Ternary Gamma-Semirings

• Ternary Gamma-Semirings are algebraic structures defined by a commutative monoid addition and a parameterized ternary operation that generalizes classical semirings.
• They facilitate advanced ideal theory, enabling prime, semiprime, and radical decompositions analogous to those in ring theory.
• Applications span chemical transformations, multi-objective optimization, and symbolic reasoning, demonstrating the framework's interdisciplinary impact.

A ternary Gamma-semiring (TGS) is an algebraic structure defined by a set equipped with a commutative monoid addition and a ternary operation coordinated by a parameter set Γ, designed to encode triadic, parameter-dependent interactions. TGS generalizes classical semiring and Γ-ring frameworks by replacing binary multiplication with a fundamental ternary law, enabling the modeling of multi-state, multi-parameter systems such as chemical transformations, multi-objective optimizations, and higher-arity symbolic reasoning. The structural and ideal theory of TGS provides a foundation for associated module categories, lattice-theoretic decompositions, radical theory, categorical constructions, and emergent applications in both pure and applied mathematics.

1. Formal Definition and Axiomatic Structure

Let $T$ be a nonempty set (often with a commutative monoid $(T, +, 0)$), and let $\Gamma$ be a nonempty parameter set. A commutative ternary $\Gamma$-semiring is specified by the data

$(T, +, 0, \Gamma, [\cdot, \cdot, \cdot]_\gamma)$

where, for each $\gamma \in \Gamma$, the ternary product

$$[\cdot, \cdot, \cdot]_\gamma : T \times T \times T \to T$$

satisfies the following axioms (all $a, b, c, d, e \in T$, all $\alpha, \beta, \gamma, \delta \in \Gamma$) (Gokavarapu et al., 16 Nov 2025, Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 27 Oct 2025):

• Additivity (Distributivity in Each Slot):

$$[a+a', b, c]_\gamma = [a, b, c]_\gamma + [a', b, c]_\gamma$$

and similarly for the second and third arguments.

• Zero Absorption:

$$[0, b, c]_\gamma = [a, 0, c]_\gamma = [a, b, 0]_\gamma = 0.$$

• Ternary Associativity:

$$[[a, b, c]_\alpha, d, e]_\beta = [a, b, [c, d, e]_\beta]_\alpha$$

• Symmetry (in the commutative case):

$$[a, b, c]_\gamma = [b, a, c]_\gamma = [a, c, b]_\gamma$$

(full symmetry over all argument permutations, if imposed).

Additional structural variants (e.g., partial symmetry or idempotent addition) define subvarieties with application-specific semantics (Gokavarapu et al., 15 Nov 2025).

2. Ideal Theory: Ideals, Lattices, and Radicals

A (two-sided) $\Gamma$-ideal $I \subseteq T$ is a submonoid $(I, +)$ closed under ternary products in every slot: $$a, b \in T,\; i \in I,\; \gamma \in \Gamma \implies [a, b, i]_\gamma \in I$$ and likewise when $i$ is in the first or second argument (Gokavarapu et al., 27 Oct 2025, Gokavarapu et al., 3 Nov 2025).

Prime and Semiprime Ideals

• A prime ideal $P \subset T$ is proper and

$$[a, b, c]_\gamma \in P \implies a \in P \text{ or } b \in P \text{ or } c \in P$$

for all $a, b, c, \gamma$.

• A semiprime ideal $Q$ is defined by

$[a, a, a]_\gamma \in Q \implies a \in Q$

for all $a, \gamma$.

• The radical (or prime radical) of $I$ is

$$\sqrt{I} = \bigcap_{\text{prime }P \supseteq I} P = \{\, a \mid \exists \alpha, \beta, [a, a, a]_\gamma \in I\,\}$$

(Gokavarapu et al., 27 Oct 2025).

Distributivity and modularity characterize the lattice of $\Gamma$-ideals in the finite case (Gokavarapu et al., 3 Nov 2025).

