A Novel Algebraic Structure of (α,β)-Complex Fuzzy Subgroups.

Hanan Alolaiyan,Dragan Pamucar,Halimah A Alshehri,Muhammad Gulzar,Muhammad Haris Mateen

doi:10.3390/e23080992

Hanan Alolaiyan, Dragan Pamucar + Show 3 more

Open Access

https://doi.org/10.3390/e23080992

Copy DOI

Abstract

A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of -complex fuzzy sets and then define -complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an -complex fuzzy subgroup and define -complex fuzzy normal subgroups of given group. We extend this ideology to define -complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that -complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of -complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the -complex fuzzy subgroup of the classical quotient group and show that the set of all -complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of -complex fuzzy subgroups and investigate the -complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.

Highlights

In 1965, Zadeh [1] presented the theory of fuzzy sets and discussed their initiatory results
We prove that every complex fuzzy subgroup (CFSG) is an (α, β)-complex fuzzy subgroup ((α, β)-complex fuzzy subgroups (CFSGs)) and discuss fundamental properties of this newly defined CFSG
We initiate the definition of the index of (α, β)-CFSG and develop the (α, β)-complex fuzzification of Lagrange’s Theorem

Summary

Introduction

In 1965, Zadeh [1] presented the theory of fuzzy sets and discussed their initiatory results. The abstraction of fuzzy subrings was proposed by Liu [3] Later, these notions were discussed in [4,5,6]. With the development of science and technology, the decision making problems are becoming increasingly difficult To overcome this drawback, Ramot et al [12,13] presented the generalized form of fuzzy set by combining a phase term, called a complex fuzzy set. The efficiency of complex fuzzy logic in the respect of membership has a powerful role to deal with concrete problems It is highly valuable for calculating unevenness, and it is very useful way to address ambiguous ideas. Presented a new abstraction of complex fuzzy subgroup. We initiate the definition of the index of (α, β)-CFSG and develop the (α, β)-complex fuzzification of Lagrange’s Theorem

Preliminaries

Conclusions