Ternary Γ-Semiring Theory

• Ternary Γ-semiring is an algebraic structure generalizing classical semirings by replacing binary multiplication with a symmetric, Γ-indexed ternary product.
• Its ideal and radical theory extend classical algebra by forming distributive lattices and enabling subdirect decompositions for structural analysis.
• Algorithmic classification and categorical frameworks facilitate applications in coding theory, fuzzy logic, and multi-agent systems.

A commutative ternary $\Gamma$-semiring is an algebraic structure generalizing classical semiring and $\Gamma$-ring frameworks by substituting the binary multiplicative operation with a $\Gamma$-indexed, symmetric ternary product. The resulting structure forms the cornerstone of a rapidly growing theoretical and computational literature, driving developments in ideal theory, radical theory, universal algebra, categorical logic, and multi-agent modeling applications.

1. Formal Structure and Axioms

Let $T$ be a nonempty set, and let $\Gamma$ be a nonempty index (parameter) set. A commutative ternary $\Gamma$-semiring is defined by the triple $(T,+,\{\cdot,\cdot,\cdot\}_\Gamma)$, where:

• $(T,+)$ is a commutative monoid with identity $0$;
• For each $\gamma \in \Gamma$, the map $$\{\cdot,\cdot,\cdot\}_\gamma: T \times T \times T \to T$$ specifies the $\gamma$-labeled ternary product.

The following axioms are satisfied for all $a, b, c, d, e \in T$ and $\alpha, \beta \in \Gamma$ (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 15 Nov 2025):

• (T1) Additive Monoid: $(T,+)$ is a commutative monoid.
• (T2) Distributivity: The ternary product is distributive in each argument:

$$\{a+b, c, d\}_\gamma = \{a, c, d\}_\gamma + \{b, c, d\}_\gamma$$

and analogously in the remaining two slots.

• (T3) Zero Absorption: $0$ is absorbing under ternary multiplication:

$$\{0,a,b\}_\gamma = \{a,0,b\}_\gamma = \{a,b,0\}_\gamma = 0$$

• (T4) Ternary Associativity: Nested products can be rebracketed:

$$\{\{a,b,c\}_\alpha, d, e\}_\beta = \{a, b, \{c, d, e\}_\beta\}_\alpha$$

• (T5) Symmetry: The ternary product is fully symmetric:

$$\{a,b,c\}_\gamma = \{\sigma(a), \sigma(b), \sigma(c)\}_\gamma$$

for every permutation $\sigma \in S_3$.

Homomorphisms between ternary $\Gamma$-semirings preserve both the additive and all ternary structures.

2. Ideal Theory, Congruences, and Lattice Structure

A subset $I \subseteq T$ is a $\Gamma$-ideal if:

1. $I+I \subseteq I$;
2. For any $\gamma\in\Gamma$ and any $x \in I$ (resp.\ any $y$ or $z \in I$), $y, z \in T$,

$\{x, y, z\}_\gamma \in I$

and analogously for the other two slots.

$\Gamma$-ideals form a distributive lattice under $I \wedge J = I \cap J$ and $I \vee J = \langle I \cup J \rangle_\Gamma$, where $\langle S \rangle_\Gamma$ denotes the smallest $\Gamma$-ideal containing $S$. A $\Gamma$-congruence is an equivalence relation $\rho \subset T \times T$ such that

$$a\, \rho\, b,\, c\, \rho\, d,\, e\, \rho\, f \implies \{a, c, e\}_\gamma \, \rho\, \{b, d, f\}_\gamma$$

for all $\gamma \in \Gamma$. There is an inclusion-reversing bijection between $\Gamma$-ideals and $\Gamma$-congruences: $I \mapsto \rho_I,\, \rho \mapsto I_\rho$. The set of $\Gamma$-congruences also forms a lattice (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 27 Oct 2025).

3. Subdirect Decomposition and Radical Theory

Every finite commutative ternary $\Gamma$-semiring admits a unique, up to isomorphism, decomposition into subdirectly irreducible components. Specifically, given all maximal proper $\Gamma$-congruences $\{\rho_i\}_{i\in I}$ on $T$, the canonical map

$T \hookrightarrow \prod_{i\in I} T/\rho_i$

is a subdirect embedding (Gokavarapu et al., 3 Nov 2025). The maximal congruences correspond to maximal proper $\Gamma$-ideals. Each quotient by a maximal $\Gamma$-ideal is simple or subdirectly irreducible.

Radical theory generalizes the classical Jacobson radical and nilpotent structure. For $x \in T$, $x$ is nilpotent if nested products of $x$ with itself (under various $\gamma$-parameters) eventually yield $0$. The nilradical $\Nil(T)$ is the set of all nilpotent elements. The prime radical $\Rad(T)$ is the intersection of all prime $\Gamma$-ideals. In finite settings, $\Nil(T) = \Rad(T)$, and a Wedderburn-type structure theorem holds:

$T \cong \Rad(T) \times (T/\Rad(T))$

where $T/\Rad(T)$ is semisimple (contains no nonzero nilpotents) (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 27 Oct 2025).