Table: Ideal-theoretic Properties

Property	Definition	Key Result
Prime Ideal	$$[a, b, c]_\gamma \in P \implies a \in P \vee b \in P \vee c \in P$$	$P$ is prime $\Leftrightarrow$ $T/P$ is domain
Semiprime	$[a, a, a]_\gamma \in Q \implies a \in Q$	Intersections of semiprimes are semiprime
Radical	$$\sqrt{I} = \bigcap_{P \supseteq I,\;P\text{ prime}} P$$	$\sqrt{I}$ is semiprime

3. Structural Decomposition, Congruences, and Spectra

Congruences and Subdirect Representations

A $\Gamma$-congruence is an equivalence relation compatible with both $+$ and $[\cdot, \cdot, \cdot]_\gamma$. For finite $T$, the lattice of ideals and that of congruences are in order-reversing bijection (Gokavarapu et al., 3 Nov 2025):

$$I \leftrightarrow \rho_I = \{(a, b) | [a, b, c]_\gamma \in I\;\forall c, \gamma \}$$

Every finite commutative ternary $\Gamma$-semiring $T$ admits a subdirect decomposition into irreducible factors, uniquely determined up to isomorphism (Gokavarapu et al., 3 Nov 2025).

The $\Gamma$-Spectrum and Zariski-Type Topology

The set of prime ideals $\operatorname{Spec}_\Gamma(T)$ carries a Zariski-type topology:

$$V(I) = \{\, P \in \operatorname{Spec}_\Gamma(T) : I \subseteq P \, \}$$

Closed sets $V(I)$ form a topology, and the spectrum supports geometric concepts analogous to those of classical algebraic geometry (Gokavarapu et al., 27 Oct 2025, Gokavarapu et al., 18 Nov 2025).

4. Module and Homological Theory

A (left) $\Gamma$-module over $T$ is a commutative monoid $(M, +)$ with a compatible action: $$T \times \Gamma \times M \times \Gamma \times T \to M,\quad (a, \alpha, m, \beta, b) \mapsto a_\alpha m_\beta b$$ with additive, distributive, associativity, and absorption properties matching the semiring structure (Gokavarapu et al., 4 Nov 2025, Gokavarapu et al., 18 Nov 2025). Schur-type lemmas, exactness, monoidal closure (tensor-Hom adjunctions), and derived functors ($\operatorname{Ext}$, $\operatorname{Tor}$) all extend to the TGS framework.

Supports and annihilators of modules yield a bijection between simple modules and primitive ideals, supporting dualities between submodule lattices and the spectral topology (Gokavarapu et al., 4 Nov 2025).

5. Computational Classification and Small Models

Finite commutative ternary $\Gamma$-semirings with $|T| \leq 4$ and $|\Gamma| \leq 2$ have been classified via constraint-driven enumeration, automorphism analysis, and canonical labeling (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 15 Nov 2025). This confirms structural predictions and subvariety distinctions.

$|T|$	$|\Gamma|$	# Isomorphism Classes	Representative Examples
2	1	1	Boolean-idempotent, group-like
3	1	2–4	Cyclic, truncated, tropical
4	2	4–24	Modular, Boolean, hybrid types

The enumeration algorithm exploits closure, distributivity, and symmetry constraints, achieving practical tractability for small orders (Gokavarapu et al., 15 Nov 2025).

6. Applications and Mathematical Significance

The TGS framework fundamentally generalizes classical semirings and $\Gamma$-rings to systems where operations are triadic and parameter-indexed, with applications including:

