Polyadic Rings: Higher-Arity Algebra

• Polyadic rings are higher-arity generalizations of classical rings, replacing binary operations with totally associative m-ary and n-ary operations.
• They rely on arity-shape invariants and Diophantine closure conditions, resulting in a rich classification scheme with applications in cryptography and coding theory.
• Their construction using direct products and polyadization extends traditional algebraic frameworks to p-adic, algebraic, and quantum domains.

A polyadic ring is a higher-arity generalization of classical ring theory, where addition and multiplication are replaced by totally associative, distributive operations of arity $m \geq 2$ (“$m$-ary addition”) and $n \geq 2$ (“$n$-ary multiplication”). Polyadic rings have emerged as a foundational platform for new constructions in algebra, number theory, coding theory, and more recently, cryptography, due to the combinatorial complexity and “arity shape” phenomena that are absent in the binary case. Their structure theory relies on arity-shape invariants and Diophantine closure conditions, giving rise to rich classification types and applications beyond ordinary ring-theoretic frameworks.

1. Definition and Fundamental Properties

A commutative polyadic $(m,n)$-ring $$\mathcal{R}_{m,n} = \langle X \mid \nu_m, \mu_n \rangle$$ consists of a set $X$ equipped with:

• an $m$-ary addition $\nu_m: X^m \to X$, making $(X, \nu_m)$ a commutative $m$-ary group: totally associative, commutative (symmetric under $S_m$), with a unique querelement (a generalized inverse) for each $x$;
• an $n$-ary multiplication $\mu_n: X^n \to X$, making $(X, \mu_n)$ an associative $n$-ary semigroup;
• polyadic distributivity: $\mu_n$ distributes over $\nu_m$ in each slot.

Closure conditions in concrete settings, especially for integer representatives in a single congruence class $[[a]]_b$, enforce the existence of integer shape invariants $I^{(m)} = (a m - a)/b$ and $J^{(n)} = (a^n - a)/b$; only when these are integers does the structure close as a polyadic ring on $[[a]]_b$ (Duplij et al., 14 Dec 2025, Duplij, 2017, Duplij, 2022). Polyadic rings are called nonderived when neither operation decomposes into compositions of lower-arity (e.g., binary) operations.

Table 1. Polyadic ring operations and closure criteria

Operation	Formal type	Closure criterion
$\nu_m$	$X^m \to X$	$I^{(m)}(a,b) \in \mathbb{Z}$
$\mu_n$	$X^n \to X$	$J^{(n)}(a,b) \in \mathbb{Z}$

The polyadic ring axioms generalize the binary case, recovering ordinary rings when $m=n=2$.

2. Classification via Arity-Shape and Parameter-to-Arity Maps

The classification of polyadic rings, especially over the integers, is controlled by congruence classes and their admissible arities through a parameter-to-arity map $\Phi(a,b) = (m,n)$. This map is:

• non-injective (different $(a,b)$ may yield the same $(m,n)$);
• non-surjective (not every $(m,n)$ occurs);
• multi-valued (many $(m,n)$ may arise for fixed $(a,b)$).

Given a congruence class $[[a]]_b$, admissible $m$ and $n$ are given by

$$m = 1 + u \cdot \frac{b}{\gcd(a,b)}, \quad n = 1 + v \cdot \mathrm{ord}_{b/\gcd(a,b)}(a \cdot \gcd(a,b)),$$

for $u, v \in \mathbb{N}$, where $\mathrm{ord}_g(x)$ denotes the multiplicative order (Duplij et al., 14 Dec 2025). This combinatorial “arity-shape” complexity is central for applications, especially cryptographic security.

Examples:

• $(a,b) = (2,3)$: $m \equiv 1 \mod 3$, $n$ odd, so $(m,n) = (4+3u, 3+2v)$;
• $(a,b) = (1,2)$: $m$ odd, every $n\ge2$ works.

3. Structure: Identities, Inverses, and Polyadic Arithmetic

Every commutative $m$-ary group $(R, \nu_m)$ has a unique neutral element $e_+$, characterized by $\nu_m(e_+, \ldots, e_+, x) = x$. Each $x$ has a unique querelement $q(x)$ such that $\nu_m(x, \ldots, x, q(x)) = e_+$. For $(R, \mu_n)$, the existence of a multiplicative unit $e_{\times}$ is not automatic and may require additional constraints on $(a, b, n)$ (Duplij et al., 14 Dec 2025, Duplij, 2017). Polyadic rings can be zeroless (no $0$ element), nonunital, or possess multiple or even all units — phenomena impossible in binary rings (Duplij, 2017).

