On Structural Properties of ξ -Complex Fuzzy Sets and Their Applications

Abstract

Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept of $\xi$ -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of $\left(\alpha ,\delta \right)$ -cut sets of $\xi$ -complex fuzzy sets and justify the representation of an $\xi$ -complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of $\xi$ -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an $\xi$ -complex fuzzy set is $\xi$ -complex fuzzy subgroup. Furthermore, we extend the idea of $\xi$ -complex fuzzy normal subgroup to define the quotient group of a group $G$ by this particular $\xi$ -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of $G$ by a specific normal subgroup $G_{A^{\xi }}$ .