with Φ ≠∈ ∧q, and related properties are investigated. Aslam and Abdullah [38] introduced the concept of (Φ, Ψ)-intuitionistic fuzzy ideals of semigroups and obtained some properties of (Φ, Ψ)-intuitionistic fuzzy ideals. Recently, Abdullah et.al., initiated the concept of (α, β)-intuitionistic fuzzy ideals of hemirings by using the ”belong to” relation (∈) and ”quasi-coincident with” relation (q) between an intuitionistic fuzzy point and an intuitionistics fuzzy set, and they defined prime (semi-prime) (α, β)-intuitionistic fuzzy ideals of hemirings [39].

In this paper, we introduce the concept of interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of semigroup and (α, β)-intuitionistic fuzzy bi-ideal of semigroup where Φ,Ψ are any two of {∈, q, ∈ ∨q, ∈ ∧q} with α ≠ ∈ ∧q, by using belong to relation (∈) and quasi-coincidence with relation (q) between intuitionistic fuzzy point and intuitionistic fuzzy set, and investigated related properties. We also prove that in regular semigroup, every (∈, ∈ ∨q)-intuitionistic fuzzy (1, 2) ideal of semigroup S is an (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of semigroup S.

 

2. Preliminaries

Definition 1 ([1,2]). Let X be a non-empty fixed set. An interval valued intuitionistic fuzzy set (briefly, IVIFS) A is an object having the form

where the functions denote the degree of membership and the degree of non-membership of each element x ∈ X to the set A, respectively, and for all x ∈ S for the sake of simplicity, we use the symbol for the IVIFS

Definition 2 ([4]). Let c be a point in a non-empty set X. If are two interval numbers such that , and at same time both values does not less than . Then, the IFS

is called an interval valued intuitionistic fuzzy point (IVIFP for short) in X, where is the degree of membership (resp, non-membership) of and c ∈ X is the support of be an IVIFP in X. and let be an interval valued IFS in X. Then, is said to belong to A, written . We say that is quasi-coincident with A, written . To say that means that means that does not hold and

Definition 3. An IVIFS in S is called an intuitionistic fuzzy subsemigroup of S if the following conditions hold:

Definition 4. An IVIFS in S is called an intuitionistic fuzzy right ideal of S if it satisfy and λA(xy) ≤ λA(x) for all x, y ∈ S.

Definition 5. An IVIFS in S is called an intuitionistic fuzzy left ideal of S if it satisfy and λA(xy) ≤ λA(y) for all x, y ∈ S.

Theorem 1 ([13]). An IVIFS in a semigroup S is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy left (resp. right) ideal of a semigroup S if and only if the following conditions hold.

 

3. Interval valued (α, β)-Intuitionistic Fuzzy bi-ideals

Definition 6. An IVIFS in a semigroup S is said to be an interval valued (α, β)-intuitionistic fuzzy subsemigroup of a semigroup S if the following condition holds:

Definition 7. An IVIFS in a semigroup S is said to be an interval valued (α, β)-intuitionistic fuzzy left (resp, right) ideal of semigroup S if ∀x, y ∈ S and or , the following hold.

An IVIFS in a semigroup S is said to be an interval valued (α, β)-intuitionistic fuzzy ideal of a semigroup S, if is an interval valued (α, β)-intuitionistic fuzzy left ideal and interval valued (α, β)-intuitionistic fuzzy right ideal of a semigroup S.

Definition 8. An IVIFS in a semigroup S is said to be an interval valued (α, β)-intuitionistic fuzzy bi-ideal of a semigroup S, where α, β are any two of {∈, q, ∈ ∨q, ∈ ∧q} with α ≠ ∈ ∧q, if for all x, y, z ∈ S, or , the following conditions hold:

Definition 9. An IVIFS in semigroup S is said to be an interval valued (α, β)-intuitionistic fuzzy (1, 2) ideal of semigroup S if for all or , the following conditions hold.

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

Proof. Let . Suppose that , then but

So, , which is a contradiction. Now, let for β ∈ {∈, q, ∈ ∨q, ∈ ∧q}, which is a contaradiction. Hence, is a subsemigroup of S. □

Theorem 3. Let be a non-zero interval valued (α, β)-intuitionistic fuzzy subsemigroup of S. Then, the set is a subsemigroup of S.

