Generalized Fuzzy Soft Structure Group And Ring

Theories of fuzzy sets and soft sets can be considered as the most important mathematical tools for dealing with uncertainties. By fusing their advantages, professor Maji introduced the concept of fuzzy soft sets in2001. In the present study, we concentrate on the fundamental theory of fuzzy soft sets and discuss the generalized fuzzy soft structures of groups and rings. The main content we have done is as follows:First of all, fuzzy soft sets and groups have been combined. Based on the concepts of "belongs to∈" and "quasi-coincident with q ", we define (∈,∈Vqk) fuzzy soft groups and (∈,∈Vqk) fuzzy normal soft groups which are generalized fuzzy soft groups with the parameter k, then discuss the related properties about the operations of fuzzy soft sets. By using the level cut set (F, A)1, we propose the equivalence characterization of (∈,∈Vqk) fuzzy (normal) soft groups, that is, fuzzy soft set (F,A) is an (∈,∈Vqk) fuzzy (normal)’soft group if and only if (F, A)t is a (normal) soft group over group G, Vt∈(0,(1-k)/2]-Moreover, the image of (∈,∈Vqk) fuzzy (normal) soft groups under fuzzy soft homomorphism are investigated.Secondly, we promote the structure from group to ring and use the subin-terval (λ,μ) of unit interval to blur soft ring more broadly, then introduce the concepts of (λ,μ) fuzzy soft rings and (λ,μ) fuzzy soft ideals which are gener-alized fuzzy soft rings. Similarly to (∈,∈Vqk) fuzzy soft groups, the properties of (λ,μ) fuzzy soft rings (ideals) are studied and the relationship between (λ,μ) fuzzy soft rings (ideals) and soft rings (ideals) is discussed. We get that fuzzy soft set (F, A) is a (λ,μ) fuzzy soft ring (ideal) if and only if (F, A)t is a soft ring (ideal) over ring R, Vt∈(λ,μ]. Then the image of (λ,μ) fuzzy soft rings (ideals) under fuzzy soft homomorphism are discussed.At last,(λ,μ) fuzzy soft quotient ring is introduced by structuring ring R/(F,A). We prove that if fuzzy set B is a (λ,μ) fuzzy subring (ideal) of ring R, then fuzzy soft set B/(F, A) is a (λ,μ) fuzzy soft ring (ideal) of ring R/(F, A). Furthermore, the basic fuzzy soft isomorphism theorem of (λ,μ) fuzzy soft ring is established.