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

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.

💡 Research Summary

The paper presents a comprehensive study of finite commutative ternary Γ‑semirings, integrating algebraic theory, algorithmic enumeration, categorical modeling, and potential industrial applications. A ternary Γ‑semiring is defined as a triple (T,+,{·,·,·}γ)γ∈Γ where (T,+) is a commutative monoid with zero and each γ∈Γ supplies a ternary operation that is symmetric, distributive in every argument, and satisfies the absorbing rule with zero. The authors restrict attention to finite Γ, and focus on structures of size |T|≤4 and |Γ|≤2.

The core computational contribution is a constraint‑driven enumeration algorithm. The ternary tables are stored as a collection of 3‑dimensional tensors Mγ