Proof. The proof follows from Theorem 2. □

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

Proof. Let Let be a non-zero interval valued (α, β)-intuitionistic fuzzy bi-ideal of S. Then, by Theorem 2, is a subsemigroup of S. Now, let x, z ∈ and y ∈ S. Then, . Suppose that , then but

which implies that for β ∈ {∈, q, ∈ ∨q, ∈ ∧q}, which is a contradiction. Now, let for β ∈ {∈, q, ∈ ∨q, ∈ ∧q}, which is a contaradiction. Hence is a bi-ideal of S. □

Theorem 5. Let be a non-zero interval valued (α, β)-intuitionistic fuzzy (1, 2) ideal of S. Then, the set is a (1, 2) ideal of S.

Proof. Straightforward. □

Theorem 6. Let L be a left (resp. right) ideal of S and let be an IVIFS such that

Then, is an interval valued (q, ∈ ∨q)-intuitionistic fuzzy left (resp. right) ideal of S.

Proof. (For α = q), let x, y ∈ S and be such that . So, y ∈ L. Therefore, xy ∈ L. Thus, if and so and . Therefore, does not occur. From the fact that , it follows that the case does not occur. Hence, is an interval valued (q, ∈ ∨q)-intuitionistic fuzzy left ideal of S. □

Theorem 7. Let B be a subsemigroup of S and let be an IVIFS such that

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

Proof. Straightforward. □

Theorem 8. Let B be a bi-ideal of a semigroup S and let be an IVIFS of S such that

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

Proof. (i) (For α = q), let x, y ∈ S and be such that . Then, , and . Thus, x, y ∈ B. Since B is subsemi-group. So, xy ∈ B. Thus, and and . So, and . Since does not occur. From the fact that and it follows that does not occur. Hence, is an interval valued (q, ∈ ∨q)-intuitionistic fuzzy subsemigroup of S. Let x, y, z ∈ S and be such that . Then, , and . Thus, x, z ∈ B. Since B is a bi-ideal. So, xyz ∈ B. Thus, and . and . Since does not occur. From the fact that , it follows that does not occur. Hence, is an interval valued (q, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S. □

Theorem 9. Let B be a (1, 2) ideal of a semigroup S and let be an IVIFS of S such that

Then, is an interval valued (q, ∈ ∨q)-intuitionistic fuzzy (1, 2) ideal of S.

Proof. Proof follow from Theorem 8. □

 

4. Interval Valued Intutionistic Fuzzy Bi-Ideals of type (∈, ∈ ∨q)

Definition 10. An IVIFS in semigroup S is said to be an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of semigroup S if ∀x, y, a ∈ S, or , the following conditions hold.

Definition 11. An IVIFS in a semigroup S is said to be an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy (1, 2) ideal of a semigroup S if for all x, y, z, a ∈ or , the following conditions hold:

Proposition 1. An IVIFS of a semigroup S is an interval valued intuitionistic fuzzy subsemigroup if and only if it satisfies for all x, y ∈ S and or ,

Proof. Let us suppose that be an interval valued intuitionistic fuzzy subsemigroup of S. Let x, y ∈ S, and let . Then, , and . Since by given condition

So, is an interval valued (∈, ∈)-intuitionistic fuzzy subsemigroup of S.

Conversely, suppose that is satisfied the given condition. We show that and . On contrary assume that there exist x, y ∈ S such that and . Let t ∈ D(0, 1] and s ∈ D[0, 1) be such that and

Then, , which contradicts our hypothesis. Hence, and . Thus, interval valued intuitionistic fuzzy subsemigroup of S. □

Proposition 2. An IVIFS of a semigroup S is an interval valued intuitionistic fuzzy bi-ideal of S if and only if it satisfy for all x, y, z ∈ S and or

Proof. Proof follows from Proposition 1. □

Theorem 10. Let be an IVIFS in semigroup S. Then, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of semigroup S if and only if the following condition hold;

Proof. Suppose that is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of semigroup S.

(a) Let x, y ∈ S. We consider the following cases:

Case (1) Assume that

Then, and

Choose such that

Then, , a contradicts.

