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Finite Structure and Radical Theory of Commutative Ternary $Γ$-Semirings

Chandrasekhar Gokavarapu, D Madhusudhana Rao

TL;DR

The paper develops a finite, coherent theory of commutative ternary Γ-semirings by establishing a modular and distributive ideal lattice, a subdirect decomposition framework, and a radical theory in which $\mathrm{Rad}(T)=\mathrm{Nil}(T)$. It proves a Wedderburn-type decomposition $T\cong\mathrm{Rad}(T)\times S$ with a semisimple factor $S$, and provides complete classifications for $|T|\le 4$, validating the invariants $\mathcal{I}(T)=(|T|,|\Gamma|,|\mathrm{Id}(T)|,|\mathrm{Con}(T)|,|\mathrm{Rad}(T)|,|\mathrm{Nil}(T)|)$ as effective fingerprints. An algorithmic framework enumerates all nonisomorphic finite structures, coupling algebraic axioms with canonical labeling to produce representative operation tables and classification tables. The work integrates radical–spectrum theory with computational enumeration, and outlines applications to coding, cryptography, fuzzy logic, path problems, and categorical semantics, while proposing concrete future directions such as prime avoidance, Krull-type dimensions, and module theory in the ternary Γ-context.

Abstract

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.

Finite Structure and Radical Theory of Commutative Ternary $Γ$-Semirings

TL;DR

The paper develops a finite, coherent theory of commutative ternary Γ-semirings by establishing a modular and distributive ideal lattice, a subdirect decomposition framework, and a radical theory in which . It proves a Wedderburn-type decomposition with a semisimple factor , and provides complete classifications for , validating the invariants as effective fingerprints. An algorithmic framework enumerates all nonisomorphic finite structures, coupling algebraic axioms with canonical labeling to produce representative operation tables and classification tables. The work integrates radical–spectrum theory with computational enumeration, and outlines applications to coding, cryptography, fuzzy logic, path problems, and categorical semantics, while proposing concrete future directions such as prime avoidance, Krull-type dimensions, and module theory in the ternary Γ-context.

Abstract

Purpose: To develop the algebraic foundation of finite commutative ternary -semirings by identifying their intrinsic invariants, lattice organization, and radical behavior that generalize classical semiring and -ring frameworks. Methods: Finite models of commutative ternary -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 -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 confirms the structural consistency of the theory. Conclusion: The paper provides a compact algebraic framework linking ideal theory and decomposition in finite ternary -semirings, establishing the basis for later computational and categorical developments.

Paper Structure

This paper contains 32 sections, 26 theorems, 28 equations, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structural Properties of Finite Ternary $\Gamma$-Semirings
  4. Basic structural results
  5. Ideal lattice and decompositions
  6. Algebraic invariants and classification hints
  7. Algorithmic Classification and Enumeration Methods
  8. Computational strategy
  9. Enumeration algorithm
  10. Correctness and complexity
  11. Illustrative enumeration examples
  12. Data representation and isomorphism testing
  13. Computational Data, Enumeration Results, and Classification Tables
  14. Summary of enumerated structures
  15. Representative operation tables
  16. ...and 17 more sections

Key Result

Lemma 1

Let $(T,+)$ be the additive reduct of a finite commutative ternary $\Gamma$-semiring. If there exists $\gamma\in\Gamma$ such that $\{a\,a\,a\}_\gamma=a$ for all $a\in T$, then $(T,+)$ is idempotent, i.e. $a+a=a$ for every $a\in T$.

Theorems & Definitions (70)

  • Lemma 1: Additive idempotence criterion
  • proof
  • Theorem 1: Existence of zero and unit elements
  • proof
  • Remark 1
  • Definition 1
  • Proposition 2
  • proof
  • Theorem 3: Subdirect decomposition
  • proof
  • ...and 60 more