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

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.