The Universal Coefticient Theorem in the Category of Fuzzy Soft Modules

Sebuhi Abdullayev,S A Bayramov

doi:10.24297/jam.v14i2.7599

Sebuhi Abdullayev, S A Bayramov

Open Access

https://doi.org/10.24297/jam.v14i2.7599

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Abstract

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.

Highlights

The concept of fuzzy sets was introduced by Lotfi A
Later several researchers have studied fuzzy modules and Lopez-Permouth and Malik introduced the category of R − fz modof fuzzy left R − modules over a ring R [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]

Summary

Introduction

The concept of fuzzy sets was introduced by Lotfi A. Let us define morphisms of the chain complexes of fuzzy soft modules. Let {( Fn, A), ∂n}, {(Gn, B), ∂n′ } be chain complexes of soft modules over {Mn} and {Nn}, respectively, {fn: Mn → Nn}n be homomorphism of modules and g: A → B is a mapping of sets. (Nn, Gn(g(a)) →∂n′ (Nn−1, Gn−1(g(a))) ({fn}, g): {( Fn, A), ∂n} → {(Gn, B), ∂n′ } is said to be morphism of chain complexes of fuzzy soft modules. Chain complexes of fuzzy soft modules and morphisms of their forms a category. Let ({φn}, g): ( {ψn}, g): {(Fn, A), ∂n} → {(Gn, B), ∂n′ } be morphisms of chain complex of fuzzy soft modules and let D = {(Dn, g): (Fn, A) → (Gn+1, B)} be a family of homomorphisms of fuzzy soft modules.

Let define the homomorphism

Is a fuzzy soft chain complexes of fuzzy soft modules?