Higher Power Polyadic Group Rings

Abstract

This paper introduces and systematically develops the theory of polyadic group rings, a higher arity generalization of classical group rings. We construct the fundamental operations of these structures, defining the m-ary addition and n-ary multiplication for a polyadic group ring built from an $(m_r,n_r)$-ring and an $n_g$-ary group. A central result is the derivation of the “quantization” conditions that interrelate these arities, governed by the arity freedom principle, which also extends to operations with higher polyadic powers. We establish key algebraic properties, including conditions for total associativity and the existence of a zero element and identity. The concepts of the polyadic augmentation map and augmentation ideal are generalized, providing a bridge to the classical theory. The framework is illustrated with explicit examples, solidifying the theoretical constructions.

1. Introduction

The theory of group rings, which constructs a ring $\mathcal{R}\left[\mathsf{G}\right]$ from a given ring $\mathcal{R}$ and a group $\mathsf{G}$, is a cornerstone of modern algebra. Its applications permeate various fields, including representation theory, homological algebra, and algebraic topology [1,2,3]. The standard construction leverages the binary operations of the constituent ring and group to define an associative algebra, providing a rich framework for studying the interplay between ring-theoretic and group-theoretic properties [4,5,6].

A significant and modern generalization of classical algebraic structures arises from increasing the arity of their fundamental operations. This leads to the theory of polyadic algebraic structures [7,8], where operations map n elements to a single one, with $n\ge 3$. This framework reveals phenomena absent in the binary case; for instance, polyadic groups (n-ary groups) can exist without a unique identity element or inverses in the classical sense, with their structure governed by the more general concept of a querelement [7]. Similarly, polyadic rings, defined by an m-ary addition and an n-ary multiplication linked by generalized distributivity laws, exhibit a more complex and nuanced structure [9].

While the theories of binary group rings and polyadic structures are individually well-established, the synthesis of the theory of polyadic group rings remains largely unexplored. Constructing such an object, denoted ${\mathcal{R}}^{[m_r,n_r]}\left[{\mathsf{G}}^{\left[n_g\right]}\right]$ from an $(m_r,n_r)$-ring and an $n_g$-ary group, presents fundamental challenges. The arities of the initial structures are not independent; they are constrained by the requirement that the resulting object must itself be a ring-like structure with well-defined ${\mathit{m}}_r$-ary addition and ${\mathit{n}}_r$-ary multiplication. This interplay is governed by what has been termed the “arity freedom principle” [10], leading to “quantization” conditions that determine the admissible arities ${\mathit{m}}_r$ and ${\mathit{n}}_r$ of the resulting polyadic group ring. In an earlier study, a special case of polyadic group rings, where $m_r=2$ and $n_r=2$, known as the n-group ring, with the binary initial ring $\mathcal{R}={\mathcal{R}}^{[2,2]}$, was briefly considered in [11,12,13].

In this article, we introduce and develop the theory of polyadic group rings for the first time with the general initial (polyadic) $(m_r,n_r)$-ring ${\mathcal{R}}^{[m_r,n_r]}$. Our primary objective is to generalize the classical group ring construction to the higher-arity setting and establish its foundational properties. The main contributions of this work are as follows:

• We provide a rigorous definition of a polyadic group ring, formally constructing its ${\mathit{m}}_r$-ary addition and ${\mathit{n}}_r$-ary multiplication operations, carefully accounting for the arities of the underlying ring and group.
• We derive the precise “quantization” conditions that link the arities $({\mathit{m}}_r,{\mathit{n}}_r)$ of the group ring to the arities $(m_r,n_r)$ of the initial ring and $n_g$ of the initial group, including the novel case of operations with higher polyadic powers.
• We establish key properties of these structures, proving under which conditions the polyadic group ring is totally associative and possesses analogs of a zero element and, when applicable, an identity.
• We define and investigate the concepts of the polyadic augmentation map and the polyadic augmentation ideal, generalizing central tools from the classical theory.
• We illustrate the theory with concrete, non-trivial examples involving nonderived polyadic rings and finite polyadic groups, explicitly computing products and demonstrating the workings of the constructed operations.

