Intuitionistic Fuzzy Soft Γ-Semigroup

Objectives: This study deals with the algebraic structure intuitionistic fuzzy soft (IFS) Г-semi group is defined that is a natural extension of IFS semi group and fuzzy soft Г-semi groups. The notion of IFS Г-ideals over Г-semi groups are introduced. Also, the basic concepts of the ideals are studied. Methods: Using approaches given in IFS semi groups to enlarge fuzzy soft Г-semi groups offer the required algebraic structure. Findings: The ideaI F Ssoft Г-semi group is the natural consequence of viewing collectively the IFSsemigroup and fuzzy soft Г-semi groups. Applications: Fuzzy soft set theory, IF set theory, etcetera are well celebrated mathematical concepts handling the uncertainties in the fields of engineering, economy, social science.


Introduction
Fuzzy sets have important applications in control theory, data analysis, artificial intelligence, computational intelligence, decision theory, medicine, logic, management science, and expert systems, see 1 . Fuzzy sets were even generalized and these new structures are called soft set. It was a new approach to counter the uncertainty in the modeling problems.
The notion of fuzzy subset was initiatedin 2 in 1965, also see 3,4 . The concept of a Г-semi group was presented in 5 -8 to generalize the notion of semi group. In 1986 9 introduced the IF subset as generalization of fuzzy subsets, see also 9 -13 . Soft set was familiarizedin 14 in 1999. Fuzzy soft set and IFSset were introduced by 12 in 2001 gave IFS semi groups in 13 2014 defined Fuzzy soft Г-semi groups, see 15 . This work is the extension of the structures. Here we revise some important algebraic structures that are necessary for our paper. 6 A nonempty set G is said to be semi group if it is closed with respect to an associative binary operation. A subset ⊆ N G is known as a left ideal in G when ⊆ GN N  is the set of integers. Denoted

Г-semigroup
G the set of all 2 × 2 matrices over is a Г-semigroup, Г where is the following set  and I 2 ∈Г. Definition: A subset N of a Г-semi group G is known as a left Г-ideal in G when GГN ⊆ N, provided N is non-empty. Similarly, N is said to be right Г-ideal when NГG ⊆ N. If a subset of G is left as well as right Г -ideal then it is called Г-ideal in Г-semi group.
Example: Consider the above example. Let Here R  is a right and L  is a left ideal of 2 ( ) M  , respectively. 12,14 Definition: Suppose we have a universe given by the symbol Ũ and the set of parameters E. The P(Ũ) represents the collection of the subsets of Ũ. Also suppose A is contained in E. Then we call (F, A)a soft set over Ũ, here F is a function :

Soft Set
Definition: Suppose we have soft sets (J, C) and 2) The ∨ operation is (J, C) ∨ (K, D) = (H, C × D) here 3) The union is where L = C ∪ D and for every n∈L, The intersection is the soft set (H,L) Where L = C ∩ D an or K(n) (as both are same set). Definition: We call soft set (J,C) over a semi group G a soft ideal (respectively, left or right soft ideal) over G, if

IFS semigroup13
IF set generalizes fuzzy set. In fuzzy set addition of values of inclusion and exclusion equals to one, but in intuitionistic Fuzzy Set the addition is always less than or equal to 1. It is defined as follows: Definition: Suppose we have a non-empty set G. An IF Set in G iswritten as are the inclusion and exclusion mappings of the element in A respectively.
be the collection of IFSsets of Ũ.
Definitions: Suppose we are given If the universe Ũ is semi group G then we can define the ideal as follows.
Definition: IFS set ( , ) f S  on a semi group G is IFS ideal over G if the following inequalities are true: for all s, t∈ G and α∈S.
Here we give two theorems from. The proof of the first theorem is also given, because it is the motivation for the proof of the same theorems in the case of IFS Г-semi group.
Theorem: Suppose G is semi group and ( , ) are IFS ideals over G, it follows that for any ( , ) P γ θ ∈ , we have

IFS Г-Semi group
This section is dealing with the IFS Г-ideal over Г-semi group. The concept of IFSset over semi group and fuzzy soft set over Г-semi group is given in the section of preliminaries. The proofs of the theorem related to IFS Г-ideals over Г-semi group are proved in this section. Definition: An IFS set ( , ) f S  over a Г-semi group G is called an IFS Г-ideal over G if it satisfies: for every p, q ∈G, a∈A and ζ∈ Г .
Here note that f  is the function given by f S  are two IFS Г-ideals on a Г-semi group G, then AND operation ( , ) ( , ) are also fuzzy soft ideals over G. Proof: