Ternary G-Products: Algebraic & Geometric Insights

• Ternary G-products are fully symmetric ternary operations that generalize classical multiplication by preserving invariance under all permutations of input values.
• They model key geometric phenomena like discrete volume counts in hexagonal lattices and underpin structural analyses in groupoids, semirings, and graded algebras.
• Their algebraic properties enable efficient algorithms for 3-primality testing and factorization, linking arithmetic geometry with combinatorial and computational methods.

A ternary G-product is a fully symmetric ternary operation generalizing classical multiplication, appearing in diverse algebraic, geometric, and computational frameworks. These operations arise in lattice point enumeration, the arithmetic of unique factorization domains, semiring theory, cubic-graded algebras, and the structure theory of groupoids and inverse semigroups. Canonical constructions of ternary G-products include discrete volume counts in lattices, formal algebraic operations in ternary semirings, S₃-classified cubic products in graded algebras, and naturally associated operations in groupoid theory. The broad applicability of ternary G-products connects them to primality notions, spectral geometry, higher arity homological algebra, combinatorial optimization, coding/information theory, and logical inference on fuzzy sites.

1. Canonical and Geometric Models of the Ternary G-Product

The archetypal ternary G-product, as introduced in the context of arithmetic, encodes the discrete volume of an equiangular hexagon in the hexagonal lattice. Given positive integers $a,b,c$, the product is defined as the number of lattice points inside a hexagon aligned along three non-collinear lattice directions with opposite side lengths $a-1, b-1, c-1$ (Bingham, 2020). This operation is totally symmetric under any permutation of the arguments:

$$\langle a, b, c\rangle = a b + b c + c a - (a + b + c) + 1 = a b c - (a-1)(b-1)(c-1)$$

When a single argument is 1, the hexagon degenerates to a parallelogram, recovering ordinary multiplication. This construction supplies a geometric ground for ternary "multiplication" beyond binary operations and gives rise to arithmetic phenomena and classification of "3-primality." The full symmetry reflects the underlying $D_6$ symmetry of the hexagonal lattice.

Analogous ternary products arise in algebraic settings as natural or standard ternary operations on groupoids and inverse semigroups, e.g., $[x y z] = (x \cdot y) \cdot z$ or $\{x y z\} = x \cdot y^{-1} \cdot z$ (Monzo, 2015). In $Z_3$-graded Grassmann algebra, ternary G-products are defined via cyclic $S_3$-skew commutation rules, with relations such as $x^a x^b x^c = j\, x^b x^c x^a$ for $j = e^{2\pi i/3}$ (Abramov et al., 2015).

2. Algebraic and Structural Properties

The ternary G-product in the arithmetic context satisfies the following algebraic identities for all $x, y, z$:

• Total commutativity: $\langle x, y, z \rangle$ is invariant under all permutations of $(x, y, z)$.
• Binary reduction: $\langle 1, y, z \rangle = y z$ (degeneration to binary multiplication).
• Identity: $\langle 1, 1, n\rangle = n$ for any $n$.

In the abstract setting of commutative ternary $\Gamma$-semirings, a ternary G-product is a parametric ternary operation $\{-,-,-\}_{\gamma}$ indexed by $\gamma\in\Gamma$, subject to the following axioms (Gokavarapu et al., 25 Dec 2025):

