New Structure of Intuitionistic Fuzzy Subgroups under Complex Realm

Complex intuitionistic fuzzy sets (CIFS) can represent human information that contains uncertainty and periodicity semantics concurrently. In this research, the generalization concepts of CIF-subgroup (CIFSG) and anti-CIFSG of a given group are introduced, and their significance lies in assigning the grade of membership and non-membership functions in the complex plane. Also, CIFSG and anti-CIFSG are an inspiring expansion of the intuitionistic fuzzy subgroup (IFSG) and anti-IFSG. Our modification is highlighted in the phase term for membership and non-membership functions. Subsequently, a numerical example illustrates the inherent conditions of CIFSG is given. Also, we prove the relation between the complement of CIFSG and anti-CIFSG. After applying the operations ◇, and ▢ to a CIFSG, the results are CIFSG. Also, the intersection of two CIFSGs is as well a CIFSG. Moreover, the complex normal intuitionistic fuzzy subgroup (CNIFSG) and its algebraic properties are presented and investigated. Some equivalent conditions to show that a CIFSG is CNIFSG are obtained.