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

TL;DR

This paper develops the algebraic theory of finite commutative ternary Gamma-semiring structures, focusing on their invariants, lattice organization, and radicals, extending classical semiring and Gamma-ring theories.

Contribution

It introduces a decomposition framework for finite commutative ternary Gamma-semiring systems, generalizing classical ideal and radical theories.

Findings

Unique decomposition into subdirectly irreducible components

Radical and ideal correspondences mirror classical results

Classification of systems with order 4 confirms structural consistency

Abstract

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.