Computational and Categorical Frameworks of Finite Ternary $Γ$-Semirings: Foundations, Algorithms, and Industrial Modeling Applications

TL;DR

This work extends finite commutative ternary Γ-semiring theory into a constructive computational and categorical framework, focusing on explicit classification for small orders. It combines constraint-driven enumeration with automorphism-based canonical labeling and a categorical interpretation via Spec_Γ(T), producing data-driven invariants such as structural entropy $H(T)$, radical proportion $\rho(T)$, and congruence density $\kappa(T)$, along with predictive models and asymptotic insights. The paper reports complete classifications for $|T|\le 4$, establishes polynomial-time feasibility, and develops a unified synthesis linking algebra, topology, and computation, with implications for coding theory, fuzzy logic, and industrial modeling. Overall, it provides a computationally grounded, philosophically informed framework that advances automated reasoning about multi-parameter, higher-arity algebraic systems.

Abstract

Purpose: This study extends the structural theory of finite commutative ternary $Γ$-semirings into a computational and categorical framework for explicit classification and constructive reasoning.Methods: Constraint-driven enumeration algorithms are developed to generate all non-isomorphic finite ternary $Γ$-semirings satisfying closure, distributivity, and symmetry. Automorphism analysis, canonical labeling, and pruning strategies ensure uniqueness and tractability, while categorical constructs formalize algebraic relationships. \textit{Results:} The implementation classifies all systems of order $|T|\!\le\!4$ and verifies symmetry-based subvarieties. Complexity analysis confirms polynomial-time performance, and categorical interpretation connects ternary $Γ$-semirings with functorial models in universal algebra. \\ Conclusion: The work establishes a verified computational theory and categorical synthesis for finite ternary $Γ$-semirings, integrating algebraic structure, algorithmic enumeration, and symbolic computation to support future industrial and decision-model applications.

Figures (1)

• Figure 1: Schematic PCA clustering of invariant vectors $\Sigma(T)$.

Theorems & Definitions (50)

• Theorem 3.1: Entropy–simplicity principle
• proof
• Remark 3.2
• Theorem 3.3: Radical–congruence correlation
• proof
• Definition 3.4: Invariant signature
• Theorem 3.5: Canonical labeling algorithm
• proof
• Remark 3.6
• Proposition 3.7