Finite Ternary Γ-Semirings

• Finite ternary Γ-semirings are finite algebraic structures combining a commutative monoid with a family of ternary operations that satisfy distributivity, symmetry, and associativity.
• They exhibit complex ideal and congruence lattices, subdirect decompositions, and radical theories that extend classical semiring frameworks through unique ternary phenomena.
• Enumeration algorithms and categorical approaches enable precise classification and practical applications in optimization, coding theory, and decision modeling.

A finite ternary $\Gamma$-semiring is an algebraic structure generalizing both binary semirings and $\Gamma$-rings by replacing the multiplication with a family of parametrized ternary operations. These objects are defined on a finite set, equipped with a commutative monoid addition and a collection of ternary products indexed by a finite parameter set $\Gamma$. Structural results, computational classification, and categorical organization of finite ternary $\Gamma$-semirings demonstrate robust parallels to the classical theory of finite semirings while introducing distinct phenomena inherent to the ternary and multi-parameter setting (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 15 Nov 2025, Gokavarapu et al., 27 Oct 2025).

1. Formal Definition and Core Axioms

Let $T$ be a finite set, $\Gamma$ a finite parameter set, $+:T\times T\to T$ a binary operation with identity $0\in T$, and $\{\cdot,\cdot,\cdot\}_\gamma:T^3\to T$ a family of ternary operations indexed by $\gamma\in\Gamma$. Then $(T,+,0,\{\cdot,\cdot,\cdot\}_\Gamma)$ is a finite commutative ternary $\Gamma$-semiring if the following axioms hold for all $a,b,c,d,e\in T$ and all $\gamma,\alpha,\beta\in\Gamma$:

• (T1) Commutative Monoid: $(T,+,0)$ is a commutative monoid: $a+b = b+a$, $(a+b)+c = a+(b+c)$, $a+0 = a$.
• (T2) Distributivity: Each ternary product is additive in every argument:

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

and similarly for the second and third coordinates.

• (T3) Absorbing Zero: $$\{0,a,b\}_\gamma = \{a,0,b\}_\gamma = \{a,b,0\}_\gamma = 0$$.
• (T4) Ternary Associativity: For all $\alpha,\beta\in\Gamma$,

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

• (T5) Symmetry (Ternary Commutativity): Each $\{a,b,c\}_\gamma$ is invariant under permutations of $(a,b,c)$.

A $\Gamma$-homomorphism is a function $f:T\to T'$ preserving $+$ and each ternary product: $f(a+b) = f(a)+f(b)$, $f(\{a,b,c\}_\gamma) = \{f(a),f(b),f(c)\}_\gamma$ for all $\gamma\in\Gamma$ (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 15 Nov 2025, Gokavarapu et al., 27 Oct 2025).

2. Ideal, Congruence, and Lattice Structure

A subset $I\subseteq T$ is a $\Gamma$-ideal if (i) $I+I\subseteq I$ and (ii) $\{I,T,T\}_\Gamma\subseteq I$. Let $\Id(T)$ denote the set of all $\Gamma$-ideals ordered by inclusion. For $I,J\in\Id(T)$,

$$I\vee J = \langle I\cup J \rangle_\Gamma,\quad I\wedge J = I\cap J.$$

$\Id(T)$ thus forms a finite distributive (modular) lattice.

Congruences on $T$ are in bijection with $\Gamma$-ideals via the order-reversing maps:

$$I \mapsto \rho_I = \{(a,b):\forall c,\,\forall \gamma,\, \{a,b,c\}_\gamma \in I\},\qquad \rho \mapsto I_\rho = \{a: a\,\rho\,0\}$$

Therefore, the congruence lattice $\Con(T)$ is anti-isomorphic to $\Id(T)$.

Small ternary $\Gamma$-semirings exhibit $\Gamma$-ideal lattice types such as one-point chains (simple case), $M_3$ diamonds, and Boolean squares, confirming a richer lattice landscape than in the binary semiring case. For finite $T$, all ideals, congruences, and their lattices can be algorithmically computed (Gokavarapu et al., 3 Nov 2025).

3. Subdirect Decomposition and Semisimplicity

Every finite commutative ternary $\Gamma$-semiring admits a unique (up to isomorphism) subdirect decomposition into subdirectly irreducible quotients. Let $\{\rho_i:i\in I\}$ be the maximal proper $\Gamma$-congruences of $T$. Then $T$ embeds subdirectly into $\prod_{i\in I}T/\rho_i$, and this embedding is injective since $\bigcap_i\rho_i = \Delta_T$ (the diagonal).

If each $T/\rho_i$ is simple, then $T$ is semisimple. The semisimple case further admits a Wedderburn-Artin type structure theorem: any finite commutative ternary $\Gamma$-semiring with radical $J(T)=0$ decomposes as a direct sum of simple quotients over its minimal prime ideals (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 27 Oct 2025).

4. Radical and Prime Ideal Theory

An element $x\in T$ is nilpotent if, for some $a_1,\dots,a_k\in T$ and $\gamma_1,\dots,\gamma_k\in\Gamma$,

$$\{\dots\{x,x,a_1\}_{\gamma_1},x,a_2\}_{\gamma_2}\dots,a_k\}_{\gamma_k} = 0.$$

The nil-radical $\Nil(T)$ is the set of nilpotent elements. A prime ideal $P$ satisfies: if $\{a,b,c\}_\gamma \in P$, then $a\in P$ or $b\in P$ or $c\in P$. The prime radical (intersection of all prime ideals) coincides with $\Nil(T)$ in the finite case:

$\Rad(T) = \Nil(T).$

For an ideal $P$, $P$ is prime iff $T/P$ has no nonzero zero-divisors under the ternary $\Gamma$-operation. Semiprime ideals are stable under intersections and coincide with their radicals. Every maximal ideal is prime; every prime is semiprime (Gokavarapu et al., 27 Oct 2025).

