Finite Structure and Radical Theory of Commutative Ternary $Γ$-Semirings
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| Format: | Preprint |
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2025
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| author | Gokavarapu, Chandrasekhar Rao, D Madhusudhana |
| author_facet | Gokavarapu, Chandrasekhar Rao, D Madhusudhana |
| contents | Purpose: To develop the algebraic foundation of finite commutative ternary $Γ$-semirings by identifying their intrinsic invariants, lattice organization, and radical behavior that generalize classical semiring and $Γ$-ring frameworks.
Methods: Finite models of commutative ternary $Γ$-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 $Γ$-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 $Γ$-semirings, establishing the basis for later computational and categorical developments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_01789 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite Structure and Radical Theory of Commutative Ternary $Γ$-Semirings Gokavarapu, Chandrasekhar Rao, D Madhusudhana Rings and Algebras 16Y60, 08A30, 06B23, 16D25 Purpose: To develop the algebraic foundation of finite commutative ternary $Γ$-semirings by identifying their intrinsic invariants, lattice organization, and radical behavior that generalize classical semiring and $Γ$-ring frameworks. Methods: Finite models of commutative ternary $Γ$-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 $Γ$-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 $Γ$-semirings, establishing the basis for later computational and categorical developments. |
| title | Finite Structure and Radical Theory of Commutative Ternary $Γ$-Semirings |
| topic | Rings and Algebras 16Y60, 08A30, 06B23, 16D25 |
| url | https://arxiv.org/abs/2511.01789 |