Neutrosophic Ideals of Neutrosophic TM-Algebras

Abstract

The study of algebraic structures under uncertainty has gained increasing attention with the development of neutrosophic logic, which extends classical and fuzzy frameworks by introducing three independent membership functions: truth, indeterminacy, and falsity. This paper aims to investigate the nature and properties of neutrosophic ideals in the setting  of neutrosophic TM-algebras, a generalization of BCK/BCI-algebras. The primary objectives are to define neutrosophic ideals in TM-algebras, analyze their behavior under fundamental algebraic operations such as intersection and union, and distinguish them from neutrosophic subalgebras.Methodologically, we establish formal definitions and prove several theorems regarding closure, level sets, and stability under homomorphisms. Illustrative examples and comparative tables are provided to highlight the structural differences between classical, fuzzy, and neutrosophic ideals.The novelty of this work lies in its systematic characterization of neutrosophic ideals within TM-algebras and the demonstration that intersections and level sets preserve ideal structure, while not every subalgebra qualifies as an ideal. This provides a richer framework compared to classical and fuzzy algebraic theories.The implications of these results extend beyond abstract algebra: neutrosophic ideals offer a flexible mathematical tool for modeling indeterminacy and inconsistency in real-world c ontexts such as decision-making, artificial intelligence, and computational logic. The findings underline the potential of neutrosophic structures to unify and extend multiple uncertainty models, suggesting promising directions for future applications in soft computing and information systems.