@article{Ali2024, 
author = {Arshad Ali and Muhammad Haris Mateen and Qin Xin and Turki Alsuraiheed and Ghaliah Alhamzi},
title = {(  ϵ  ,  δ  )-complex anti fuzzy subgroups and their applications},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {5},
pages = {11580-11595},
keywords = {complex anti-fuzzy set, (ϵ, δ)-complex anti-fuzzy set, (ϵ, δ)-complex anti-fuzzy subgroup, (ϵ, δ)-complex anti-fuzzy normal subgroup},
url = {https://www.sciopen.com/article/10.3934/math.2024568},
doi = {10.3934/math.2024568},
abstract = {The complex anti-fuzzy set (CAFS) is an extension of the traditional anti-fuzzy set with a wider range for membership function beyond real numbers to complex numbers with unit disc aims to address the uncertainty of data. The complex anti-fuzzy set is more significant because it provides two dimensional information and versatile representation of vagueness and ambiguity of data. In terms of the characteristics of complex anti-fuzzy sets, we proposed the concept of    (  ϵ  ,  δ  )-CAFSs that offer a more comprehensive representation of the uncertainty of data than CAFSs by considering both the magnitude and phase of the membership functions and explain the        (          ϵ      ,      δ        )  -complex anti fuzzy subgroups (CAFS) in the context of CAFSs. Moreover, we showed that everyCAFSGis a    (  ϵ  ,  δ  )-CAFSG. Also, we used this approach to define    (  ϵ  ,  δ  )-complex anti-fuzzy(CAF) cosets and    (  ϵ  ,  δ  )-CAF normal subgroups of a certain group as well as to investigate some of their algebraic properties. We elaborated the    (  ϵ  ,  δ  )-CAFSG of the classical quotient group and demonstrated that the set of all    (  ϵ  ,  δ  )-CAF cosets of such a particular CAFs normal subgroup formed a group. Furthermore, the index of        (          ϵ      ,      δ        )  -CAFSG was demonstrated and    (  ϵ  ,  δ  )-complex anti fuzzification of Lagrange theorem corresponding to the Lagrange theorem of classical group theory was briefly examined.}
}