Computational and Categorical Frameworks of Finite Ternary $\mathrm{\Gamma }$-Semirings: Foundations, Algorithms, and Industrial Modeling Applications

Purpose: This study extends the structural theory of finite commutative ternary $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$-semirings satisfying closure, distributivity, and symmetry. Automorphism analysis, canonical labeling, and pruning strategies ensure uniqueness and tractability, while categorical constructs formalize algebraic relationships. \ $\mathit{\text{Results:}}$ The implementation classifies all systems of order $\mathrm{\mid }T\mathrm{\mid }\text{\hspace{0.17em}⁣}\le \text{\hspace{0.17em}⁣}4$ and verifies symmetry-based subvarieties. Complexity analysis confirms polynomial-time performance, and categorical interpretation connects ternary $\mathrm{\Gamma }$-semirings with functorial models in universal algebra. \ Conclusion: The work establishes a verified computational theory and categorical synthesis for finite ternary $\mathrm{\Gamma }$-semirings, integrating algebraic structure, algorithmic enumeration, and symbolic computation to support future industrial and decision-model applications.