Horizontal and Vertical Generalized n-Fold Algebra: Formal Construction and Applications in Multi-Dimensional Modeling

Two-Fold Algebra (TFA) was recently developed to bridge classical algebraic operations with fuzzy and fuzzy-extension (especially neutrosophic) components, allowing for the simultaneous modeling of objects and their associated uncertainty descriptors. However, as real-world systems increasingly demand the integration of multiple, independent qualification dimensions—such as risk, sustainability, and reliability, the binary nature of TFA becomes a limiting factor. This paper introduces two generalized frameworks: the Horizontal and respectively Vertical Generalization n-Fold Algebra (n-FA), and from 2-valued to m-values operations, m ≥ 2. We formally define the n-FA structure as a coupling of a classical backbone (1) with (n-1) independent or interdependent component sub-laws. We provide rigorous systematic construction, explore various specializations (including fuzzy and intuitionistic-fuzzy cases), and derive the essential algebraic properties—such as closure, associativity, and monotonicity—required for coherent multi-component operations. Finally, we demonstrate the versatility of n-FA through numerical examples in supply-chain risk and multi-criteria decision-making, establishing it as a robust mathematical language for complex, high-dimensional uncertainty modeling.