The normal intuitionistic fuzzy subgroups

In this paper,we present the concept of (α, β)-normal intuitionistic fuzzy subgroup. And we show that, in 16 kinds of (α, β)-normal intuitionistic fuzzy subgrous, the significant ones are the (∈,∈)-normal intuitionistic fuzzy subgroup, the (∈, ∈ ⋁q)- normal intuitionistic fuzzy subgroup and the (∈ ∧q, ∈)- normal intuitionistic fuzzy subgroup. We also show that A is a (∈, ∈)- normal intuitionistic fuzzy subgroup of G if and only if, for any a ∈ (0, 1], the cut set A_a of A is a 3-valued normal fuzzy subgroup of G, and A is a (∈, ∈ ⋁q)-normal intuitionistic fuzzy subgroup (or (∈ ∧q, ∈)- normal intuitionistic fuzzy subgroup) of G if and only if, for any a ∈ (0, 0.5] (or for any a ∈ (0.5,1]), the cut set A_a of A is a 3-valued normal fuzzy subgroup of G. At last, we generalize the (∈, ∈)- normal intuitionistic fuzzy subgroup, the (∈, ∈ ⋁q)-normal intuitionistic fuzzy subgroup and the (∈ ∧q, ∈)- normal intuitionistic fuzzy subgroup to normal intuitionistic fuzzy subgroup with thresholds,i.e.,(s, t]- normal intuitionistic fuzzy subgroup. We show that A is a (s, t]- normal intuitionistic fuzzy subgroup of G if and only if,for any a ∈ (s, t], the cut set A_a of A is a 3-valued normal fuzzy subgroup of G. we also characterize the (s, t]- normal intuitionistic fuzzy subgroup by the neighborhood relations between a fuzzy point x_a and an intuitionistic fuzzy set A.