A New Constrain Group Operating on Neutrosophic Fuzzy Subgroup and Normal Subgrou

Abstract

This research presents an innovative concept known as the Neutrosophic  group 
operating on fuzzy subsets, extending the traditional algebraic framework to include elements of 
fuzziness and neutrosophy. The study defines the structure and algebraic properties of 
Neutrosophic  group operating fuzzy subgroups, highlighting how these subsets behave 
under group operations influenced by uncertainty and indeterminacy. Several foundational 
algebraic characteristics are discussed, including closure, associativity, identity, and inverses within 
this specialized fuzzy context. In addition, the notion of homomorphisms is incorporated by 
examining how fuzzy subgroups within the Neutrosophic  group behave under 
structure-preserving mappings. Specifically, we analyze the homomorphic image and preimage of 
these subgroups, establishing essential results that contribute to understanding their structural 
consistency and transformation properties. Furthermore, the concept of the direct product of 
Neutrosophic group operating fuzzy subgroups is introduced. We demonstrate that the 
direct product maintains the integrity of the fuzzy subgroup structure, preserving its key features 
across multiple components. This idea is not only applied to the direct product of two such 
subgroups but is also extended to a finite number of them, reinforcing the robustness and 
generalizability of the model. These developments open new avenues in fuzzy algebra and 
Neutrosophic logic-based group theory.

DOI 10.5281/zenodo.17594633