$$\begin{aligned} & \text{(T1) Additivity:} && (T, +) \text{ is a commutative semigroup with } 0 \ & \text{(T2) Parametric product:} && \{-, -, -\}_{\gamma}: T^3 \to T,\, \forall\,\gamma\in\Gamma \ & \text{(T3) Distributivity:} && \{a+b, c, d\}_{\gamma} = \{a, c, d\}_{\gamma} + \{b, c, d\}_{\gamma} \ & && \{a, b, c+d\}_{\gamma} = \{a, b, c\}_{\gamma} + \{a, b, d\}_{\gamma} \ & \text{(T4) Ternary associativity:} && \{\{a, b, c\}_{\gamma_1}, d, e\}_{\gamma_2} = \{a, \{b, c, d\}_{\gamma_1}, e\}_{\gamma_2} \ & && = \{a, b, \{c, d, e\}_{\gamma_1}\}_{\gamma_2} \ & \text{(T5) Parameter compatibility:} && \{\{a, b, c\}_{\gamma_1}, d, e\}_{\gamma_2} = \{ a, b, c \}_{\gamma_1 \gamma_2 } \ & \text{(T6) Zero absorption:} && \{0, a, b\}_\gamma = \{a, 0, b\}_\gamma = \{a, b, 0\}_\gamma = 0 \end{aligned}$$

A finite example: $T = \{0,1,2\}$, $\Gamma = \{1,2\}$ with addition and product modulo 3, and $\{a, b, c\}_\gamma = \gamma(a + b + c) \bmod 3$ (Gokavarapu et al., 25 Dec 2025).

In graded algebras, the structural variety is captured by various S₃-symmetry classes, with ternary G-products classified as commutative or anticommutative (Λ-type), determined by cyclic, full, or j-skew symmetry under the symmetric group action (Abramov et al., 2015).

3. Primality, Factorization, and Algorithmic Aspects

A natural number $n \geq 2$ is termed "3-prime" if it cannot be decomposed as $\langle x, y, z\rangle$ with at least two arguments exceeding 1, except for the trivial $\langle1,1,n\rangle$ form (Bingham, 2020). The full list of 3-primes is exceptionally sparse:

$1,\;2,\;3,\;5,\;11,\;17,\;41$

These correspond precisely to the discriminants of imaginary quadratic fields of class number one (Heegner numbers), establishing a direct link between arithmetic geometry and the structure of the ternary G-product. The scarcity of 3-primes reflects the exceptional uniqueness of factorization in the corresponding ring of integers.

Algorithms for 3-primality and 3-factorization (Bingham, 2020):

• Ternary-sieve enumeration: A super-linear $O(N^{3/4})$ combinatorial method traverses a Sieve-of-Eratosthenes augmented for ternary operation.
• Ternary factorization: For each admissible $k$, compute $M = n + 2T_{k-1}$, factor $M$ as $a \cdot b$, and check if $a, b > k-1$ to obtain $(x, y, z) = (a-(k-1), b-(k-1), k)$ such that $n = \langle x, y, z \rangle$.

For Γ-semiring models, primality testing proceeds by enumerating ideals and verifying the implication $$\{a, b, c\}_\gamma \in I \implies a \in I \vee b \in I \vee c \in I$$. Complexity for finite $T$ with $|T| = n$: $O(n^{3|\Gamma|})$, improved in practical cases (Gokavarapu et al., 25 Dec 2025).

4. Spectrum, Topological, and Homological Frameworks

Ternary G-products enable construction of enriched algebraic geometry and cohomology. For a commutative ternary Γ-semiring $T$, the spectrum $\operatorname{Spec}_\Gamma(T)$ comprises all prime ideals, equipped with a Zariski-type topology generated by basic opens $$D(I) = \{ \mathfrak{p} \mid I \nsubseteq \mathfrak{p}\}$$.

The structure presheaf $\mathcal{O}_T$ assigns to each $D(I)$ its ternary localization $S_I^{-1}T$, resulting in a ringed space $(\operatorname{Spec}_{\Gamma}(T),\mathcal{O}_T)$ in which all ternary operations remain valid (Gokavarapu et al., 25 Dec 2025).

The covering notion on this spectrum is extended to a weighted (fuzzy) Grothendieck topology. For a covering family $\{D(I_\alpha)\}$, assign weights $w_\alpha \in (0,1]$ such that $\sum_\alpha w_\alpha \geq 1$. Sheafification and stalk localizations proceed analogously to the classical case but accommodate fuzzy data and weighted supports, crucial for applications in logic and data science.

