Structural Theory of Interval-Valued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures

This paper presents a structural theory of Interval-Valued Neutrosophic (IVN) Ẑ-ideals in Ẑ-algebraic structures to address an ongoing problem of representing uncertainty and indeterminacy. The available fuzzy and Neutrosophic ideals provide useful methods, but they lack the expressive power to portray simultaneous fluctuations in truth, falsehood, and indeterminacy. To deal with this, presented an IVN membership functions, in which each component is represented as an interval rather than a single value, resulting in a more flexible and realistic representation of uncertain phenomena. The work defines IVN Ẑ-ideals and discusses their structural, closure, and include relationships. It has been demonstrated step by step to enhance transparency in result development and are accompanied by illustrated examples to show how they differ from classical fuzzy and Neutrosophic ideals. A comparative investigation reveals that IVN-based techniques outperform current techniques in representing complex uncertainty and indeterminacy. Though the proposed framework offers an important generalisation, it has limits in terms of computational complexity and scalability. Additional study may be done regarding algorithmic optimisation, its application to large-scale algebraic systems, decision-making, and artificial intelligence. In summary, the study strengthens theoretical principles of algebraic models with uncertainty and leads towards more advanced applications.