η−Intuitionistic Fuzzy Soft Groups

The fuzzy set theory was introduced by Zadeh in 1965 and the soft set theory was introduced by Molodtsov in 1999. Recently, many researchers have developed these two theories and combined the theory of fuzzy set and soft set became the fuzzy soft set. In this research, we present the idea of the η −intuitionistic fuzzy soft group defined on the η −intuitionistic fuzzy soft set. The main purpose of this research is to create a new concept, which is an η −intuitionistic fuzzy group. To achieve this, we combine the concept of η −intuitionistic fuzzy group and intuitionistic fuzzy soft group. As the main result, we prove the correlation between intuitionistic fuzzy soft group and η −intuitionistic fuzzy soft group along with some properties of η −intuitionistic fuzzy soft group. Also, we prove some properties of subgroup of an η −intuitionistic fuzzy soft group. An η −intuitionistic fuzzy soft homomorphism is also proved.


INTRODUCTION
The theory of fuzzy has been studied by many researchers in various fields. Zadeh introduced the fuzzy set theory in [1] by defining a membership function that maps each member of a set to a closed interval of 0 and 1. Then Atanassov formed the intuitionistic fuzzy set that consist of membership function and nonmembership function in [2]. The theory of fuzzy set and intuitionistic fuzzy set was developed into group theory became fuzzy subgroup in [3] and intuitionistic fuzzy subgroup in [4]. The intuitionistic fuzzy subgroup was studied in various types. For example, the intuitionistic L-fuzzy subgroups formed in [5], the ( , ] −intuitionistic fuzzy subgroups defined in [6], definition of ( , )cut of intuitionistic fuzzy subgroups in [7], and t-intuitionistic fuzzy subgroups in [8]. Doda and Sharma studied the finite groups of different orders and gave the idea of recording the count of intuitionistic fuzzy subgroups in [9]. Zhou and Xu extended the intuitionistic fuzzy sets based on the hesitant fuzzy membership in [10]. The concept of the ( , ) −intuitionistic fuzzy subgroups and normal subgroups were defined in [11]. Then the fundamental properties of t-intuitionistic fuzzy abelian subgroup along with the homomorphism of t-intuitionistic fuzzy abelian subgroup were studied in [12]. Latif, et al. studied the fundamental theorems of t-intuitionistic fuzzy isomorphism of t-intuitionistic fuzzy subgroup in [13]. Moreover, the concept of −intuitionistic fuzzy subgroup, −intuitionistic fuzzy cosets, and −intuitionistic fuzzy normal subgroup were characterized in [14]. Based on those research, Shuaib, et al. in [15] formed a concept called an −intuitionistic fuzzy subgroup on an −intuitionistic fuzzy set along with −intuitionistic fuzzy homomorphism. While the theory of soft set is introduced in [16] which is an ordered pair of parameter and function that maps each member of parameter to the power set of an empty set. The soft sets constructed in the form of membership function became fuzzy soft sets defined in [17]. Maji, et al. introduced the concept of intuitionistic fuzzy soft set which is a generalization of intuitionistic fuzzy set and soft set in [18]. Furthermore, the operation properties and algebraic structure of intuitionistic fuzzy soft set were discussed in [19]. Soft group on the soft set is defined in [16]. Then Aygünoglu and Aygun in [20] developed the concept of soft group in the form of membership function became fuzzy soft group. The concept of intuitionistic fuzzy soft set to semigroup was applied in [21] and intuitionistic fuzzy soft ideals over ordered ternary semigroup was defined in [22]. The concept of soft group is developed into an intersection called soft int-group in [23]. Moreover, Karaaslan, et al. in [24] applied the concept of soft int-group into intuitionistic fuzzy soft set by forming the intuitionistic fuzzy soft group and investigate some properties of intuitionistic fuzzy soft group.
Based on [15] and [24], we introduce the notion of the −intuitionistic fuzzy soft group on the −intuitionistic fuzzy soft set and give some basic properties. Moreover, we define the notion of the −intuitionistic fuzzy subgroup and investigate the properties of homomorphism of an −intuitionistic fuzzy soft group.

METHODS
The method of this research is literature review, data collecting techniques by conducting review studies of books, notes, and other scientific research results related to the object of the problem. In this paper, we begin by forming the definition of −intuitionistic fuzzy soft set based on the definition of intuitionistic fuzzy soft set and −intuitionistic fuzzy set. Then, we form the definition of −intuitionistic fuzzy soft group based on the definition of intuitionistic fuzzy soft group. We continue to prove some properties of the −intuitionistic fuzzy soft group along with subgroup of an −intuitionistic fuzzy soft group. Moreover, we continue to define the definition of image and pre-image of an −intuitionistic fuzzy soft group and prove the homomorphism of an −intuitionistic fuzzy soft group. Here is shown the definition of intuitionistic fuzzy soft set, −intuitionistic fuzzy set, and intuitionistic fuzzy soft group.  Definition 3 [24]. Let be an arbitrary group and Γ be an intuitionistic fuzzy soft set over universe , then Γ is called intuitionistic fuzzy soft group on over if

RESULTS AND DISCUSSION
In this section, we introduce the notion of −intuitionistic fuzzy soft group on −intuitionistic fuzzy soft set. Inspiring from Definition 1 and Definition 2, we define the notion of −intuitionistic fuzzy soft set. Some basic properties such as intersection and union of −intuitionistic fuzzy soft set is proved by the following proposition. Since −intuitionistic fuzzy soft group is defined based on intuitionistic fuzzy soft group, so there is a correlation between those concepts. The following theorem, we prove that every intuitionistic fuzzy soft group is an −intuitionistic fuzzy soft group.  The map : 1 → 2 of a group 1 to a group 2 is called homomorphism if for all , ∈ 1 then ( ) = ( ) ( ). The following theorems, we prove that every image and preimage of the −intuitionistic fuzzy group under the homomorphism function are the −intuitionistic fuzzy soft group.

CONCLUSIONS
Based on the result, it is concluded that −intuitionistic fuzzy soft group depends on intuitionistic fuzzy soft group. It is proved that every intuitionistic fuzzy soft group is an −intuitionistic fuzzy soft group. The definition of −intuitionistic fuzzy soft subgroup and it's properties are presented. The −intuitionistic fuzzy soft homomorphism show that image and pre-image of an −intuitionistic fuzzy soft group are also −intuitionistic fuzzy soft groups. For the future research, it is suggested to notion the −intuitionistic fuzzy soft ring along with the properties.