Cohomological invariants, derived functors $\operatorname{Ext}$ and $\operatorname{Tor}$, as well as the fuzzy cohomology groups $H^n_w(X, \mathcal{F})$, are defined on this weighted site. Comparison theorems establish perfect correspondences between cohomological supports, Schur-density loci, and radical/primitive spectral strata. Ext and Tor can be computed via projective/injective resolutions; localization is exact and commutes with these functors (Gokavarapu et al., 25 Dec 2025).

5. S₃-Symmetry, Graded G-Products, and Combinatorial Classification

The symmetry group $S_3$ underlies the classification of ternary G-products in $Z_3$-graded and $Z_2 \oplus Z_3$-graded algebras. Four classes arise in both the commutative (S-type) and anticommutative (Λ-type/Grassmann-type) categories:

• Commutative: S (full cyclic with $j$-weights), $\bar S$ (conjugate), $S_1$ (weak cyclic), $S_0$ (strong, all permutations equal)
• Anticommutative: Λ₀ (full S₃ antisymmetry), Λ₁ (cyclic antisymmetry), Λ ($j$-skew), $\bar \Lambda$ ($j^2$-skew)

In the physically relevant $Z_3$-graded Grassmann algebra, the canonical ternary G-product is chosen among the $j$-skew types, such as $x^a x^b x^c = j x^b x^c x^a$. All symmetry classes admit concise cubic or quartic defining relations, and higher power monomials vanish due to associativity and S₃-symmetry (Abramov et al., 2015).

6. Applications, Computational Methods, and Open Directions

Ternary G-products are foundational in areas including fuzzy logic (as graded many-valued connectives), coding theory (via ternary linear codes with minimum distance linked to $\operatorname{Tor}_1$), and the algebraic topology of computational spectra (where Laplacian spectra encode cohomology ranks of weighted knowledge-graph embeddings) (Gokavarapu et al., 25 Dec 2025). Cohomological invariants arising from ternary G-products are leveraged in data science for topological data analysis and structural learning in multi-relational frameworks.

The global algorithmic pipeline for finite ternary Γ-semiring models proceeds by enumerating ternary algebraic structures, computing spectra, Ext and Tor, assigning fuzzy weights, and outputting algebraic "fingerprints." Global computational complexity is $O(n^{5|\Gamma|})$ time and $O(n^{3|\Gamma|})$ space, with fuzzy closure and cohomology groups efficiently computable by linear algebra on finite injective resolutions.

Open problems and research directions include developing derived tensor categories for higher-monad structures, extending the framework to noncommutative or graded ternary $\Gamma$-semirings, exploring topos-theoretic generalizations of fuzzy sites, analyzing computational complexity phase transitions, and integrating these techniques into industrial, reliability, and AI knowledge-graph systems (Gokavarapu et al., 25 Dec 2025).

7. Connections to Groupoid Theory and Ternary Varieties

Ternary G-products naturally arise as the "natural" ternary operation on groupoids: $[x y z] = (x\cdot y)\cdot z$, yielding a ternary algebra structure. In right-modular groupoids with left identity $\ell$, this operation is laterally commutative and bi-unital, and the class of such ternary algebras (semiheaps) is categorically equivalent to the class of right-modular groupoids with left identity. Inverse semigroups admit both natural and standard ternary products, yielding algebraic varieties of signature $(3,1,1)$, with characteristic heap/parapara-associative identities and full compatibility with inverse and idempotent unary operations (Monzo, 2015).

The ternary G-product thus serves as both an invariant of and a classifier for algebraic structures (groupoids, semigroups) and underpins a broader variety theory for higher arity operations in abstract algebra. The full data of the groupoid is recoverable from the structure of its associated ternary G-product. This categorical correspondence is foundational for the analysis of semilinear and heap-like algebraic systems.