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

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