Case (2) Assume that . Then, but , a contradicts. Therefore, and

(b) Now, let x, y, a ∈ S. We consider the following case’s

(1) Assume that

Choose such that

Then, , which is a contradiction.

Case (2) Assume that and , which is a contradiction. Therefore, and

Conversely, assume that satisfy (a) and (b). Let x, y ∈ S, , be such that . Then, , and . Now we have

Then, we have the following cases

Case (1) If , then, which implies that

Case(2) If , then , which implies that and . Therefore, . Hence,

Let x, y, a ∈ S and such that . Then, , and . Now we have

Then, we have the following case’s

Case (3) If , then , which implies that

Case(4) If , then , which implies that and . Therefore, . Hence, . This completes the proof. □

Remark 1. Every intuitionistic fuzzy bi-ideal is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of semigroup S. But the converse is not true, see example.

Example 1. Let S = {1, 2, 3, 4, 5} be a semigroup defined by the following Cayley table.

Let be an IVIFS in a semigroup S defined by (3) = [0.7, 0.75], (3) = [0.2, 0.25], . Then, by routine calculation is an interval valued (∈,∈ ∨q)-intuitionistic fuzzy bi-ideal of S but not intuitionistic fuzzy bi-ideal. i.e.,

Remark 2. From above Remark and Example, we can say that interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S is a generalization of an intuitionistic fuzzy bi-ideal of S.

Theorem 11. Let be an IVIFS in a semigroup S. Then, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy (1, 2) ideal of a semigroup S if and only if the following conditions hold,

Proof. Straightforward. □

Proposition 3. Every interval valued (∈, ∈ ∨q)-intuitionistic fuzzy left (right) ideal of S is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S.

Remark 3. If is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S, then need not to be an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy left (right) ideal of S.

Example 2. Let S = {1, 2, 3, 4} be a semigroup with the following Cayley table.

(1) Let be an IVIFS defined as; and . Then, clearly is an interval valued (∈,∈ ∨q)-intuitionistic fuzzy bi-ideal of S but not interval valued (∈, ∈ ∨q)-intuitionistic fuzzy right ideal of S because

(1) Let be an IVIFS defined as; and . Then, clearly is an interval valued (∈,∈ ∨q)-intuitionistic fuzzy bi-ideal of S but not interval valued (∈, ∈ ∨q)-intuitionistic fuzzy left ideal of S because

Proposition 4. (1)Every (∈ ∨q, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S is in-terval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal.

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

Proof. Straightforward. □

Example 1 shows that the converse of Proposition 4, is not true in general.

Theorem 12. If {A}i∈∧ is a family of interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideals of S, then ∩i∈∧ Ai is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S, where

Proof. Strightforward. □

Remark 4. If {A}i∈∧ is a family of interval valued (∈,∈ ∨q)-intuitionistic fuzzy bi-ideals of S, then ∪i∈∧ Ai is not an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S, where . See example below.

Example 3. Let S = {1, 2, 3, 4} be a semigroup defined by the following Cayley table.

Let be IVIFS’s of semigroup S defined by and and and

Then, both are an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideals of S. But A∪B is not an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S. i.e.

Hence,

Theorem 13. If {Ai}i∈∧ is a family of interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideals of S such that Ai ⊆ Aj or Aj ⊆ Ai for all i, j ∈ I, then is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S.

Proof. For all x, y ∈ S, we have

It is clear that

Assume that

Then, there exists t such that

Since for all i, j ∈ I, so there exists k ∈ I such that . On other hand for all i ∈ I, a contradiction. Hence,

and

It is clear that

Assume that

Then there exists t such that

Since for all i, j ∈ I, so there exists k ∈ I such that . On the other hand, for all i ∈ I, a contradiction. Hence,

Let x, a, y ∈ S, we obtain

and

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

Definition 12. Let S be a semigroup and be IVIFSs of S. Then, the -product of A and B is defined by:

Remark 5. If S is a semigroup and A, B, C, D are IVIFSs of S such that A ⊆ B and C ⊆ D, then

Proposition 5. Let S be a semigroup and be interval valued (∈,∈ ∨q)-intuitionistic fuzzy bi-ideals of S. Then, is an interval valued (∈,∈ ∨q)-intuitionistic fuzzy bi-ideal of S.

