On the definition of the intuitionistic fuzzy subgroups

AbstractIn this paper, a new kind of intuitionistic fuzzy subgroup theory, which is different from that of Ma, Zhan and Davvaz (2008) [22,23], is presented. First, based on the concept of cut sets on intuitionistic fuzzy sets, we establish the neighborhood relations between a fuzzy point xa and an intuitionistic fuzzy set A. Then we give the definitions of the grades of xa belonging to A, xa quasi-coincident with A, xa belonging to and quasi-coincident with A and xa belonging to or quasi-coincident with A, respectively. Second, by applying the 3-valued Lukasiewicz implication, we give the definition of (α,β)-intuitionistic fuzzy subgroups of a group G for α,β∈{∈,q,∈∧q,∈∨q}, and we show that, in 16 kinds of (α,β)-intuitionistic fuzzy subgroups, the significant ones are the (∈,∈)-intuitionistic fuzzy subgroup, the (∈,∈∨q)-intuitionistic fuzzy subgroup and the (∈∧q,∈)-intuitionistic fuzzy subgroup. We also show that A is a (∈,∈)-intuitionistic fuzzy subgroup of G if and only if, for any a∈(0,1], the cut set Aa of A is a 3-valued fuzzy subgroup of G, and A is a (∈,∈∨q)-intuitionistic fuzzy subgroup (or (∈,∈∨q)-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 Aa of A is a 3-valued fuzzy subgroup of G. At last, we generalize the (∈,∈)-intuitionistic fuzzy subgroup, (∈,∈∨q)-intuitionistic fuzzy subgroup and (∈∧q,∈)-intuitionistic fuzzy subgroup to intuitionistic fuzzy subgroups with thresholds, i.e., (s,t]-intuitionistic fuzzy subgroups. We show that A is a (s,t]-intuitionistic fuzzy subgroup of G if and only if, for any a∈(s,t], the cut set Aa of A is a 3-valued fuzzy subgroup of G. We also characterize the (s,t]-intuitionistic fuzzy subgroup by the neighborhood relations between a fuzzy point xa and an intuitionistic fuzzy set A