Inverse System in The Category of Intuitionistic Fuzzy Soft Modules

This paper begins with the basic concepts of soft module. Later, we introduce inverse system in the category of intutionistic fuzzy soft modules and prove that its limit exists in this category. Generally, limit of inverse system of exact sequences of intutionistic fuzzy soft modules is not exact ([12]). Then we define the notion ) 1 ( lim which is first derived functor of the inverse limit functor. Finally, using methods of homology algebra, we prove that the inverse system limit of exact sequence of intutionistic fuzzy soft modules is exact.


Introduction
Many practical problems in economics, engineering, environment, social science, medical science etc. cannot be dealt with by classical methods, because classical methods have inherent difficulties. Proba bility theory, fuzzy sets rough sets, and other mathematical tools have their inherent difficulties ( [14,19,20]). The reason for these difficulties may be due to the inadequacy of the theories of parameterization tools.
Molodtsov [12] initiated the concept of soft set theory as a new mathematical tool for dealing with uncertainties. Later, work on the soft set theory is progressing rapidly. Maji et al. [10,11] have published a detailed theoretical study on soft sets. After Molodtsov's work, some differe nt applications of soft sets were studied in [16]. H. Aktaş and N. Cagman [2] has established a connection betwen soft sets and fuzzy sets and they introduced soft groups. At the same time, they gave a definition of soft groups, soft rings and derived their basic properties ( [1,7,8]). Qiu-Mei Sun et al. [20] defined soft modules and investigated their basic properties.
U. Acar and F. Koyuncu introduced soft rings. Qiu-Mei Sun and his friends introduced soft modules [15].
The problem which obtained in new categories are closed according to algebraic operations is very important. Since inverse limit and direct limit contain most of the operations, the proof of presence of the limits is actual problem.
The inverse (direct) limit is not only an important concept in category theory, but also plays an important role in topology, algebra, homology theory etc. To the date, inverse and direct saystems and their limits were defined in different categories. Furthermore, some of their properties were investigated [5,8,10,12].
In this paper begins with the basic concepts of soft module. We introduce inverse system in the category of intuitionistic fuzzy soft modules and prove that its limit exists in this category. Generally, limit of inverse system of exact sequences of intuitionistic fuzzy soft modules is not exact. Then we define the notion ) 1 ( lim which is first derived functor of the inverse limit functor. Finally, using methods of homology algebra, we prove that the inverse system limit of exact sequence of intuitionistic fuzzy soft modules is exact.

Preminilaries
In this section, we recall necessary information commonly used in intuitionistic fuzzy soft module.
It is denoted as It is written as A be any nonemply set.  (1) g is a mapping from A onto B , and is a family of intuitionistic fuzzy soft modules over  

Inverse system of intuitionistic fuzzy soft modules
This category of intuitionistic fuzzy soft modules denoted as IFSM. , where  is a directed set, is called an inverse system of intuitionistic fuzzy soft modules.
Now we consider the following any inverse system It is clear that parameter sets in (3.1) consist of the following inverse system of sets Similarly,       M in (3.1) consist of the following inverse system of modules is an inverse system of intuitionistic fuzzy modules?
We denote inverse limit of (3.4) as be a family of intuitionistic fuzzy soft homomorphisms of intuitionistic fuzzy soft modules, the conditions is an intuitionistic fuzzy soft homomorphism of intuitionistic fuzzy soft modules. It is clear that for all    , the following diagram is commutative: The proof is completed.
Now we consider the following inverse system of intuitionistic fuzzy soft modules over    be an isotone mapping and following mapping It is clear that inverse systems of intuitionistic fuzzy soft modules and morphisms of them consist of a category. This category is denoted as Inv(IFSM).
be a morphism of inverse systems of intuitionistic fuzzy soft modules. Here is an inverse system of sets and 7491   , lim is a mapping of limit sets of this inverse systems. Similarly, is a morphism of inverse systems of modules?
 is a morphism of limits of inverse systems of intuitionistic fuzzy soft modules?
Proof. Since the product operation of intuitionistic fuzzy soft modules is a factor, the following diagram is commutative: is a morphism of inverse systems of intuitionistic fuzzy modules? Then is an intuitionistic fuzzy soft homomorphism of intuitionistic fuzzy modules and the following diagram being commutative: is a covariant factor from the category Inv (IFSM) to the category of IFSM.
is a family of inverse systems of intuitionistic fuzzy soft modules, then Proof. The proof of the theorem is straightforward.

Derivative Factor of lim factor
Let us review the problem of exact limit for inverse system of exact sequence of intuitionistic fuzzy soft modules 1 are inverse systems of modules and?
are morphisms of inverse systems. The following sequence is short exact sequence of inverse systems of intuitionistic fuzzy soft modules. Taking the limits of this sequence is not exact.
As it seen the limit of inverse system of exact sequence of intuitionistic fuzzy soft modules is not exact. So, it is necessary to define derivative factor of inverse limit factor in category of intuitionistic fuzzy soft modules. We get inverse system in (3.1). We define the following homomorphism of modules d is a homomorphism of intuitionistic fuzzy modules. Indeed, We investigate another property of Proof. The proof of lemma is trivial.
We accept natural numbers set which is index set of inverse system.
. We may define is a homomorphism of fuzzy soft modules? By using simplicity of calculation, it is shown that D is a chain homotropy between