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

Florentin Smarandache

Authors

Florentin Smarandache 1University of New Mexico, Mathematics, Physics, and Natural Sciences Division, 705 Gurley Ave., Gallup, NM, USA

Keywords:

n-Fold Algebra, Horizontal n-Fold Algebra, Vertical n-Fold Algebra, Fuzzy, Fuzzy Extensions, Neutrosophic Logic, Algebraic Structures, Uncertainty Modeling, Multi-Criteria Decision Making (MCDM), Fuzzy Sets, Hybrid Algebraic Laws, Information Fusion, Neutrosophic Two-Fold Algebra, Decision Support Systems

Abstract

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.

DOI 10.5281/zenodo.18262728