A new generalization of fuzzy subgroup and level subsets

In this paper, by using the notions of ”not belonging” (∈) and ”non quasi-k-coincidence” (qk) of a fuzzy point with a fuzzy set, we define the notion of (∈,∈ ∨ qk)-fuzzy subgroups of a group which is a generalization of fuzzy subgroups and (∈,∈ ∨ q)-fuzzy subgroups. Aslo, we generalized the concept of ∈-level set, (∈ ∨ q)-level set and (∈,∈ ∨ q)level set by using ”not belonging” (∈) and ”non quasi-k-coincidence” (qk) of a fuzzy point with a fuzzy set to abtain the notions (∈ ∨ qk)-level set and (∈,∈ ∨ qk)-level set. We give characterizations of an (∈,∈ ∨ qk)-fuzzy subgroup by the properties of these generalized level sets. The important achievement of the study with an (∈,∈ ∨ qk)-fuzzy subgroup and (∈,∈ ∨ qk)-level sets is the generalization of that the notions of fuzzy normal subgroups and level sets. 2010 AMS Classification: 20N20, 20M19