On the definition of the intuitionistic fuzzy subgroups

In this paper, a new kind of intuitionistic fuzzy subgroup theory, which is different from that of Ma, Zhan and Davvaz (2008) [22] and [23], is presented. First, based on the concept of cut sets on intuitionistic fuzzy sets, we establish the neighborhood relations between a fuzzy point xaxa and an intuitionistic fuzzy set AA. Then we give the definitions of the grades of xaxa belonging to AA, xaxa quasi-coincident with AA, xaxa belonging to and quasi-coincident with AA and xaxa belonging to or quasi-coincident with AA, respectively. Second, by applying the 3-valued Lukasiewicz implication, we give the definition of (α,β)(α,β)-intuitionistic fuzzy subgroups of a group GG for α,β∈{∈,q,∈∧q,∈∨q}α,β∈{∈,q,∈∧q,∈∨q}, and we show that, in 16 kinds of (α,β)(α,β)-intuitionistic fuzzy subgroups, the significant ones are the (∈,∈)(∈,∈)-intuitionistic fuzzy subgroup, the (∈,∈∨q)(∈,∈∨q)-intuitionistic fuzzy subgroup and the (∈∧q,∈)(∈∧q,∈)-intuitionistic fuzzy subgroup. We also show that AA is a (∈,∈)(∈,∈)-intuitionistic fuzzy subgroup of GG if and only if, for any a∈(0,1]a∈(0,1], the cut set AaAa of AA is a 3-valued fuzzy subgroup of GG, and AA is a (∈,∈∨q)(∈,∈∨q)-intuitionistic fuzzy subgroup (or (∈,∈∨q)(∈,∈∨q)-intuitionistic fuzzy subgroup) of GG if and only if, for any a∈(0,0.5]a∈(0,0.5](or for any a∈(0.5,1]a∈(0.5,1]), the cut set AaAa of AA is a 3-valued fuzzy subgroup of GG. At last, we generalize the (∈,∈)(∈,∈)-intuitionistic fuzzy subgroup, (∈,∈∨q)(∈,∈∨q)-intuitionistic fuzzy subgroup and (∈∧q,∈)(∈∧q,∈)-intuitionistic fuzzy subgroup to intuitionistic fuzzy subgroups with thresholds, i.e., (s,t](s,t]-intuitionistic fuzzy subgroups. We show that AA is a (s,t](s,t]-intuitionistic fuzzy subgroup of GG if and only if, for any a∈(s,t]a∈(s,t], the cut set AaAa of AA is a 3-valued fuzzy subgroup of GG. We also characterize the (s,t](s,t]-intuitionistic fuzzy subgroup by the neighborhood relations between a fuzzy point xaxa and an intuitionistic fuzzy set AA.