N-ary Gamma Semirings Overview

• N-ary Gamma semirings are multi-parameter, multi-ary algebraic structures that generalize classical semirings and Gamma rings via an n-ary multiplication dependent on a semigroup of parameters.
• Their ideal theory distinguishes positional and threshold ideals, unifying prime, semiprime, and radical concepts across commutative and non-commutative cases.
• They exhibit a rich spectral topology and decomposability, establishing a triadic spectral geometry that connects to classical Wedderburn-Artin decompositions in finite settings.

An n-ary Gamma semiring is a multi-parameter, multi-ary algebraic structure generalizing classical semirings and Gamma rings by allowing the multiplicative law to depend on both an arity parameter $n \ge 3$ and a semigroup $\Gamma$ of “parameters” that act as operation indices. The theory encompasses non-commutative, commutative, and higher arity settings, providing a unified foundation for prime/semiprime ideal theory, radical theory, and spectral geometry in polyadic algebraic systems. The central objects of study are quadruples $(T, +, \Gamma, \mu)$, with $(T,+)$ a commutative semigroup with identity, $\Gamma$ an additive semigroup, and $\mu : T^n \times \Gamma^{n-1} \to T$ an $n$-ary Gamma-multiplication satisfying distributivity, zero absorption, and $n$-ary associativity. Their radical and spectral theories naturally generalize those for binary or ternary Gamma semirings and underlie a “triadic” spectral geometry unifying the commutative, non-commutative, and higher-arity cases (Gokavarapu et al., 18 Nov 2025, Gokavarapu et al., 3 Nov 2025).

1. Formal Definition and Basic Structure

Let $(T,+)$ be a commutative semigroup with zero and $\Gamma$ an additive semigroup. For $n\ge 3$, an $n$-ary $\Gamma$-semiring is defined by

$(T, +, \Gamma, \mu)$

where

$\mu : T^n \times \Gamma^{n-1} \to T,$

written as

$$\mu(x_1,\alpha_1,\ldots,\alpha_{n-1},x_n) = [x_1]_{\alpha_1}[x_2]_{\alpha_2}\cdots[x_{n-1}]_{\alpha_{n-1}}[x_n],$$

and satisfying:

• Distributivity: For all $i=1,\ldots,n$, $\alpha_j \in \Gamma$, $x_k,y_k \in T$,

$$\mu(\ldots, x_i+y_i, \ldots) = \mu(\ldots, x_i, \ldots) + \mu(\ldots, y_i, \ldots)$$

• Zero absorption: If any $x_i=0$ then $\mu(x_1,\ldots,x_n)=0$.
• $n$-ary associativity: All legal iterates of $\mu$ give the same outcome; substitution of $\mu$-values into any slot yields results independent of evaluation order.

Commutative $n$-ary $\Gamma$-semirings further require symmetry of $\mu$ in the $T$-slots; the non-commutative theory omits this.

For $n=3$, this recovers the commutative ternary $\Gamma$-semirings studied in (Gokavarapu et al., 3 Nov 2025).

2. Ideal Theory: Positional and $(n,m)$-Ideals

Ideals in $n$-ary $\Gamma$-semirings are indexed both by position and by the “threshold” $m$ counting slots required for closure. For non-commutative or higher-arity cases:

Positional Ideals $((n,S)$-ideals):

Let $S\subseteq \{1,\ldots,n\}$ with $S\neq \emptyset$. A subset $I\subseteq T$ is an $(n,S)$-ideal if

$$(I,+) \le (T,+) \quad\text{and}\quad x_i\in I\;\forall\,i\in S \Rightarrow \mu(x_1,\ldots,x_n)\in I$$

for all $x_j\in T,\,\alpha_k\in\Gamma$.

For $n=3$:

• Left ideals $\approx (n,\{2\})$
• Right ideals $\approx (n,\{3\})$
• Two-sided ideals $\approx (n,\{2,3\})$

Threshold Ideals $((n,m)$-ideals):

Given $1\le m\le n$, a nonempty $I\subseteq T$ is an $(n,m)$-ideal if $(I,+)$ is a subsemigroup and

$$\#\{i : x_i \in I\} \geq m \implies \mu(x_1,\ldots,x_n) \in I,$$

for all $x_j\in T,\, \alpha_k\in\Gamma$.

• Intersections and sums of $(n,m)$-ideals remain $(n,m)$-ideals.
• Every $(n,m)$-ideal is the intersection of positional $(n,S)$-ideals with $|S|=m$.

This refinement is fundamental in stratifying the lattice of ideals by arity and closure thresholds (Gokavarapu et al., 18 Nov 2025).

3. Prime, Semiprime Ideals and The Radical Theories

Prime and semiprime notions generalize as follows.

n-ary Primes:

A proper $(n,1)$-ideal $P$ is $n$-ary prime if

$$\mu(x_1,\alpha_1,\ldots,x_n)\in P \implies x_i\in P \text{ for some }i,$$

equivalently,

$$\mu(\bar{x}_1,\ldots,\bar{x}_n)=\bar{0} \implies \bar{x}_i=\bar{0} \text{ for some }i$$

in $T/P$.

n-ary Semiprimes:

A two-sided ideal $Q$ is $n$-ary semiprime if

$$\Delta_n(a;\alpha_1,\ldots,\alpha_{n-1}) := \mu(\underbrace{a,\alpha_1,a,\ldots,a}_{n~\text{copies}}) \in Q \implies a \in Q.$$

$Q$ is semiprime iff $Q = \sqrt[n,\Gamma]{Q}$ where: $$\sqrt[n,\Gamma]{I} = \bigcap_{P \supseteq I~P~n\text{-ary prime}} P = \{ a : \Delta_n(a;\vec{\alpha}) \in I ~\text{for some } \vec{\alpha} \}$$

(Gokavarapu et al., 18 Nov 2025).