Proof. Straightforward. □

Definition 13. An interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S is called -idemoptent if

Proposition 6. Let S be a semigroup and A is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy subsemigroup of S. Then

Proof. Straightforward. □

Lemma 1. Let S be a semigroup, be interval valued (∈,∈ ∨q)-intuitionistic fuzzy bi-ideals of S. Then,

Theorem 14. Let S be a semigroup, be an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S. Then, for all x ∈ S.

Proof. Let x ∈ S. Then, we have two cases. (1) If x ≠ yz for every y, z ∈ S. (2) If x = yz for some y, z ∈ S.

Case 1 : If x ≠ yz, then clearly

Thus,

Case 2 :If x = yz for some y, z ∈ S, then we have

Since x = yz = y(tr) = ytr and is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S, so we have

Thus,

and

Since x = yz = y(tr) = ytr and is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S, so we have

Thus,

Hence, □

Theorem 15. Let S be a semigroup and be an IVIFS of S. Then, is an interval valued (∈,∈ ∨q)-intuitionistic fuzzy subsemigroup of S if and only if

Theorem 16. Let S be a semigroup and be an IVIFS of S. Then, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of if and only if the following hold:

Proof. Let be an interval valued (∈,∈ ∨q)-intuitionistic fuzzy biideal of S. Then, by Proposition 6 and Theorem 14, we have

Conversely, suppose that the given conditions hold. Now, let x, y ∈ S such that a = xy. Then, we have

and

Now, let x, y, z ∈ S such that a = xyz. Then, we have

and

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

Theorem 17. Let S be a semigroup and be an IVIFS of S. Then, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy left (resp. right, two sided) ideal of S if and only if the following hold:

Proof. Straightforward. □

Theorem 18. Let be interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideals of S. Then, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S.

Proof. Let be interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideals of S and let x ∈ S. Then, we have two cases

(1) If x ≠ yz for any y, z ∈ S. (2) If x = yz for some y, z ∈ S.

Case 1 : If x ≠ yz for any y, z ∈ S, then

and

Thus, in this case.

Case 2 : If x = yz for some y, z ∈ S, then

Since x = yz, y = ab and z = pq. So, x = (ab) (pq) = (abp) q and we have

Since is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S we have

So,

Therefore, . Now,

Since x = yz, y = ab and z = pq. So, x = (ab) (pq) = (abp) q and we have

Since is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S we have

So,

Therefore, . Thus, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy subsemigroup of S.

Now, let x, y, z ∈ S. Then,

Since x = ab and z = pq, so xyz = (ab) y (pq) = (a (by) p) q and we have

Since is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S we have

So,

Thus,

and

Since x = ab and z = pq, so xyz = (ab) y (pq) = (a (by) p) q and we have

Since is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S we have

So,

Thus,

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

For any interval valued intuitionistic fuzzy set , s ∈ D[0, 1), we denote

Obviously, are called ∈-level set, q-level set and ∈ ∨q-level set of , respectively.

Theorem 19. Let S be a semigroup and an IVIFS of S. Then, , is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy left (resp. right) ideal of S if and only if for all , the set is a left (resp. right) ideal of S.

Proof. Let be an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy left ideal of S and for any . Let y ∈ and x ∈ S. Then, . Since

So, is a left ideal of S.

Conversely, Let us suppose that is an IVIFS of S such that is a left ideal of S. Suppose on contrary there exist x, y ∈ S such that

Let us choose t ∈ D(0, 0.5] and s ∈ D[0.5, 1). Then,

Thus, , which is a contradiction. Hence,

This completes the proof. □

Theorem 20. Let S be a semigroup and an IVIFS of S. Then, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S if and only if for all , the set is a b-ideal of S.

Proof. The proof follows from Theorem 19. □

Theorem 21. Let S be a semigroup and an IVIFS of S. Then, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy (1, 2)-ideal of S if and only if for all , the set is a (1, 2)-ideal of S.

Proof. The proof follows from Theorem 19. □

Theorem 22. Let S be a semigroup and an IVIFS of S. Then, is an interval valued (∈,∈ ∨q)-intuitionistic fuzzy subsemigroup ideal of S if and only if for all , or , the set is a subsemigroup of S.