This work not only broadens the landscape of ring theory by introducing a new class of algebraic objects but also provides a framework for future investigations into their representation theory, homological invariants, graded rings and algebras. For instance, in [14], a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures, was developed. Furthermore, the complex, non-binary operations inherent to polyadic group rings present a promising algebraic platform for applications in coding theory [15,16] and post-quantum cryptography [17]. A compelling demonstration of this practical potential can be found in [18,19]. The authors construct a novel encryption and decryption procedure that directly leverages polyadic algebraic structures alongside signal processing methods [20], which represents a tangible and promising application of polyadic theory to cryptography, moving beyond purely theoretical constructs.

2. Preliminaries

Here we present the notation and the general properties of polyadic structures (for more details and references, consult [10]).

Let $S^{\times n}$ be the n-fold Cartesian product of a non-empty set S. Elements of the form $(x_1,\dots ,x_n)\in S^{\times n}$ are termed polyads or n-tuples $\left(\mathrm{x}\right)$. An n-tuple consisting of identical elements is denoted $\left(x^n\right)$. A polyadic operation (or n-ary operation) is defined as a mapping ${\mathsf{\mu }}_n:S^{\times n}\to S$, denoted by ${\mathsf{\mu }}_n\left[\mathrm{x}\right]$. A polyadic structure $⟨S\mid {\mathsf{\mu }}_{n_i}⟩$ consists of a set S that is closed under a family of polyadic operations ${\mathsf{\mu }}_{n_i}$.

The fundamental one-operation polyadic structure is the n-ary magma $\mathcal{M}=⟨S\mid {\mathsf{\mu }}_n⟩$. The imposition of additional axioms results in various group-like structures. For instance, a polyadically associative magma constitutes an n-ary semigroup ${\mathcal{S}}_n=⟨S\mid {\mathsf{\mu }}_n\mid assoc⟩$. Polyadic associativity is defined through the invariance relation ${\mathsf{\mu }}_n[\mathrm{x},{\mathsf{\mu }}_n\left[\mathrm{y}\right],\mathrm{z}]=invariant$, wherein the embedded multiplication may appear in any of the $n-1$ positions (resulting in $n-1$ relations), which enables the omission of parentheses in compositions. The polyads $\mathrm{x},\mathrm{y},\mathrm{z}$ have appropriate lengths such that the total number of elements is $2n-1$. This iterated product

$${\mathsf{\mu }}_n^{\circ {\mathcal{\ell }}_{\mathsf{\mu }}}\left[\mathrm{x}\right]=\stackrel{{\mathcal{\ell }}_{\mathsf{\mu }}}{\overbrace{{\mathsf{\mu }}_n\left[{\mathsf{\mu }}_n[\dots {\mathsf{\mu }}_n\left[\mathrm{x}\right]]\right]}},\mathrm{x}\in S^{{\mathcal{\ell }}_{\mathsf{\mu }}(n-1)+1},$$

(1)

where ${\mathcal{\ell }}_{\mathsf{\mu }}$ denotes the number of multiplication operations. From (1), a fundamental distinction between polyadic and conventional binary ($n=2$) structures arises: the length of a word in a composition of n-ary multiplications is not arbitrary but “quantized”, assuming only the following admissible values are possible:

$$L^{\mathrm{admiss}}(n,{\mathcal{\ell }}_{\mathsf{\mu }})={\mathcal{\ell }}_{\mathsf{\mu }}(n-1)+1.$$

(2)

This viewpoint facilitates the classification of polyadic operations into two categories: those iterated from binary or lower-arity operations and those that are noniterated, or equivalently, derived and nonderived. For instance, the ternary product ${\mathsf{\mu }}_3^{+}[x,y,z]=xyz$ on positive integers ${\mathbb{Z}}^{+}$ is derived since it equals to the composition ${\mathsf{\mu }}_3^{+}={\mathsf{\mu }}_2\circ {\mathsf{\mu }}_2$ of binary products ${\mathsf{\mu }}_2\left[{\mathsf{\mu }}_2\left[x,y\right],z\right]=\left(x·y\right)·z$. On negative numbers ${\mathbb{Z}}^{-}$, however, the ternary multiplication ${\mathsf{\mu }}_3^{-}$ is nonderived: two negatives multiply to a positive, breaking closure under binary multiplication, so the above binary iteration fails. Ternary associativity holds for both, making ${\mathcal{S}}_3^{+}=⟨{\mathbb{Z}}^{+}\mid {\mathsf{\mu }}_3^{+}⟩$ and ${\mathcal{S}}_3^{-}=⟨{\mathbb{Z}}^{-}\mid {\mathsf{\mu }}_3^{-}⟩$ ternary semigroups, but only ${\mathcal{S}}_3^{-}$ is nonderived. Below we consider only nonderived polyadic operations and structures.

