On the definition of the intuitionistic fuzzy subgroupsI

In 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 rela- tions 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 belong- ing 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 defi- nition of . ; /-intuitionistic fuzzy subgroups of a group G for ; 2 f2; q; 2 ^q; 2 _qg, and we show that, in 16 kinds of . ; /-intuitionistic fuzzy subgroups, the significant ones are the .2; 2/-intuitionistic fuzzy subgroup, the .2; 2 _q/-intuitionistic fuzzy sub- group and the .2 ^q; 2/-intuitionistic fuzzy subgroup. We also show that A is a .2; 2/- intuitionistic fuzzy subgroup of G if and only if, for any a 2 .0; 1U, the cut set Aa of A is a 3-valued fuzzy subgroup of G, and A is a .2; 2 _q/-intuitionistic fuzzy subgroup (or .2; 2 _q/-intuitionistic fuzzy subgroup) of G if and only if, for any a 2 .0; 0:5U(or for any a 2 .0:5; 1U), the cut set Aa of A is a 3-valued fuzzy subgroup of G. At last, we general- ize the .2; 2/-intuitionistic fuzzy subgroup, .2; 2 _q/-intuitionistic fuzzy subgroup and .2 ^q; 2/-intuitionistic fuzzy subgroup to intuitionistic fuzzy subgroups with thresholds, i.e., .s; tU-intuitionistic fuzzy subgroups. We show that A is a .s; tU-intuitionistic fuzzy sub- group of G if and only if, for any a 2 .s; tU, the cut set Aa of A is a 3-valued fuzzy subgroup of G. We also characterize the .s; tU-intuitionistic fuzzy subgroup by the neighborhood relations between a fuzzy point xa and an intuitionistic fuzzy set A.