The Universal Coefticient Theorem in the Category of Fuzzy Soft Modules

This paper begins with the basic concepts of chain comlexes of fuzzy soft modules. Later, we introduce short exact sequence of fuzzy soft modules and prove that split short exact sequence of fuzzy soft chain complex. Naturally, we want to investigate whether or not the universal coefficient theorems are satisfied in category of fuzzy soft chain complexes. However, in the proof of these theorems in the category of chain complexes, exact sequence of homology modules of chain complexes is used. Generally, sequence of fuzzy soft homology modules is not exact in fuzzy chain complexes. Therefore in this study, we construct exact sequence of fuzzy soft homology modules under some conditions. Universal coefficients theorem is proven by making use of this idea.


Introduction
The concept of fuzzy sets was introduced by Lotfi A. Zadeh in 1965 [22]. Since then tht fuzzysets and fuzzy logic have been applied in many real life problems in uncertain, ambiguous environment. The idea of extending the concepts of fuzzy sets to algebra dates back to the introduction in 1971 by Rosenfeld of fuzzy subgroups of a group [11]. Later several researchers have studied fuzzy modules and then Lopez-Permouth and Malik introduced the category of − fuzzy left − modules over a ring [21]. Ameri and Zahedi defined the concept of fuzzy exact sequence in the category of fuzzy modules, and obtained some results related to these notions [15]. Same researchers have previously introduced the category of fuzzy chain complexes and determined fuzzy homology functor in the category. It was proved that this functor is invariant with respect to fuzzy homotopy given in [14]. Molodtsov [10] 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. [8,9] have published a detailed theoretical study on soft sets. After Molodtsov's work, some different applications of soft sets were studied in [9]. 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 ( [1,4,7]). Qiu-Mei Sun et al. [20] defined soft modules and investigated their basic properties.
L. Jin-Liang [7] presented fuzzy soft sets and fuzzy soft groups. C. Gunduz and S. Bayramov [5] presented fuzzy soft and intuitionistic fuzzy soft modules. Universal coefficient theorem in the category of fuzzy and intuitionistic fuzzy modules proved in [13,19]. Naturally , we want to investigate whether or not the universal coefficient theorems are satisfied in category of fuzzy chain complexes. However, in the proof of these theorems in the category of chain complexes, exact sequence of homology modules of chain complexes is used. Generally, sequence of fuzzy homology modules is not exact in fuzzy chain complexes. Therefore, in this study, we construct exact sequence of fuzzy homology modules under some conditions. Universal coefficients theorem is proven by making use of this idea.

Preminilaries
In this section, we recall necessary information commonly used in intuitionistic fuzzy soft module.
, is a fuzzy set on X .
It is denoted as It is written as ,   (1) g is a mapping from A onto B , and is a fuzzy submodule of M Definition 2.11. Let μ A be a right fuzzy ⋀ -module and let v B be a left fuzzy ⋀-module. Given a fuzzy projective presentation Thus, the fuzzy sequence is exact.
Proof: Firstly, we prove that the sequence of homology modules of chain complexes is exact and homomorphism is fuzzy homomorphism. Since the homomorphism * : (``, " ( ) ) → −1 ( −1 ′ , −1 ′ ( )) is not homomorphism of fuzzy modules. Sequence of fuzzy homology modules (3) is not generally exact. Since fuzzy short exact sequence (2)  These ̅ : (`,`( )) → −1 (`,`( )) is a fuzzy homomorphism of fuzzy modules. Therefore, the sequence (2) is exact. Proof: In order to prove proposition, it is enough to demonstrate that the fuzzy soft homomorphism ̅ ⊗ 1 ̅ has a left inverse. The fuzzy soft homomorphism ̅ has a left inverse ̅. Hence the fuzzy homomorphism ̅ ⊗ 1 ̃ is the left inverse of the fuzzy soft homomorphism ̅` ⊗ 1 ̅ . Since tensor product in category of fuzzy soft modules is a functor for each fuzzy chain complex is a split fuzzy soft short exact sequence. Then by using Theorem 2.9, we can easily prove the following theorem. is a fuzzy homomorphism.