The Wedderburn-type decomposition applies: letting $R = \Rad(T)$ and $S = T/R$, it follows that $S$ is semisimple and $T \cong R\times S$. $\Gamma$-ideals, congruences, and radical theory parallel much of the classical semiring framework, but the ternary setting displays unique behaviors such as nontrivial Jacobson radicals and nontrivial intersection patterns for maximal ideals in small orders (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 27 Oct 2025).

5. Classification and Examples for Orders $\lvert T\rvert \le 4$

Complete classification for $\lvert T\rvert \le 4$ and $|\Gamma| \le 2$ has been obtained via constraint-driven enumeration algorithms. Table 1 summarizes the number of non-isomorphic structures:

$|T|$	$|\Gamma|$	# Structures	Dominant Type
2	1	1	Boolean idempotent
3	1	2	Modular vs. truncated
3	2	4	Mixed actions
4	1	3	Truncated/cyclic hybrids
4	2	4	Boolean, tropical, hybrid

Canonical examples:

• $|T|=2, |\Gamma|=1$ (Boolean): $T = \{0,1\}$, $+$ and ternary product $\min$ (or addition modulo $2$).
• $|T|=3, |\Gamma|=1$ (Modular): $T = \{0,1,2\}$, $a+b \equiv a+b \pmod{3}$, $\{a,b,c\}_\gamma = a+b+c \pmod{3}$.
• $|T|=4, |\Gamma|=1$ (Truncated min–max): $T = \{0,1,2,3\}$, $+$ as $\max$, ternary product as $\min$.
• $|T|=4, |\Gamma|=2$ (Hybrid): one ternary law as $\min(a+b+c,3)$, another as $\max(a,b,c)$.

Order 4 introduces phenomena absent in lower orders, such as the median law and saturating (floor/ceil-average) laws, resulting in nontrivial prime ideal spectra and radical structure (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 15 Nov 2025, Gokavarapu et al., 27 Oct 2025).

6. Enumeration Algorithms and Computational Aspects

Enumeration proceeds by fixing $n=|T|$, $g=|\Gamma|$, and iterating through all possible additive monoid structures and families of ternary operation tables. Constraint-driven recursive algorithms generate all non-isomorphic models, verifying all axioms with polynomial or brute-force complexity dependent on $n$ and $g$. For $n\leq 4$, $g \leq 2$, classification is completed in seconds to minutes, and canonical labeling ensures unique representatives (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 15 Nov 2025).

Let $C(n,g)$ be the number of valid completions. The enumeration runs in expected time $O(C(n,g)\,n^3g)$, worst-case $O(n^{3g+3})$. The isomorphism checking (canonical labeling) requires $O(n^3|\Gamma|)$ (Gokavarapu et al., 15 Nov 2025). These algorithms have enabled comprehensive classification and precise automorphism group determination for all small structures.

7. Categorical and Topological Framework

Finite commutative ternary $\Gamma$-semirings with $\Gamma$-homomorphisms form a category $\mathbf{T\Gamma S}_{\mathrm{fin}}$. Key categorical properties include:

• The existence of forgetful functors to commutative semigroups and to ternary semirings, both admitting adjoint functors.
• Wedderburn-type decompositions yield a categorical equivalence:

$$\mathbf{T\Gamma S}_{\mathrm{fin}} \simeq \mathbf{T\Gamma S}_{\mathrm{nil}} \times \mathbf{T\Gamma S}_{\mathrm{ss}}$$

where the nilpotent and semisimple parts split (Gokavarapu et al., 3 Nov 2025, Gokavarapu et al., 15 Nov 2025).

A Zariski-like spectral functor is defined $\Spec_\Gamma: \mathbf{T\Gamma S}_{\mathrm{fin}}^{\mathrm{op}} \to \mathbf{Top}$ by associating to $T$ its space of prime ideals, equipped with closed sets $V(I) = \{ P : I \subseteq P \}$, making $\Spec_\Gamma(T)$ a finite $T_0$-space. Semisimple finite quotients yield discrete spectra (Gokavarapu et al., 27 Oct 2025).

Applications: Finite ternary $\Gamma$-semirings are observed in supply-chain optimization, multi-parameter decision modeling, fuzzy and tropical coding theory, and automated configuration search, leveraging their explicit computational models and categorical tractability (Gokavarapu et al., 15 Nov 2025).

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References:

• (Gokavarapu et al., 3 Nov 2025): "Finite Structure and Radical Theory of Commutative Ternary $\Gamma$-Semirings"
• (Gokavarapu et al., 15 Nov 2025): "Computational and Categorical Frameworks of Finite Ternary $\Gamma$-Semirings: Foundations, Algorithms, and Industrial Modeling Applications"
• (Gokavarapu et al., 27 Oct 2025): "Prime and Semiprime Ideals in Commutative Ternary $\Gamma$-Semirings: Quotients, Radicals, Spectrum"