Bipolar Valued Fuzzy α-Ideal of BF-Algebra

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.


Introduction
In the traditional fuzzy sets [7], the membership degrees of elements range over the interval [0,1].Lee [3] has introduced an extension of fuzzy sets named bipolar valued-valued fuzzy sets in 2000 and in [4], he compares it with other fuzzy settings .Bipolar valued fuzzy sets are an extension of fuzzy sets whose membership degree range is enlarged from the interval [0,1] to [-1,1].Bipolar valued fuzzy sets have membership degrees that represent the degree of satisfication to the property corresponding to a fuzzy set and its counter-property in a bipolar valued fuzzy set, the membership degree 0 means that the elements are irrele-vant to the corresponding property.Further the membership degrees on (0,1] indicate that the elements somewhat satisfy the property, and the membership degrees on [-1,0) indicate that elements somewhat satisfy the implicit counter property.
In the definition of bipolar valued fuzzy sets, there are two kinds of representations, such as canonical representation and reduced representation.Here, the canonical representation of a bipolar valued fuzzy sets is used.Let X be the universe of discourse.
In [5] and [6], the authors dealt the concepts of Homomorphic image of Intuitionistic L-Fuzzy ideals and N-ideals of BF-algebras respectively.In 2011, Farhat Nisar [2], discussed the Bipolar-valued Fuzzy K-subalgebras.Motivated by this, in this paper, the notion of Bipolar Valued Fuzzy Set (BVFS) is applied in BF-algebras [1].The concept of bipolar valued fuzzy subalgebras/ideals of a BF-algebra is introduced and several properties are investigated.The relations between a bipolar valued fuzzy subalgebra and a bipolar valued fuzzy ideal are given.A condition is provided for a bipolar valued fuzzy subalgebra to be a bipolar valued fuzzy ideal.The characterizations of a bipolar valued fuzzy ideal is also given.The concept of eqivalence relations on the family of all bipolar valued fuzzy ideals of BF-algebra is considered and some related properties are analysed.
The paper has been organised as follows: Section 2 provides the preliminaries.Section 3 deals with the notion of Bipolar valued Fuzzy Subalgebra.Section 4 discusses the Bipolar valued Fuzzy ideal of BF-algebras and the conclusion is presented in Section 5.

Preliminaries
In this section some basic definitions and results that are needed in the sequel are recalled.The following notations are also used in the rest of the paper:

Basic Results on BF-Algebras Definition 1. [1]
A BF-algebra is a non-empty set X with a constant 0 and a single binary operation * which satisfies the following axioms: Then (X, * , 0) is BF-algebra.

Definition 3. [1]
A BG-algebra is a non-empty set X with a constant 0 and a single binary operation * which satisfies the following axioms: A binary relation ≤ on a BF-algebra X can be defined as x ≤ y, if and only if x * y = 0.A subset S of a BF-algebra X is called a subalgebra of X if x * y ∈ S for all x, y ∈ S.An ideal of a BF-algebra X is a subset I of X containing 0 such that if x * y ∈ I and y ∈ I, then x ∈ I.Note that every ideal of a BF-algebra X has the following property:x ≤ y and y ∈ I imply x ∈ I.
A fuzzy set µ in a BF-algebra X is said to be a fuzzy subalgebra of X, if it satisfies: A fuzzy set µ in a BF-algebra X is said to be a fuzzy ideal of X, if it satisfies: Note that every fuzzy ideal µ of a BF-algebra X is order reversing, i.e., if x ≤ y,then µ(x) ≥ µ(y).

Basic Results on Bipolar Valued Fuzzy Set
Fuzzy sets are useful mathematical structure to represent a collection of objects and whose boundary is vague.There are several kinds of fuzzy set extensions in the fuzzy set theory, for example, intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets,etc.
Definition 4. Let X be a non empty set.A Bipolar Valued Fuzzy Set (BVFS) B in X is an object with the form The positive membership degree µ + (x) denotes the satisfication degree of an element x to the property corresponding to a bipolar valued fuzzy set B = {(x; µ + (x), ν − (x)) | x ∈ X} and the negative membership degree ν − (x) denotes the satisfication degree of an element x to some implicit counter-property corresponding to a bipolar valued-valued fuzzy set B = {(x : µ + (x), ν − (x)) | x ∈ X} .If µ + (x) = 0 and ν − (x) = 0, it is the situation that x is regarded as having only positive satisfication for B = {(x; µ + (x), ν − (x)) | x ∈ X} If µ + (x) = 0 and ν − (x) = 0, it is the situation that x does not satisfy the property of B = {(x; µ + (x), ν − (x)) | x ∈ X} but somewhat satisfies the counter property of B = {(x; µ + (x), ν − (x)) | x ∈ X} .It is possible for an element x to be such that µ + (x) = 0 and ν − (x) = 0, 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 B = (X; µ + , ν − ) shall be used for the bipolarvalued fuzzy set B = {(x; µ + (x), ν − (x)) | x ∈ X} .

Bipolar Valued Fuzzy Subalgebra
In this section, the notion of Bipolar Valued Fuzzy Subalgebra of a BF-algebra is introduced and some elegant results are also discussed.In the forth coming sections, X is denoted as BF-algebra unless otherwise specified.Definition 5. A BVFS B = (X; µ + , ν − ) in X is called a bipolar valued fuzzy subalgebra of X, which satisfies: Then, B = (X; µ + , ν − ) is a bipolar valued fuzzy subalgebra of X.
This completes the proof.
Theorem 9. Let B = (X, µ + , ν − ) be a bipolar valued fuzzy subalgebra of X.Then, the following assertions are valid.

For all
and so x * y ∈ B − s .Hence, B − s is a subalgebra of X.

Bipolar Valued Fuzzy Ideal of BF-Algebras
This section deals with the ideas of Bipolar Valued Fuzzy Ideal of BF-algebras.
Definition 13.A BVFS B in a BF-algebra X is said to be a Bipolar valued fuzzy Closed-BF-ideal (BVFC-BF-ideal) of X, if Example 14.The BF-algebra X = {0, 1, 2, 3} with the Cayley table given below is considered.
* 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 if and only if the fuzzy subsets µ + and −ν − are fuzzy BF − ideals of X, by above theorem, if and only if A and ♦A are also BVF-ideals of X by the definition of A and ♦A.

Conclusion
In this work, the investigation of the Bipolar Valued fuzzy structures on BFalgebras is started and several interesting results are arrived.The surprising point is that Andrzej Walendziak [1] says that the structure of BF algebra becomes a BG-algebra, by the proof following directly from the definition.Hence, it is concluded that whatever result is proved for BF-algebras, can directly be carried over to BG-algebras.