We now recall the definitions of key elements in polyadic structures. For an element $x\in S$, its ${\mathcal{\ell }}_{\mathsf{\mu }}$-polyadic power (or higher polyadic power) is defined by

$$x^{⟨{\mathcal{\ell }}_{\mathsf{\mu }}⟩}={\mathsf{\mu }}_n^{\circ {\mathcal{\ell }}_{\mathsf{\mu }}}\left[x^{{\mathcal{\ell }}_{\mathsf{\mu }}(n-1)+1}\right],$$

(3)

which, in the binary case $n=2$, yields $x^{⟨{\mathcal{\ell }}_{\mathsf{\mu }}⟩}=x^{{\mathcal{\ell }}_{\mathsf{\mu }}+1}$, differing by unity from the conventional power.

A polyadic idempotent $x_{id}$ (if existent) satisfies

$$x_{id}^{⟨{\mathcal{\ell }}_{\mathsf{\mu }}⟩}=x_{id},x_{id}\in S.$$

(4)

A polyadic zero z is uniquely defined by the $n-1$ conditions

$${\mathsf{\mu }}_n[z,\mathrm{x}]=z,\mathrm{x}\in S^{n-1},$$

(5)

with z positioned in any of the n argument slots. A polyadic nilpotent element $x_{nil}$ is defined by

$$x_{nil}^{⟨{\mathcal{\ell }}_{\mathsf{\mu }}⟩}=z,x_{nil}\in S.$$

(6)

A neutral $(n-1)$-polyad e satisfies

$${\mathsf{\mu }}_n[x,\mathrm{e}]=x,\mathrm{e}\in S^{n-1},$$

(7)

which is typically non-unique. If all components of the neutral polyad are identical, $\mathrm{e}=e^{n-1}$, then

$${\mathsf{\mu }}_n[x,e^{n-1}]=x,$$

(8)

and e is termed an identity of $⟨S\mid {\mathsf{\mu }}_n⟩$; it may appear in any of the n positions within the operation. From (5) with $\mathrm{x}=z^{n-1}$ and (8) with $x=e$, it follows that both the polyadic zero z and the identity e are idempotents satisfying (4). Certain exotic polyadic structures may lack idempotents, a zero, or an identity altogether, or may feature multiple identities [10].

In the polyadic case ($n\ge 3$), the notion of invertibility is not linked to the identity (8) but is determined by the querelement $\overline{x}=\overline{x}\left(x\right)$, defined via the $n-1$ relations [7]

$${\mathsf{\mu }}_n[\overline{x},x^{n-1}]=x,x\in S,$$

(9)

which must hold for $\overline{x}$ in each of the n possible positions. Such an element x is termed polyadically invertible. If every element of an n-ary semigroup ${\mathcal{S}}_n$ is polyadically invertible, then ${\mathcal{S}}_n$ constitutes an n-ary (polyadic) group ${\mathcal{G}}_n=⟨S\mid {\mathsf{\mu }}_n\mid assoc⟩$. Notably, the presence of an identity is not a prerequisite for polyadic groups.

