In the mathematical applications, ideal concepts are involved. They have been studied and analyzed in various ways. Already ideal and α- ideal concepts were discussed in BF-algebras. In this paper the idea of bipolar valued fuzzy α- ideal of BF algebra is proposed. The relationship between bipolar valued fuzzy ideal and bipolar valued fuzzy α- ideal is studied. Some interesting results are also discussed.
After the concept of fuzzy sets of Zadeh [
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 [
Inspired by the concepts recently, the concept of Filters of BCH-Algebras Based on Bipolar-Valued Fuzzy Sets [
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
Definition 2.1. [
1.
2.
3.
Example 2.2. Let
Then (X, *, 0) is BF-algebra.
Definition 2.3. [
A binary relation in a BF-algebra X can be defined as
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
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
An α-ideal I of X is called closed, if
A fuzzy set
A fuzzy set
A fuzzy set
Definition 2.4. [
Remark 2.5. If
Definition 2.6. [
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
The positive membership degree
polar valued fuzzy set
Definition 2.8. A BVFS B in a set X with the positive membership
Definition 2.9. Let
and
Definition 2.10. Let
such that
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
Definition 3.2. A BVFS
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
Example 3.5. The BF-algebra X = {0, 1, 2, 3} is considered with the Cayley table as given below.
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 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 a BVF-α-ideal of X but not BVFC-α-ideal.
Since
Proposition 3.10. If A is Bipolar valued fuzzy α-ideal of X with
Proof: Let
Then, by the partial ordering if
Thus,
And
It completes the proof.
Theorem 3.11. If A is BVFC-α-ideal of X, then the sets
Proof: Clearly,
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
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
Proof: Let A be a Bipolar valued fuzzy BF-ideal of X and assign
So, we have
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
Consider
where
Let
Now,
Similarly,
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.
Theorem 3.18. A BVFS
Proof: Let
Further, clearly
Also
Therefore,
Conversely, assume
It is enough to prove that,
For,
It fulfills the proof.
The following can be obtained using this theorem.
Theorem 3.19. A BVFS
⟡B
Proof:
That is, if and only if,
The following is analogously true.
Theorem 3.20. A BVFS
⟡B
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,
Proof: Let
and
Now, by the definitions 2.8, 2.9 and 2.4, the following is framed
and
Now,
Hence, the image
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,
Proof: Let
Then, we have
Hence, by the above theorem, the image
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,
Proof: Let
Now, it is clear that
Then,
Also
Then, the inverse image of B,
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,
Proof: Let
Hence, through the above theorem, the inverse image
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,
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,
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,
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,
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 [
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