After the concept of fuzzy sets of Zadeh [1] , Lee [2] proposed an extension of fuzzy sets namely Bipolar Valued Fuzzy Sets (BVFS). Their range of membership degree has been extended from the interval [0, 1] to [−1, 1] and in [3] . He made a comparison with other fuzzy settings. These Bipolar valued fuzzy sets possess degrees of membership that denote the degree of satisfaction to the property corresponding to a fuzzy set and its counter- property in a bipolar valued fuzzy set. The membership degree 0 refers that the elements are irrelevant to the corresponding property. Further, the membership degrees (0, 1] show that the elements somewhat satisfy the property, and the membership degrees [−1, 0) denote that the elements somewhat satisfy the implicit counter property.

There are two kinds of representations in the definition of bipolar valued fuzzy sets. They are canonical representation and reduced representation. In this work, the canonical representation of bipolar valued fuzzy sets is utilized.

In 2011, Bipolar valued fuzzy K-subalgebras are discussed by Farhat Nisar [4] . The authors of [5] studied the concepts of Intuitionistic L-fuzzy p-ideals of BF-algebras and their related results.

Inspired by the concepts recently, the concept of Filters of BCH-Algebras Based on Bipolar-Valued Fuzzy Sets [6] has been discussed. In this paper, these concepts are intended to α-ideal of BF-algebras and bipolar valued fuzzy α-ideal of a BF-algebra is proposed. The nature of the homomorphic images of bipolar valued fuzzy α-ideal of a BF-algebra is also analyzed.

The paper is organized as follows: Section 2 provides the preliminaries. In Section 3, Bipolar valued fuzzy α-ideal is discussed and in Section 4, homomorphism on Bipolar valued fuzzy α-ideal is studied. Section 5 gives the conclusion.

In this section, some basic definitions and results that are required in the sequel are recalled. The notations and are used.

In this section, Bipolar valued fuzzy α-ideal of a BF-algebra is defined. It is also proved that any Bipolar valued fuzzy α-ideal in X is a Bipolar valued fuzzy BF-ideal and the sufficient condition is derived for the converse.

Definition 3.1. A BVFS in X is called a bipolar valued fuzzy subalgebra of X, if it satisfies:

Definition 3.2. A BVFS in X is called a bipolar valued fuzzy ideal (BVF-ideal) of X, if it satisfies:

Definition 3.3. A BVFS B in a BF-algebra X, is to be a Bipolar Valued Fuzzy Closed BF-ideal (BVFC-BF- ideal) of X, if

Ÿ

Ÿ

Ÿ

Ÿ

Definition 3.4. A BVFS A in a BF-algebra X is called to be a Bipolar Valued Fuzzy α-ideal (BVF-α-ideal) of X, if

Ÿ and

Ÿ

Ÿ

Example 3.5. The BF-algebra X = {0, 1, 2, 3} is considered with the Cayley table as given below.

is the BVFS of X defined as

and

is a BVF-α-ideal of X.

Definition 3.6. A BVFS A in a BF-algebra X is considered to be a Bipolar valued fuzzy closed α-ideal (BVFC-α-ideal) of X, if

Ÿ

Ÿ

Ÿ

Ÿ

Example 3.7. Consider the BF-algebra X = {0, 1, 2, 3} with the Cayley table given below.

is the BVFS of X defined as

and

is a BVFC-α-ideal of X.

Trivially, the following can be proved:

Proposition 3.8. Every BVFC-α-ideal is a BVF-α-ideal.

In general, the converse of the above proposition is not true from the following:

Example 3.9. Consider the BF-algebra X = {0, 1, 2, 3} with the Cayley table given below

is the BVFS of X defined as

and

is a BVF-α-ideal of X but not BVFC-α-ideal.

Since and

Proposition 3.10. If A is Bipolar valued fuzzy α-ideal of X with for any then and That is is order-reversing and is order-preserving.

Proof: Let such that

Then, by the partial ordering if is defined in X, and

Thus,

And

It completes the proof.

Theorem 3.11. If A is BVFC-α-ideal of X, then the sets and are α-ideals of X.

Proof: Clearly, and. Hence, and.

Let

But

Hence, J is an α-ideal of X. Similarly, it can be proved that K is an α-ideal of X.

Theorem 3.12. Any BVF- α-ideal of X is a Bipolar valued fuzzy BF-ideal of X.

Proof: It is trivial by putting in the definition of BVF-α-ideal.

The converse of the above theorem may not be true.

Now, a sufficient condition is derived for a Bipolar valued fuzzy BF-ideal to be a BVF-α-ideal as follows:

Theorem 3.13. Let A be a BVF-BF-ideal of X. If and then A is BVF- α-ideal of X.

Proof: Let A be a Bipolar valued fuzzy BF-ideal of X and assign

So, we have and

Then,

and

Hence, A is BVF- α-ideal of X.

Theorem 3.14. The intersection of any two Bipolar valued fuzzy α-ideals of X is also a Bipolar valued fuzzy α-ideal.

Proof: Let A and B be any two Bipolar valued fuzzy α-ideals of X.

Let and.

Consider

where and.

Let

Now, and

.

Similarly, and it completes the proof.

The above theorem can be generalized as follows.

