Multipolar Intuitionistic Fuzzy Ideal in B-Algebras

B-algebra is an algebraic structure which combine some properties from BCK-algebras and BCI-algebras. Some researchers have investigated the concept of multipolar fuzzy ideals in BCK/BCI-algebras and multipolar intuitionistic fuzzy set in B-algebras. In this paper, we construct a new structure which is called a multipolar intuitionistic fuzzy ideal in B-algebras. This structure is a combination of three structures such as multipolar fuzzy ideals in BCK/BCI-algebras, fuzzy B-subalgebras in B-algebras, and multipolar intuitionistic fuzzy B-algebras. We investigated and proved some characterizes of the multipolar intuitionistic fuzzy ideal, such as a necessary condition and sufficient condition.


INTRODUCTION
Zadeh [1] introduced a new idea, namely a fuzzy set as a non-empty set with a degree of membership whose value in interval [0,1] in 1965. The degree of membership of each member of the set is determined by the membership function. That notion from Zadeh became the basis for further researchers to develop fuzzy concepts in various fields such as graph theory, data analysis, decision making, and so on.
A simple example of an algebraic structure is a group. Not only groups, -algebras, -algebras and -algebras are also other examples of algebraic structures. Imai and Iseki [2] proposed the notion a new algebraic structure called -algebras in 1966.
-algebras is an important class of algebraic structure which is constructed from two different fragments, set theory and propositional calculus. In the same year, Iseki [3] continued his research to propose the notion of -algebras which is generalization from -algebras. A new idea about algebraic structure is calledalgebras which satisfies some properties from -algebras and -algebras was proposed by Neggers and Kim in [4]. They also investigated its properties.
Zhang [5] introduced the concepts of bipolar fuzzy sets which is the extension of fuzzy set. Meng [6] studied about fuzzy implicative ideals in -algebras in 1997. Moreover, Muhiuddin and Al-Kadi [7] introduced bipolar fuzzy implicative ideals in -algebras. They discussed about the relationship between a bipolar fuzzy ideal and bipolar fuzzy implicative ideal. Furthermore, Chen et al. [8] introduced the concepts of multipolar fuzzy sets which is the extension of bipolar fuzzy set. Kang et al. [9] proposed the concepts about multipolar intuitionistic fuzzy set with finite degree and its application in / -algebras. In 1999, Attanasov [10] introduced the new notion about intuitionistic fuzzy set. Jun et al. [11] defined fuzzy -algebras. Then, Al-Masarwah and Ahmad [12] discussed about multipolar fuzzy ideals in / -algebras. Ahn and Bang [13] studied fuzzy -subalgebras in -algebras. Recently, Borzooei et al. [14] proposed the concept about multipolar intuitionistic fuzzy -algebras and some properties. They constructed a simple multipolar fuzzy set. Then, they also discussed about multipolar intuitionistic fuzzy subalgebras of -algebras.
In this paper, we construct a new structure which is called a multipolar intuitionistic fuzzy ideal in -algebras. This structure is a combination of three structures which are the results of research by Al-Masarwah and Ahmad [12], Ahn and Bang [13], and Borzooei et al. [14]. Next, we investigated and proved some necessary condition and sufficient condition of the multipolar intuitionistic fuzzy ideal.

METHODS
By using literary study and analogical related concepts from [12], [13] and [14], we propose the terminology of multipolar intuitionistic fuzzy ideal in -algebras. We start to describe the structure of -algebra, fuzzy -algebra, and multipolar intuitionistic fuzzy sets. Each structure is given its definition, examples, and some of its properties. Definition 2.1 [15] -algebra is a nonempty set with 0 as identity element (right) and a binary operation * satisfying the following axioms for all , , ∈ : i.

RESULTS AND DISCUSSION
In this section, we will describe the structure of multipolar intuitionistic fuzzy ideal in -algebras. The description begins with the definition of the new structure, then examples are given, and its properties are determined and proven.  For any ∈ and multipolar intuitionistic fuzzy set (l,) in , we give the conditions for the set ( ) to be an ideal of and its example. Hence, 0 ∈ ( ). ii.
Next, we discuss some properties of multipolar intuitionistic fuzzy ideal in -algebras. Corollary If we assume that is a commutative -algebra, then the statements in Proposition 3.7 and Proposition 3.8 are equivalent.
Furthermore, we also give another condition of multipolar intuitionistic fuzzy ideal in -algebras such that make this following proposition. Proof. We assume that (l,) is a multipolar intuitionistic fuzzy ideal over . Let , , ∈ such that ( * ) * = 0. So, * ≤ . By using Definition 3.1 (i) and (ii), we have Hence, (l,) is a multipolar intuitionistic fuzzy ideal over . ∎

CONCLUSIONS
In this paper, we apply the terminology of multipolar intuitionistic fuzzy ideal in -algebras and investigate some properties. We also explain the conditions for a multipolar intuitionistic fuzzy set to be a multipolar intuitionistic fuzzy ideal and give some examples. These definitions and main results can be applied with similarly in other algebraic structure such as -algebras, -algebras and -algebras.