Neutrosophic Sets and Systems
Abstract
This paper extends the recently developed framework of Neutrosophic Logic-based Two-Fold Algebra (TFA), which was designed to integrate classical algebraic operations with fuzzy and neutrosophic components for modeling uncertainty alongside deterministic structures. Although TFA successfully combines algebraic operations with uncertainty descriptors, its binary structure becomes restrictive in applications requiring the simultaneous integration of multiple independent qualification dimensions, such as risk, sustainability, reliability, and performance assessment. To address this limitation, the paper introduces two generalized frameworks: Horizontal n-Fold Algebra and Vertical n-Fold Algebra, together forming the generalized n-Fold Algebra (n-FA), together with the extension from two-valued to multi-valued operations where m≥2. The proposed framework is formally defined as the coupling of a classical algebraic backbone with n−1 independent or interdependent component sub-laws. A rigorous mathematical construction of the n-FA structure is provided, along with an analysis of important specializations, including fuzzy and intuitionistic-fuzzy extensions. Fundamental algebraic properties such as closure, associativity, and monotonicity are systematically investigated to ensure the coherence and consistency of multi-component operations. The applicability and flexibility of the proposed framework are illustrated through numerical examples involving supply-chain risk analysis and multi-criteria decision-making systems. The results demonstrate that n-Fold Algebra provides a powerful mathematical framework for high-dimensional uncertainty modeling, information fusion, and advanced decision-support applications in complex systems.
Recommended Citation
Smarandache, Florentin. "Horizontal and Vertical Generalized n-Fold Algebra: Formal Construction and Applications in Multi-Dimensional Modeling." Neutrosophic Sets and Systems 98, 1 (2026). https://digitalrepository.unm.edu/nss_journal/vol98/iss1/11