A New Approach to Group Theory via Soft Sets and $L$-Fuzzy Soft Sets

Abstract

This paper aims to extend the notion of group to inside the algebraic structures of L-fuzzy soft sets. We firstly give some new notions such as product, extended product, restricted product of two L-fuzzy soft sets. By using these new notions we then introduce concept of L-fuzzy soft groups and study some of their properties. We also compare L-fuzzy soft groups to the related concept of soft groups. Furthermore, we show that L-fuzzy soft groups are more general concept than soft groups. We finally define L-fuzzy soft function and L-fuzzy soft group homomorphism, and then give theorem of homomorphic image and homomorphic pre-image under a L-fuzzy soft function.

Theoretical approaches on certain characterization of L‐fuzzy soft modules

“…Then a pair (ϒ, A ) is called a L‐fuzzy soft set over M , where ϒ : A → ℱ L ( M ). The set of all L‐fuzzy soft sets over M is denoted by ℱ LS ( M ). Definition [24] Let (ϒ, A ) and (Ω, B ) ∈ ℱ LS ( M ). Then (ϒ, A ) is called a L‐fuzzy soft subset of (Ω, B ), denoted by (ϒ, A ) ⊆ (Ω, B ), if A ⊆ B and ϒ( a ) ≤ Ω( a ) for all a ∈ A . Definition Let {(ϒ i , A i )| i ∈ Λ} be a family of L‐fuzzy soft sets over M .…”

“…Fuzzy soft set theory which is a generalization of soft set theory is firstly defined by Maji et al [22]. After that, this concept has been applied to different algebraic structures by some researcher [23–30]. Xiao et al [31] introduced concept of fuzzy soft module and gave some basic properties.…”