$\Gamma$-Jacobson Radical:

Let $\mathcal{M}_n$ be the modular maximal two-sided ideals of $T$. The $n$-ary $\Gamma$-Jacobson radical is: $$J_{\Gamma}^{(n)}(T) = \bigcap_{M \in \mathcal{M}_n} M$$ Properties:

• $J_{\Gamma}^{(n)}(T)$ is two-sided and $n$-ary semiprime.
• $J_{\Gamma}^{(n)}(T)=0$ iff $T$ is $n$-ary $\Gamma$-semisimple.
• If all maximals are $n$-ary prime, $J_{\Gamma}^{(n)}(T)$ is the intersection of all maximals (Gokavarapu et al., 18 Nov 2025).

In the finite commutative case, the radical equals the set of nilpotents: $\Rad(T) = \Nil(T)$ (Gokavarapu et al., 3 Nov 2025).

4. Spectral Topology, Triadic Geometry, and Decomposition

A spectral topology emerges by associating prime ideals with points in a compact $T_0$ space.

For each type $\eta \in \{L,R,2\}$ (Left, Right, Two-sided),

$\Spec_\eta(T) = \{\text{%%%%75%%%%-prime ideals}\}$

The closed sets are $V_\eta(A) = \{P \in \Spec_\eta(T): A \subseteq P\}$, and $D_\eta(A) = \Spec_\eta(T) \setminus V_\eta(A)$. This collection forms a compact $T_0$ topology.

Key properties:

• $V_\eta(0) = \Spec_\eta(T)$, $V_\eta(T) = \varnothing$.
• $V_\eta(A) \cap V_\eta(B) = V_\eta(A \cup B)$.
• $V_\eta(I) = V_\eta(\sqrt[\Gamma,\eta]{I})$.
• The closure of $\{P\}$ is $V_\eta(P)$, so

$$\sqrt[\Gamma,\eta]{I} = \bigcap_{P\supseteq I} P \iff V_\eta(I) = \overline{\{P : P\supseteq I\}}$$

When $T$ is finite and $J_\Gamma(T)=0$, a Wedderburn-Artin type decomposition holds: $T \cong \prod_{i=1}^s T/P_i,$ with $P_i$ minimal primitive ideals, each $T/P_i$ acting faithfully on its simple $n$-ary module.

The two-sided spectrum $\Spec_2(T)$ forms a discrete set of $P_i$, while $\Spec_L(T)$ and $\Spec_R(T)$ define “boundary faces”, with $\Spec_2(T)$ embedded “triadically” between them: a triadic spectral geometry (Gokavarapu et al., 18 Nov 2025).

5. Examples and Explicit Constructions

Matrix-entry semiring: Let $T = M_2(\mathbb{N}_0)$, $\Gamma=\{1\}$ with $a_1b_1c = a + b + c$ entrywise. Left ideals correspond to forcing zeros in the first row, right ideals in the last column.

Pinning/Reduction to Lower Arity: If $T$ has a central idempotent $e$ satisfying $\Delta_n(e;\vec{\alpha})=e$, “pinning” $n-3$ slots to $e$ reduces to ternary $\Gamma$-semiring structure; all $n$-ary ideals/radicals restrict to the ternary case.

Finite Toy Example: For $T=\{0,a,b\}$, $\Gamma=\{\alpha\}$: $a+a = b,\quad b+b = b,\quad 0 \text{-absorption}$ and

$$\mu(x,y,z)_\alpha = \begin{cases} 0, & \text{if any argument is }0\ b, & \text{if }x=y=z=a\ a, & \text{otherwise} \end{cases}$$

Two maximal two-sided ideals $I_1 = \{0,a\}$, $I_2 = \{0,b\}$, with $J_\Gamma(T) = I_1 \cap I_2 = \{0\}$ and $T \cong T/I_1 \times T/I_2$ (Gokavarapu et al., 18 Nov 2025).

6. Connections to Lower Arity and Unification

• When $n=3$ and $\mu$ is symmetric, the theory specializes to commutative ternary $\Gamma$-semirings (Gokavarapu et al., 3 Nov 2025).
• As $n\to2$, it recovers the Nobusawa–Barnes $\Gamma$-ring/semiring framework.
• The threshold invariants $\tau(I)$ index ideals by minimal slot occupancy for closure, refining the description of the ideal lattice.
• The triadic spectrum $(\Spec_L,\Spec_R,\Spec_2)$ reflects the interface between left, right, and two-sided primeness and encodes non-commutative geometric data (Gokavarapu et al., 18 Nov 2025).

7. Generalizations, Extensions, and Classification

All major structural and ideal-theoretic results for finite commutative ternary $\Gamma$-semirings admit generalization to arbitrary $n$:

• For $n$-ary $\Gamma$-semirings, the ideal lattice remains modular and distributive when $T$ is finite.
• Subdirect decomposition by maximal proper congruences persists.
• Radical theory and the ideal-radical correspondence generalize, with $\Rad(T)=\Nil(T)$ in the finite commutative case.
• Classification for small orders uses enumeration over commutative monoid and $n$-ary multiplication tables, subject to distributivity, zero absorption, and associativity (yielding, for example, 3 structures for $|T|=4$, $|\Gamma|=1$) (Gokavarapu et al., 3 Nov 2025).

This unification delivers a spectral and radical framework encompassing binary, commutative ternary, and all higher-arity $\Gamma$-semirings, organizing them into a single triadic geometric and algebraic structure (Gokavarapu et al., 18 Nov 2025, Gokavarapu et al., 3 Nov 2025).