On the definition of the intuitionistic fuzzy subgroups

Abstract: In this paper, a new kind of intuitionistic fuzzy subgroup theory, which is different from that of Ma, Zhan and Davvaz (2008) , is presented. First, based on the concept of cut sets on intuitionistic fuzzy sets, we establish the neighborhood relations between a fuzzy point and an intuitionistic fuzzy set . Then we give the definitions of the grades of belonging to , quasi-coincident with , belonging to and quasi-coincident with and belonging to or quasi-coincident with , respectively. Second, by applying the 3-valued Lukasiewicz implication, we give the definition of -intuitionistic fuzzy subgroups of a group for , and we show that, in 16 kinds of -intuitionistic fuzzy subgroups, the significant ones are the -intuitionistic fuzzy subgroup, the -intuitionistic fuzzy subgroup and the -intuitionistic fuzzy subgroup. We also show that is a -intuitionistic fuzzy subgroup of if and only if, for any , the cut set of is a 3-valued fuzzy subgroup of , and is a -intuitionistic fuzzy subgroup (or -intuitionistic fuzzy subgroup) of if and only if, for any (or for any ), the cut set of is a 3-valued fuzzy subgroup of . At last, we generalize the -intuitionistic fuzzy subgroup, -intuitionistic fuzzy subgroup and -intuitionistic fuzzy subgroup to intuitionistic fuzzy subgroups with thresholds, i.e., -intuitionistic fuzzy subgroups. We show that is a -intuitionistic fuzzy subgroup of if and only if, for any , the cut set of is a 3-valued fuzzy subgroup of . We also characterize the -intuitionistic fuzzy subgroup by the neighborhood relations between a fuzzy point and an intuitionistic fuzzy set . [Copyright &y& Elsevier]