Finite Structure and Radical Theory of Commutative Ternary $\Gamma$-Semirings

Purpose: To develop the algebraic foundation of finite commutative ternary $\Gamma$-semirings by identifying their intrinsic invariants, lattice organization, and radical behavior that generalize classical semiring and $\Gamma$-ring frameworks. Methods: Finite models of commutative ternary $\Gamma$-semirings are constructed under the axioms of closure, distributivity, and symmetry. Structural and congruence lattices are analyzed, and subdirect decomposition theorems are established through ideal-theoretic arguments. Results: Each finite commutative ternary $\Gamma$-semiring admits a unique (up to isomorphism) decomposition into subdirectly irreducible components. Radical and ideal correspondences parallel classical results for binary semirings, while the classification of all non-isomorphic systems of order $\lvert T\rvert\!\le\!4$ confirms the structural consistency of the theory. Conclusion: The paper provides a compact algebraic framework linking ideal theory and decomposition in finite ternary $\Gamma$-semirings, establishing the basis for later computational and categorical developments.