For division, quotient and remainder in the polyadic context involve $m$-ary and $n$-ary group laws, with division satisfying

$$x = \nu_m\left(\mu_n(y, q, \dots, q), r, \dots, r\right)$$

for unique $q$ (quotient) and $r$ (remainder) (Duplij, 2017).

The theory extends to notions of irreducibility and primitivity: prime polyadic integers exist only when a unit exists, and the structure of polyadic Euler functions and idempotent orders admits features without binary analogs (Duplij, 2017).

4. Polyadic Constructions: Direct Products, Polyadization, and Group Rings

Polyadic algebraic structures admit external products with richer behavior than in the binary case:

• Iterated direct products: coordinatewise $(m,n)$-operations, with possible mixed arities $(m', n')$ constrained by "quantization" equalities, e.g., $m' = \ell_1(m_1-1) + 1 = \ell_2(m_2-1)+1$ (Duplij, 2022).
• Hetero products: noncomponentwise ("entangled") external products defined via associativity quivers, allowing arity reduction and entanglement beyond classical field direct products (Duplij, 2022).

The polyadization process generalizes binary structures to nonderived polyadic rings via block-shift matrices, creating genuine polyadic multiplications, often represented as cyclic products in matrix blocks. Semisimple polyadic rings admit double decompositions, echoing but generalizing Wedderburn-Artin theory (2208.04695).

The polyadic group ring, $\mathcal{R}^{[m,n]}[G^{[n_g]}]$, merges an $(m,n)$-ring with an $n_g$-ary group, incorporating polyadic addition and convolution-type $n$-ary multiplication. Here, augmentation maps and ideals, as well as quantization constraints, generalize classical group ring invariants (Duplij, 15 Oct 2025).

5. Positional Arithmetic and Representability

Polyadic rings admit positional numeral systems with arity-aware constraints:

• Admissible word lengths in base-$p$ expansions are double-quantized: a string of $\ell_\nu$ $m$-ary additions must match $\ell_\mu = \ell_\nu(m-1)+1$ $n$-ary multiplications.
• For $m,n\ge3$, only subsets of elements possess finite expansions ("representability gap"), characterized precisely by the arity-shape invariants $I^{(m)}, J^{(n)}$ (Duplij, 15 Jun 2025).

This places strong restrictions on coding, arithmetic, and hardware design, as word lengths and digit counts are quantized per the underlying arity.

6. Extensions: $p$-adic, Algebraic, and Quantum Generalizations

The theory carries over to $p$-adic integers by defining polyadic residue classes in $\mathbb{Z}_p$; closure conditions are directly analogous to the integer case. These structures provide potential new symmetries for $p$-adic physics and non-Archimedean models (Duplij, 2022).

Polyadic rings underpin broader algebraic structures such as polyadic vector spaces, Hopf algebras, and quantum groups, where the arity-shape principle plays a central role. Such algebras often lack unique unital or idempotent elements and permit quantized dimension formulas. Polyadic analogs of the Yang-Baxter equation and $R$-matrix theory have been developed (Duplij, 2018, Duplij, 2013).

7. Applications and Structural Phenomena

Polyadic rings serve as platforms for:

• Cryptography: security leveraging non-injectivity, multivaluedness, and Diophantine complexity of the parameter-to-arity map; quantized operations yield systems highly resistant to attack by standard algebraic means (Duplij et al., 14 Dec 2025, Duplij, 15 Oct 2025).
• Coding theory: non-linear error-correcting codes and block ciphers exploiting higher-arity convolutions and arity freedom (Duplij, 15 Jun 2025).
• Computer arithmetic: design of multiary ALUs, efficient carry handling, ternary or higher-base hardware (Duplij, 15 Jun 2025).

A key structural feature is the profusion of non-isomorphic finite polyadic fields of identical size and arity shape, the existence of zeroless and nonunital examples, and conjectured canonical prime subfields generalizing $\mathrm{GF}(p)$ in binary field theory (Duplij, 2017).

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References:

• (Duplij et al., 14 Dec 2025) Cryptographic transformations over polyadic rings
• (Duplij, 15 Oct 2025) Higher power polyadic group rings
• (Duplij, 15 Jun 2025) Positional numeral systems over polyadic rings
• (Duplij, 2022) Polyadic analogs of direct product
• (2208.04695) Polyadization of algebraic structures
• (Duplij, 2022) Polyadic rings of $p$-adic integers
• (Duplij, 2018) Polyadic Hopf algebras and quantum groups
• (Duplij, 2017) Arity shape of polyadic algebraic structures
• (Duplij, 2017) Polyadic integer numbers and finite (m,n)-fields
• (Duplij, 2013) Polyadic systems, representations and quantum groups