Structures endowed with two polyadic operations fall within the class of ring-like polyadic structures [9]. A polyadic ring, or $(m,n)$-ring, ${\mathcal{R}}_{m,n}=⟨S\mid {\nu }_m,{\mathsf{\mu }}_n⟩$, consists of a non-empty set S equipped with an m-ary addition ${\nu }_m:S^m\to S$ and an n-ary multiplication ${\mathsf{\mu }}_n:S^n\to S$, such that: $⟨S\mid {\nu }_m\mid assoc\mid comm⟩$ forms an m-ary commutative group and $⟨S\mid {\mathsf{\mu }}_n\mid assoc⟩$ forms an n-ary semigroup. The operations ${\nu }_m$ and ${\mathsf{\mu }}_n$ are interconnected by the following n-ary distributivity relations [10]:

$$\begin{array}{cc}\hfill & {\mathsf{\mu }}_n[{\nu }_m[x_1,\dots ,x_m],y_2,y_3,\dots ,y_n]\hfill \\ \hfill & ={\nu }_m[{\mathsf{\mu }}_n[x_1,y_2,y_3,\dots ,y_n],{\mathsf{\mu }}_n[x_2,y_2,y_3,\dots ,y_n],\dots ,{\mathsf{\mu }}_n[x_m,y_2,y_3,\dots ,y_n]],\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & {\mathsf{\mu }}_n[y_1,{\nu }_m[x_1,\dots ,x_m],y_3,\dots ,y_n]\hfill \\ \hfill & ={\nu }_m[{\mathsf{\mu }}_n[y_1,x_1,y_3,\dots ,y_n],{\mathsf{\mu }}_n[y_1,x_2,y_3,\dots ,y_n],\dots ,{\mathsf{\mu }}_n[y_1,x_m,y_3,\dots ,y_n]],\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & ⋮\hfill \\ \hfill & {\mathsf{\mu }}_n[y_1,y_2,\dots ,y_{n-1},{\nu }_m[x_1,\dots ,x_m]]\hfill \\ \hfill & ={\nu }_m[{\mathsf{\mu }}_n[y_1,y_2,\dots ,y_{n-1},x_1],{\mathsf{\mu }}_n[y_1,y_2,\dots ,y_{n-1},x_2],\dots ,{\mathsf{\mu }}_n[y_1,y_2,\dots ,y_{n-1},x_m]],\hfill \end{array}$$

(12)

where $x_i,y_j\in S$, $i=1,\dots ,m$, $j=1,\dots ,n$.

If not all distributivity relations (10)–(12) or associativity relations hold, the ring is designated as partial (in contrast to total), giving rise to a multitude of possible polyadic ring variants. Further details are in [10].

3. Binary Group Rings

Here recall in brief the main constructions of the binary group rings in the standard approach [1,2,3,21,22] and then present them in the “polyadic” language, which will make their generalization to the novel higher arity approach more clear and transparent.

The most natural way to construct from two given one-set algebraic structures $\mathcal{A}\left(1\right)$ and $\mathcal{B}\left(1\right)$ a new two-set algebraic structure $\mathcal{C}\left(2\right)$ is to consider formal combinations of elements from $\mathcal{B}\left(1\right)$ having “weights” from $\mathcal{A}\left(1\right)$ is “more linear”, which is usually denoted $\mathcal{A}\left(1\right)\mathcal{B}\left(1\right)$. Without considering a product in $\mathcal{A}\left(1\right)\mathcal{B}\left(1\right)$ the result is only a free module-like structure in which $\mathcal{A}\left(1\right)$ plays a role of “scalars”, while $\mathcal{B}\left(1\right)$ being the “basis”, taking the starting analogy with a vector space. Further various definitions of a multiplication in $\mathcal{A}\left(1\right)\mathcal{B}\left(1\right)$ lead to different algebra-like structures with nontrivial properties, which are usually denoted as $\mathcal{C}\left(2\right)=\mathcal{A}\left(1\right)\left[\mathcal{B}\left(1\right)\right]$. An equivalent approach to the latter is consideration of the set of mappings $\mathcal{B}\left(1\right)\to \mathcal{A}\left(1\right)$ formally multiplied by “scalars” from $\mathcal{A}\left(1\right)$ with the pointwise addition and the product as a convolution, commonly denoted as $\mathcal{A}{\left(1\right)}^{\mathcal{B}\left(1\right)}$. We will exploit the first definition for higher arity generalizations. Typically, the role of “scalars” from $\mathcal{A}\left(1\right)$ is played by rings, fields, etc., and for the role of “vectors” from $\mathcal{B}\left(1\right)$ one takes semigroups, monoids, groups, loops, and so on.

