Bipolar Valued Fuzzy α-Ideal of BF-Algebra

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Received 24 May 2016; accepted 31 May 2016; published 19 August 2016

1. Introduction

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.

2. Preliminaries

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

2.1. Basic Results on BF-Algebras

Definition 2.1. [7] A BF algebra is a non-empty set X with a constant 0 and a single binary operation * which satisfies the following axioms:

1.

2.

3.

Example 2.2. Let be a set which comprises the following table.

Then (X, *, 0) is BF-algebra.

Definition 2.3. [7] A BG-algebra is a non-empty set X with a constant 0 and a single binary operation * satisfying the following axioms:

A binary relation in a BF-algebra X can be defined as, if and only if

A subset S of a BF-algebra X is called a subalgebra of X, if

An ideal of a BF-algebra X is a subset I of X consisting 0 such that, if and then

An ideal I of a BF-algebra X is called closed, if

A non-empty subset I of a BF-algebra X is α-ideal, if for all, and

An α-ideal I of X is called closed, if

A fuzzy set in a BF-algebra X can be called as a fuzzy subalgebra of X, if it satisfies:

(1)

A fuzzy set in a BF-algebra X can be called as a fuzzy ideal of X, if it satisfies:

(2)

(3)

A fuzzy set in a BF-algebra X can be called as a fuzzy α-ideal of X, if it satisfies:

(4)

(5)

Definition 2.4. [6] A function of BF-algebras is considered to be homomorphism of X, if

Remark 2.5. If is a homomorphism on BF-algebras,

Definition 2.6. [6] A function of BF-algebras is said to be anti-homomorphism of X if

2.2. Basic Results on Bipolar Valued Fuzzy Set

Fuzzy sets are generally useful mathematical structures which represent a collection of objects whose boundary is vague. Several kinds of fuzzy set extensions are there in the fuzzy set theory. The examples are intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets, etc. This section starts with the definition of Bipolar Valued Fuzzy Set.

Definition 2.7. Let X be a non empty set. A Bipolar Valued Fuzzy Set (BVFS) B in X is an object with the form

where and are mappings.

The positive membership degree denotes the satisfaction degree of an element x to the property corresponding to a bipolar valued fuzzy set and the negative membership degree

denotes the satisfaction degree of an element x to some implicit counter-property corresponding to a bi-

polar valued fuzzy set If and x is regarded as possessing only positive satisfaction for. If and it denotes that x does not satisfy the property of but somewhat satisfies the counter property of. It is possible for an element x to be such that and when the membership function of the property overlaps that of its counter property over some portion of X. For the sake of simplicity, the symbol shall be used for the bipolar valued fuzzy set

Definition 2.8. A BVFS B in a set X with the positive membership and negative membership is indicated to have Sup-Inf property, if for any subset T of X, there exists such that and.

Definition 2.9. Let be a function and let and be the bipolar valued fuzzy sets of X and Y, respectively. Then, the image of A under f is defined as such that

and

Definition 2.10. Let be a function and let and be the bipolar valued fuzzy sets of X and Y, respectively. Then, the inverse image of B under f is defined as

such that and

3. Bipolar Valued Fuzzy α-Ideal

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:

(6)

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

(7)

(8)

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

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

and

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

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

and

⟡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

and

⟡B are also BVFC-α-ideals of X.

4. Homomorphism on Bipolar Valued Fuzzy α-Ideal

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.

5. Conclusion

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.

Acknowledgements

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.

References

[1] Zadeh, L.A. (1965) Fuzzy Sets. Inform Control, 8, 338-353.

http://dx.doi.org/10.1016/S0019-9958(65)90241-X

[2] Lee, K.M. (2000) Bipolar Valued Fuzzy Sets and Their Applications. Proceedings of International Conference on Intelligent Technologies, Bangkok, 307-312.

[3] Lee, K.M. (2004) Comparison of Interval-Valued Fuzzy Sets, Intuitionistic Fuzzy Sets and Bipolar Valued Fuzzy Sets. Fuzzy Logic Intelligent Systems, 14, 125-129.

http://dx.doi.org/10.5391/jkiis.2004.14.2.125

[4] Farhat Nisar, (2011) Bipolar-Valued Fuzzy K-subalgebras. World Applied Sciences Journal, 14, 1914-1919.

[5] Muralikrishna, P. and Chandramouleeswaran, M. (2012) A Note on BF-Algebras with Intuitionistic L-Fuzzy p-Ideals. Scientia Magna, 8, 98-110.

[6] Jun, J.B. and Park, C.H. (2009) Filters of BCH-Algebras Based on Bipolar-Valued Fuzzy Sets. International Mathematical Forum, 4, 631-643.

[7] Walendziak, A. (2007) On BF-Algebras. Mathematica Slovaca, 57.