Prime and semiprime ideals are characterized by the absence of zero-divisors and stability under intersections, and their lattice-theoretic properties reflect a strict extension of the binary semiring case.

4. Algorithmic and Categorical Frameworks

Constraint-driven enumeration algorithms enable the explicit classification of all non-isomorphic finite commutative ternary $\Gamma$-semirings for $|T| \le 4$ and $|\Gamma| \le 2$. The core algorithm recursively populates all possible ternary product tables subject to closure, distributivity, symmetry, and absorption, prunes partial completions that violate any axiom, and utilizes canonical labeling with automorphism rejection to avoid duplication (Gokavarapu et al., 15 Nov 2025). Complexity for canonical form computation is $O(|T|^3|\,\Gamma\,|)$, and the enumeration algorithm is polynomial in the number of valid completions.

Computational results yield the following counts and representative types (examples from (Gokavarapu et al., 15 Nov 2025, Gokavarapu et al., 3 Nov 2025)):

$|T|$	$|\Gamma|$	Types (examples)	$|\mathrm{Aut}_\Gamma(T)|$
2	1	Boolean	2
3	1	Modular	3
3	2	Mixed idempotent	6
4	1	Truncated	4
4	2	Tropical (max, truncated, cyclic)	8

Categorically, the category $\mathbf{T\Gamma S}$ of commutative ternary $\Gamma$-semirings and $\Gamma$-homomorphisms supports additive and forgetful functors, splits via a split-exact sequence, and admits an adjoint triple structure. The spectrum functor $\Spec_\Gamma(-)$ assigns to each object its Zariski-type prime $\Gamma$-ideal spectrum as a topological space, and there is a full and faithful embedding into the category of computable categorical models (Gokavarapu et al., 15 Nov 2025, Gokavarapu et al., 27 Oct 2025).

5. Ternary $\Gamma$-Modules and Homological Theory

A left ternary $\Gamma$-module over $T$ is an additive commutative monoid $M$ with an action

$$T \times \Gamma \times M \times \Gamma \times T \to M,\quad (a, \alpha, m, \beta, b) \mapsto a_\alpha m_\beta b,$$

satisfying additivity in each argument, compatibility with ternary product, and zero absorption. The annihilator–primitive correspondence links simple modules to primitive ideals.

$\TGMod$, the category of left ternary $\Gamma$-modules, is additive, exact, and monoidal-closed. It supports:

• Tensor products and internal Hom functors
• Standard isomorphism theorems
• Derived functors $\mathrm{Ext}_T^n,\,\mathrm{Tor}_n^T$ defined by projective or injective resolutions
• Long exact sequences and base change spectral sequences

A geometric duality exists between submodules and closed subsets of the prime spectrum $\Spec_\Gamma(T)$, realized as an inclusion-reversing bijection (Gokavarapu et al., 4 Nov 2025). Analytic, fuzzy, and computational generalizations can be carried out via augmentation of the topological and sheaf-theoretic structure of $\Spec_\Gamma(T)$.

6. Applications and Extensions

Ternary $\Gamma$-semirings and their module categories serve as algebraic frameworks for modeling multi-parameter, triadic, or mediated processes. Applications include:

• Coding theory: modeling higher-arity codeword concatenations with parametric weights $\Gamma$
• Fuzzy logic: combining three-valued logic under semantic hedges labeled by $\gamma \in \Gamma$
• Decision systems: simultaneous process interactions in supply, production, and distribution
• Symbolic computation: formalizing three-way rewrite rules and complex transformation dynamics
• Chemical systems: representing multi-step reaction pathways, catalysis, and environmental mediation, with ideals corresponding to closed kinetic subsystems and homomorphisms reflecting consistent changes of chemical environment (Gokavarapu et al., 16 Nov 2025).

Categorical interpretations connect ternary $\Gamma$-semirings to universal algebra, with functorial relationships illuminating adjunctions and internal logics. The Zariski-type topology on the spectrum enables geometric and homological reasoning, paralleling schemes and sheaf theory in algebraic geometry.

7. Classification and Open Directions

All commutative ternary $\Gamma$-semirings of order $|T| \le 4$ and $|\Gamma| \le 2$ are classified up to isomorphism by exhaustive algorithmic enumeration (Gokavarapu et al., 15 Nov 2025, Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 27 Oct 2025). Types include Boolean, modular/cyclic, truncated, tropical, and mixed hybrid forms, with radical and decomposition theorems empirically validated in all small cases.

Emerging directions include:

• Classification for $|T|=5$ and general $|\Gamma|$
• Structural theory of noncommutative ternary $\Gamma$-semirings
• Deep homological studies (projectives, injectives, $\Ext$, $\Tor$)
• Fuzzy, graded, and computational geometries over $\Spec_\Gamma(T)$
• Connections to logic, noncommutative geometry, and machine-learning embeddings of algebraic spectra

These advances position commutative ternary $\Gamma$-semiring theory as a hub for future developments in algebra, category theory, representation theory, and multi-agent systems (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 15 Nov 2025, Gokavarapu et al., 27 Oct 2025, Gokavarapu et al., 4 Nov 2025, Gokavarapu et al., 16 Nov 2025).