In the simplest case, $\mathcal{A}\left(1\right)$ is an associative ring $\mathcal{R}$ (having the underlying set R, with the possible zero $0_R$ and unit $1_R$), and $\mathcal{B}\left(1\right)$ is a group $\mathsf{G}$ (having the underlying set G, with the identity ${\mathsf{e}}_{\mathsf{G}}$), and $\mathcal{R}\mathsf{G}$ can be built as follows (in the standard notation [2,21,22]).

Definition 1.

A free $\mathcal{R}$-module $\mathcal{R}\mathsf{G}$ with the basis $\left\{\mathsf{g}\mid \mathsf{g}\in \mathsf{G}\right\}$ is the set of finite formal sums

$$\sum _{\mathsf{g}\in \mathsf{G}}{r}_{\mathsf{g}}·\mathsf{g},r_{\mathsf{g}}\in R,\mathsf{g}\in G,$$

(13)

which endowed with the left-“componentwise” addition

$$\left(\sum _{\mathsf{g}\in \mathsf{G}}{r}_{\mathsf{g}}·\mathsf{g}\right)+\left(\sum _{\mathsf{g}\in \mathsf{G}}{r}_{\mathsf{g}}^{\prime }·\mathsf{g}\right)=\left(\sum _{\mathsf{g}\in \mathsf{G}}\left(r_{\mathsf{g}}+r_{\mathsf{g}}^{\prime }\right)·\mathsf{g}\right)$$

(14)

and left-“scalar” multiplication

$$\lambda \left(\sum _{\mathsf{g}\in \mathsf{G}}{r}_{\mathsf{g}}·\mathsf{g}\right)=\left(\sum _{\mathsf{g}\in \mathsf{G}}\left(\lambda r_{\mathsf{g}}\right)·\mathsf{g}\right),\lambda \in R.$$

(15)

Obviously, $\mathsf{G}\subset \mathcal{R}\mathsf{G}$, because for every $\mathsf{h}\in G$ one can choose $r_{\mathsf{h}}=1_R$ and $r_{\mathsf{g}}=0_R$, if $\mathsf{g}\ne \mathsf{h}$. Thus, each element of $\mathcal{R}\mathsf{G}$ can be treated as a finite sum of such $r_{\mathsf{g}}·\mathsf{g}$ for which $r_{\mathsf{g}}\ne 0$, and the subset of such $\mathsf{g}\in G_{\mathit{supp}}\subset G$ is the support of $\sum r_{\mathsf{g}}·\mathsf{g}$, i.e., $G_{\mathit{supp}}=$support$\left(\sum r_{\mathsf{g}}·\mathsf{g}\right)$. In general, if both underlying sets R and G are finite with $\left|R\right|=N_R$ and $\left|G\right|=N_{\mathsf{G}}$, the total number of elements in $\left|\mathcal{R}\mathsf{G}\right|=N_{R\mathsf{G}}$ is $N_{R\mathsf{G}}={\left(N_R\right)}^{N_{\mathsf{G}}}$.

The product of sums (13) cannot be defined in left-“componentwise” way analogously to the addition (14) simply by replacing addition with multiplication in the ring, e.g., ${\sum }_{\mathsf{g}\in \mathsf{G}}\left(r_{\mathsf{g}}{r}_{\mathsf{g}}^{\prime }\right)·\mathsf{g}$, since this entirely ignores the group structure. Instead, consider the both-“componentwise” product of two terms of different sums (13) and define their product by the natural way $\left(r_{\mathsf{g}}·\mathsf{g}\right)\left(r_{\mathsf{h}}^{\prime }·\mathsf{h}\right)=\left(r_{\mathsf{g}}{r}_{\mathsf{h}}^{\prime }\right)·\left(\mathsf{gh}\right)$. Then the multiplication of elements from $\mathcal{R}\mathsf{G}$ can be presented as follows (after reordering the terms)

$$\left(\sum _{\mathsf{g}\in G}{r}_{\mathsf{g}}·\mathsf{g}\right)\left(\sum _{\mathsf{h}\in G}{r}_{\mathsf{h}}^{\prime }·\mathsf{h}\right)=\left(\sum _{\mathsf{g}\in G}\sum _{\mathsf{h}\in G}\left(r_{\mathsf{g}}{r}_{\mathsf{h}}^{\prime }\right)·\left(\mathsf{gh}\right)\right)=\left(\sum _{\mathsf{u}\in G}{r}_{\mathsf{u}}^{″}·\mathsf{u}\right),$$