Proof. Let and and and . We can consider four cases:

For the first case, by Theorem 10 (a), implies that

and

and so , i.e., . For the case (ii), assume that . If , then

and if , then . Hence, . Suppose that . If , then

and if , then . Thus, . We have similar result for the case (iii). For final case, if . Hence,

and

and so , then . Thus,

and

which implies that

Conversely, suppose that is an IVIFS in S such that is a subsemigroup of S. Suppose that is not an (∈, ∈ ∨q)-intuitionistic fuzzy subsemigroup of S. Then, there exist x, y ∈ S such that

Let

Then,

This implies that . Hence, or , which is a contradiction. Therefore, we have

Thus, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy subsemigroup of S. □

Theorem 23. Let S be a semigroup and an IVIFS of S. Then, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy left (resp. right) ideal of S if and only if for all t ∈ D(0, 1] and s ∈ D[0, 1), the set is a left (resp. right) ideal of S.

Proof. The proof follows from Theorem 22. □

Theorem 24. Let S be a semigroup and an IVIFS of S. Then, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S if and only if for all or , the set is a subsemigroup of S.

Theorem 25. Let S be a semigroup and an IVIFS of S. Then, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy (1, 2)-ideal of S if and only if for all or , the set is a (1, 2)-ideal of S.

Theorem 26. Every interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of semigroup S is an interval valued (∈,∈ ∨q)-intuitionistic fuzzy (1, 2) ideal of semigroup S.

Proof. Straightforward. □

Theorem 27. If S is a regular semigroup, then every interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S is an interval valued (∈,∈ ∨q)-intuitionistic fuzzy right (left) ideal of a S.

Proof. Since S is regular, so every bi-ideal in S is a right (left) ideal. Let be an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S. Let x, y ∈ S, xSx is bi-ideal of S, then xSx is right ideal of S. Since S is regular. We have xy ∈ (xSx)S ⊆ xSx which implies that xy = xyx for some y ∈ S and since is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S, it follow that

Therefore, is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy rightideal of S. □

Theorem 28. If S is a regular semigroup, then every interval valued (∈, ∈ ∨q)-intuitionistic fuzzy (1, 2) ideal of S is an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy bi-ideal of S.

Proof. Assume that S is regular. Let be an interval valued (∈, ∈ ∨q)-intuitionistic fuzzy (1, 2) ideal of a semigroup S. Let x, y, a ∈ S. Since S is regular, we get xa ∈ (xSx)S ⊆ xSx, which implies that xa = xsx for some s ∈ S. Thus

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

Theorem 29. Let be an interval valued (∈,∈ ∨q)-intuitionistic fuzzy bi-ideal of a semigroup S. If S is completely regular and for all x ∈ S, then A(x) = A(x2) for all x ∈ S.

 

5. Conclusion

It is well known that semigroups are basic structures in many applied branches like automata and formal languages, coding theory, finite state machines and others. Due to these possiblities of applications, semigroups are presently extensively investigated in fuzzy setting. An intuitionistic fuzzy set is more material and concise to describe the essence of fuzziness, and the intuitionistic fuzzy set theory may be more suitable than the fuzzy set theory for dealing with imperfect knowledge in many problems. In study the structure of semigroup, we notice that intuitionistic fuzzy ideals with special properties always play an important role. The intuitionistic fuzzy point of a semigroup S are key tools to describe the algebraic subsystems of S. So, we combined the above concepts and introduced new types of intuitionistic fuzzy bi-ideals and (1,2)-ideals of semigroups which are called interval valued (α, β)-intuitionistic fuzzy bi-ideal and (1,2)-ideal. The results in the paper are generalizations of results about ordinary intuitionistic fuzzy ideals in semigroups. In future, we will focus on the following topics:

(1) Characterizations of regular semigroups by the properties of interval valued (α, β)-intuitionistic fuzzy ideals

(2) We will define (α, β)-intuitionistic fuzzy (interior, prime, generalized bi, prime bi) ideals of a semigroup and characterize different classes of semigroups by the properties of (α, β)-intuitionistic-fuzzy ideals. We will extend to our study to other algebraic structures.