Theorem 3.15.The intersection of a family of Bipolar valued fuzzy α-ideals of X is a Bipolar valued fuzzy α-ideal of X.

The following can be analogously proved.

Theorem 3.16. Intersection of any two Bipolar valued fuzzy closed α-ideal of X is also a Bipolar valued fuzzy closed α-ideal of X. Hence, the intersection of a family of Bipolar valued fuzzy closed α-ideal of X is also a Bipolar valued fuzzy closed α-ideal of X.

Remark 3.17. is a BVFS defined on any universe X, if and only if and are the fuzzy subsets of X.

Theorem 3.18. A BVFS is a BVF-α-ideal of X, if and only if the fuzzy subsets and are fuzzy α-ideals of X.

Proof: Let be a BVF-α-ideal of X.

Further, clearly is a fuzzy α-ideal of X.

Also and.

Therefore, is a fuzzy α-ideal of X

Conversely, assume and are fuzzy α-ideals of X.

It is enough to prove that, and

For,

It fulfills the proof.

The following can be obtained using this theorem.

Theorem 3.19. A BVFS is a BVF-α-ideal of X, if and only if

Ÿ and

Ÿ ⟡B are also BVF-α-ideals of X.

Proof: is a Bipolar valued fuzzy α-ideal of X, if and only if, the fuzzy subsets and are fuzzy α-ideals of X by the theorem 3.18.

That is, if and only if, and ⟡B are also Bipolar valued fuzzy α-ideal of X by the definition of and ⟡B.

The following is analogously true.

Theorem 3.20. A BVFS is a BVFC- α-ideal of X if and only if

Ÿ and

Ÿ ⟡B are also BVFC-α-ideals of X.

Here, the image and pre-image of Bipolar valued fuzzy α-ideals under the action of homomorphism and anti- homomorphism on BF-algebras are discussed.

Theorem 4.1. Let f be a homomorphism from BF-algebras X onto Y. A be a bipolar valued fuzzy α-ideal of X with Sup-Inf property. Then, the image of A, is a bipolar valued fuzzy α- ideal of Y.

Proof: Let with such that

and

Now, by the definitions 2.8, 2.9 and 2.4, the following is framed

and

Now,

Hence, the image is a bipolar valued fuzzy α-ideal of Y.

Theorem 4.2. Let f be a homomorphism from BF-algebras X onto Y and A be a Bipolar valued fuzzy closed α-ideal of X with Sup-Inf property. Then, the image of A, is a bipolar valued fuzzy closed α-ideal of Y.

Proof: Let with such that

Then, we have

Hence, by the above theorem, the image is considered as a Bipolar valued fuzzy closed α-ideal of Y.

Theorem 4.3. Let f be a homomorphism from BF-algebras X onto Y and B be a bipolar valued fuzzy α-ideal of Y. Then, the inverse image of B, is a bipolar valued fuzzy α-ideal of X.

Proof: Let

Now, it is clear that

and

Then,

Also

Then, the inverse image of B, is a bipolar valued fuzzy α-ideal of X.

Theorem 4.4. Let f be a homomorphism from BF-algebras X onto Y and B be a Bipolar valued fuzzy closed α-ideal of Y. Then the inverse image of B, is a Bipolar valued fuzzy closed α-ideal of X.

Proof: Let Then, we have

Hence, through the above theorem, the inverse image becomes a Bipolar valued fuzzy closed α- ideal of X.

In the same way, the following can be proved.

Theorem 4.5. Let f be an anti-homomorphism from X onto Y and A be a bipolar valued fuzzy α-ideal of X with Sup-Inf property. Then, the image of A, is a bipolar valued fuzzy α-ideal of Y.

Theorem 4.6. Let f be an anti-homomorphism from X onto Y and B be a bipolar valued fuzzy α-ideal of Y. Then, the inverse image of B, is a bipolar valued fuzzy α-ideal of X.

Theorem 4.7. Let f be an anti-homomorphism from X onto Y and A be a bipolar valued fuzzy closed α-ideal of X with Sup-Inf property. Then, the image of A, is a bipolar valued fuzzy closed α-ideal of Y.

Theorem 4.8. Let f be an anti-homomorphism from X onto Y and B be a bipolar valued fuzzy closed α-ideal of Y. Then, the inverse image of B, is a bipolar valued fuzzy closed α-ideal of X.

From the preliminaries of this research work, Bipolar valued fuzzy sets of various researchers are analyzed. Especially, for the present work stated in this paper, an investigation on the Bipolar valued fuzzy α-ideals of BF-algebras has been carried out. From the investigation, several interesting results are observed. As a result, the research has been focused on this way and all the possible ways are found out to prove this strategy. The surprising point is that in [7] Andrzej Walendziak, theorem 2.11 says that the structure of BF algebra becomes a BG-algebra and the proof is followed directly from the definition. Hence, it is concluded that all the results prove here for BF-algebras can directly be carried over to BG-algebras.

The authors would like to thank the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper.

Shanmugavelu Sabarinathan,David C. Kumar,Prakasam Muralikrishna, (2016) Bipolar Valued Fuzzy α-Ideal of BF-Algebra. Circuits and Systems,07,3054-3062. doi: 10.4236/cs.2016.710260