• Chemical Systems: Ternary $\Gamma$-semirings encode multi-step, multi-parameter chemical transformations where reactants, intermediates, and mediators all feature as intrinsic arguments of the transformation law. Distributivity and associativity express parallelism and path-independence; chemical ideals model reaction-closed and pathway-stable domains (Gokavarapu et al., 16 Nov 2025, Gokavarapu et al., 17 Nov 2025).
• Multi-objective Optimization: The ternary tropical $\Gamma$-semiring enables modeling and optimization of network systems with genuinely ternary cost dependencies—capturing non-separable three-way criteria that cannot be factorized into pairwise combinations. A ternary analogue of Bellman-Ford achieves $O(n^2m)$ complexity (Gokavarapu et al., 22 Nov 2025).
• Coding Theory: Error-correcting codes constructed from the lattice of ideals of finite TGSs yield new code families with decoding mechanisms governed by the ternary product's higher-arity absorption and radical structure, supporting parameter sets unattainable in classical linear or ring-linear codes (Gokavarapu et al., 24 Nov 2025).
• Symbolic Reasoning and Neural Architectures: By replacing binary products with learnable ternary fusion operators, neural ternary semirings enable direct representation and optimization for triadic logical, relational, and knowledge-graph reasoning, backed by algebraic regularizers enforcing approximate TGS axioms (Gokavarapu et al., 21 Nov 2025).
• Algebraic Geometry and Homological Algebra: The categorical and spectral theories for TGSs include the development of $\Gamma$-schemes, sheaf cohomology, derived functors, and dualities, enabling a derived $\Gamma$-geometry that unifies algebra, geometry, and mathematical physics at the triadic level (Gokavarapu et al., 18 Nov 2025).

7. Extensions, Generalizations, and Open Directions

• Noncommutative and $n$-ary Generalizations: The structural and radical theory extends to noncommutative and $n$-ary $\Gamma$-semirings, introducing more general primeness and semiprimeness criteria and yielding Wedderburn–Artin-type decompositions in the higher-arity context (Gokavarapu et al., 18 Nov 2025).
• Fuzzy, Analytic, and Computational Enrichments: Fuzzy TGSs assign membership degrees to elements, yielding fuzzy spectra and enabling topological and analytic enrichments for data- or uncertainty-tolerant models. Computationally, all axioms and structural properties can be verified by enumeration for small finite TGSs, and symbolic software libraries (e.g., TGammaRing) are anticipated (Gokavarapu et al., 4 Nov 2025, Gokavarapu et al., 15 Nov 2025).
• Categorical Synthesis: The category of commutative TGSs admits products, coproducts, adjoint functors, and spectrum functors analogous to classical algebra, supporting functorial extensions and logical semantics (Gokavarapu et al., 15 Nov 2025).

Open problems include prime-avoidance lemmas for $\Gamma$-ideals, Krull-type dimension theory, explicit classification of TGS modules and acts, improvements to isomorphism testing algorithms, and extensions to fuzzy or analytic radicals and their role in the Zariski-type spectral theory (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 27 Oct 2025, Gokavarapu et al., 18 Nov 2025).

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References:

• (Gokavarapu et al., 16 Nov 2025): "Axiomatic Foundations of Chemical Systems as Ternary Gamma-Semirings"
• (Gokavarapu et al., 3 Nov 2025): "Finite Structure and Radical Theory of Commutative Ternary $Γ$-Semirings"
• (Gokavarapu et al., 27 Oct 2025): "Prime and Semiprime Ideals in Commutative Ternary $Γ$-Semirings: Quotients, Radicals, Spectrum"
• (Gokavarapu et al., 17 Nov 2025): "Chemical Systems as Ternary $Γ$-Semirings: Theory, Case Studies, and Operational Tests"
• (Gokavarapu et al., 18 Nov 2025): "Derived $Γ$-Geometry, Sheaf Cohomology, and Homological Functors on the Spectrum of Commutative Ternary $Γ$-Semirings"
• (Gokavarapu et al., 15 Nov 2025): "Computational and Categorical Frameworks of Finite Ternary $Γ$-Semirings: Foundations, Algorithms, and Industrial Modeling Applications"
• (Gokavarapu et al., 4 Nov 2025): "Homological and Categorical Foundations of Ternary $Γ$-Modules and Their Spectra"
• (Gokavarapu et al., 21 Nov 2025): "Ternary Gamma Semirings as a Novel Algebraic Framework for Learnable Symbolic Reasoning"
• (Gokavarapu et al., 22 Nov 2025): "A Ternary Gamma Semiring Framework for Solving Multi-Objective Network Optimization Problems"
• (Gokavarapu et al., 24 Nov 2025): "Construction and Decoding of Error--Correcting Codes from Ideal Lattices of Finite Ternary Gamma Semirings"
• (Gokavarapu et al., 18 Nov 2025): "Structure and Spectral Theory of Non-Commutative and $n$-ary $Γ$-Semirings"