(16)

where

$$r_{\mathsf{u}}^{″}=\sum _{\mathsf{g}\in G}\sum _{\mathsf{h}\in G}\left(r_{\mathsf{g}}{r}_{\mathsf{h}}^{\prime }\right){|}_{\mathsf{gh}=\mathsf{u}},\mathsf{g},\mathsf{h},\mathsf{u}\in G,$$

(17)

or in terms of one sum

$$r_{\mathsf{u}}^{″}=\sum _{\mathsf{g}\in G}\left(r_{\mathsf{g}}{r}_{{\mathsf{g}}^{-1}\mathsf{u}}^{\prime }\right).$$

(18)

The product (16) is associative and satisfies distributivity with respect to the addition (14).

Definition 2.

The free module $\mathcal{R}\mathsf{G}$ (13)–(15) endowed with the product (16)–(17) becomes a ring which is called a group ring denoted by $\mathbf{R}=\mathcal{R}\left[\mathsf{G}\right]$.

If the initial ring is a field $\mathcal{R}=\mathcal{K}$, then $\mathcal{K}\left[\mathsf{G}\right]$ is called a group algebra over $\mathcal{K}$, being a vector space over $\mathcal{K}$ with the dimension $\left|G\right|$, if the group $\mathsf{G}$ is finite. Note that the group algebra concept is the starting point for representation theory (see, e.g., [23,24,25]).

Numerous properties of group rings were considered in [1,2,3,21,22], and refs therein.

Let us rewrite the main definitions of the binary group rings (13)–(17) in the more detail “polyadic” functional notation by writing the operations manifestly.

The initial (binary) group is the algebraic structure having one-set $\left\{\mathsf{g}\right\}=G$ and one main associative binary operation ${\mathsf{\mu }}_{\mathsf{G}}={\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{2}\right]}:G\times G\to G$, as $\mathsf{G}={\mathsf{G}}^{\left[2\right]}=⟨G\mid {\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{2}\right]},{\left(\right)}^{-1}⟩$ together with the identity ${\mathsf{e}}_{\mathsf{G}}$ satisfying ${\mathsf{\mu }}_{\mathsf{G}}\left[{\mathsf{e}}_{\mathsf{G}},\mathsf{g}\right]={\mathsf{\mu }}_{\mathsf{G}}\left[\mathsf{g},{\mathsf{e}}_{\mathsf{G}}\right]=\mathsf{g}$ and the inverse ${\left(\right)}^{-1}$ such that ${\mathsf{\mu }}_{\mathsf{G}}\left[\mathsf{g},{\mathsf{g}}^{-1}\right]={\mathsf{\mu }}_{\mathsf{G}}\left[{\mathsf{g}}^{-1},\mathsf{g}\right]={\mathsf{e}}_{\mathsf{G}}$ for all $\mathsf{g}\in G$. The initial ring $\mathcal{R}$ is the one-set $\left\{r\right\}=R$ algebraic structure $\mathcal{R}={\mathcal{R}}^{\left[2,2\right]}=⟨R\mid {\nu }_R^{\left[2\right]},{\mathsf{\mu }}_R^{\left[2\right]}⟩$ endowed by two binary operations: addition ${\nu }_R={\nu }_R^{\left[2\right]}:R\times R\to R$ and multiplication ${\mathsf{\mu }}_R={\mathsf{\mu }}_R^{\left[2\right]}:R\times R\to R$ which satisfy distributivity, such that $⟨R\mid {\nu }_R^{\left[2\right]}⟩$ is the additive semigroup, and $⟨R\mid {\mathsf{\mu }}_R^{\left[2\right]}⟩$ is the multiplicative group.

Note that by (15) a new operation (scalar multiplication) in $\mathcal{R}\mathsf{G}$ is quietly defined, which is possible, because any ring is a module over itself. Therefore, at first glance, the resulting group ring $\mathcal{R}\left[\mathsf{G}\right]$ is a 2-set and 4-operation algebraic structure, but we will see that it is more complicated.

