Fundamental theorems of group isomorphism under the framework of complex intuitionistic fuzzy set

Muhammad Jawad; Niat Nigar; Sarka Hoskova-Mayerova; Bijan Davvaz; Muhammad Haris Mateen; Muhammad Jawad; Niat Nigar; Sarka Hoskova-Mayerova; Bijan Davvaz; Muhammad Haris Mateen

doi:10.3934/math.2025088

AIMS Mathematics

2025, Volume 10, Issue 1: 1900-1920. doi: 10.3934/math.2025088

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Research article

Fundamental theorems of group isomorphism under the framework of complex intuitionistic fuzzy set

1.
School of Mathematics, Minhaj University Lahore, Lahore 54770, Pakistan
2.
Department of Mathematics and Physics, University of Defence, Brno, 66210, Czech Republic
3.
Department of Mathematical Sciences, Yazd University, Yazd, Iran

Received: 08 October 2024 Revised: 27 December 2024 Accepted: 02 January 2025 Published: 05 February 2025
MSC : 13E15, 08A72, 03E72I

Algebraic homomorphisms are essential mathematical structures that sustain operations across algebraic systems such as groups, rings, and fields. These mappings not only preserve the validity of algebraic operations but also make it easier to investigate structural similarities and equivalences across distinct algebraic entities. In this article, we establish the group isomorphism under the complex intuitionistic fuzzy set, an extended form of the complex fuzzy set that adds the complex degree of non-membership functions, which plays a significant role in the decision-making process. The complex algebraic structure provides effective tools for understanding complex phenomena. We discuss the more intricate features of homomorphism and isomorphism in the framework of a complex intuitionistic fuzzy set. In addition, we introduce the complex intuitionistic fuzzy normal subgroups. We establish the relationship between two complex intuitionistic fuzzy subgroups and analyze of complex intuitionistic fuzzy isomorphisms among these subgroups, proving the important theorems. Furthermore, we establish examples to explore the concept of complex intuitionistic fuzzy subgroups.
Citation: Muhammad Jawad, Niat Nigar, Sarka Hoskova-Mayerova, Bijan Davvaz, Muhammad Haris Mateen. Fundamental theorems of group isomorphism under the framework of complex intuitionistic fuzzy set[J]. AIMS Mathematics, 2025, 10(1): 1900-1920. doi: 10.3934/math.2025088

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Abstract

Algebraic homomorphisms are essential mathematical structures that sustain operations across algebraic systems such as groups, rings, and fields. These mappings not only preserve the validity of algebraic operations but also make it easier to investigate structural similarities and equivalences across distinct algebraic entities. In this article, we establish the group isomorphism under the complex intuitionistic fuzzy set, an extended form of the complex fuzzy set that adds the complex degree of non-membership functions, which plays a significant role in the decision-making process. The complex algebraic structure provides effective tools for understanding complex phenomena. We discuss the more intricate features of homomorphism and isomorphism in the framework of a complex intuitionistic fuzzy set. In addition, we introduce the complex intuitionistic fuzzy normal subgroups. We establish the relationship between two complex intuitionistic fuzzy subgroups and analyze of complex intuitionistic fuzzy isomorphisms among these subgroups, proving the important theorems. Furthermore, we establish examples to explore the concept of complex intuitionistic fuzzy subgroups.

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