Using this notation we present the definition (13)–(17) of the group ring $\mathbf{R}=\mathcal{R}\left[\mathsf{G}\right]$ in the “polyadic” functional form. Instead of the abstract sum ∑ in (13) we use the concrete summation $\mathsf{\Sigma }$ by indices manifestly (see, e.g., [21]).

In this way, the $\alpha$th element $\mathit{r}\left(\alpha \right)$ of the group ring $\mathbf{R}=\mathcal{R}\left[\mathsf{G}\right]$ (with the underlying set R) can be written as the formal sum

$$\mathbf{r}\left(\alpha \right)=\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left(\alpha \right),\overrightarrow{\mathsf{g}}\right)=\underset{i}{\mathsf{\Sigma }}{r}_{{\mathsf{g}}_i}\left(\alpha \right)·{\mathsf{g}}_i,r_{{\mathsf{g}}_i}\left(\alpha \right)\in R,{\mathsf{g}}_i\in G,\mathbf{r}\left(\alpha \right)\in R,$$

(19)

where $r_{{\mathsf{g}}_i}\left(\alpha \right)$ are ring elements and ${\mathsf{g}}_i$ are group elements. For conciseness, we denote the ordered families ${\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left(\alpha \right)=\left(r_{{\mathsf{g}}_1}\left(\alpha \right),r_{{\mathsf{g}}_2}\left(\alpha \right),\dots \right)$, $\overrightarrow{\mathsf{g}}=\left({\mathsf{g}}_1,{\mathsf{g}}_2,\dots \right)$ as “formal vectors”. In case the group ring $\mathcal{R}\left[\mathsf{G}\right]$ is finite, $\alpha =1,2\dots \left|R\right|$.

Denote the binary addition in $\mathbf{R}=\mathcal{R}\left[\mathsf{G}\right]$ (14) as ${\mathit{\nu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]}:\mathrm{R}\times \mathrm{R}\to \mathrm{R}$, then

$${\mathit{\nu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]}\left[\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_1\right),\overrightarrow{\mathsf{g}}\right),\mathbf{r}\left({\overrightarrow{r}}_{\tilde{\mathsf{g}}}\left({\alpha }_2\right),\overrightarrow{\mathsf{g}}\right)\right]=\underset{i}{\mathsf{\Sigma }}{\nu }_R^{\left[2\right]}\left[r_{{\mathsf{g}}_i}\left({\alpha }_1\right),r_{{\mathsf{g}}_i}\left({\alpha }_2\right)\right]·{\mathsf{g}}_i,r_{{\mathsf{g}}_i}\left({\alpha }_{1,2}\right)\in R,{\mathsf{g}}_i\in G.$$

(20)

The binary multiplication in $\mathbf{R}=\mathcal{R}\left[\mathsf{G}\right]$ is denoted by ${\mathsf{\mu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]}:\mathrm{R}\times \mathrm{R}\to \mathrm{R}$, using (16) we define the convolution-like operation

$$\begin{array}{cc}\hfill {\mathsf{\mu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]}\left[\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_1\right),\overrightarrow{\mathsf{g}}\right),\mathbf{r}\left({\overrightarrow{r}}_{\tilde{\mathsf{g}}}\left({\alpha }_2\right),\overrightarrow{\mathsf{g}}\right)\right]& =\underset{i}{\mathsf{\Sigma }}\underset{j}{\mathsf{\Sigma }}{\mathsf{\mu }}_R^{\left[2\right]}\left[r_{{\mathsf{g}}_i}\left({\alpha }_1\right),r_{{\mathsf{g}}_j}\left({\alpha }_2\right)\right]·{\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{2}\right]}\left[{\mathsf{g}}_i,{\mathsf{g}}_j\right],\hfill \\ \hfill r_{{\mathsf{g}}_{i,j}}\left({\alpha }_{1,2}\right)& \in R,{\mathsf{g}}_i\in G.\hfill \end{array}$$

(21)

Denote

$${\mathsf{g}}_k={\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{2}\right]}\left[{\mathsf{g}}_i,{\mathsf{g}}_j\right],k=k\left(i,j\right),{\mathsf{g}}_{i,j,k}\in G,$$

(22)

then (16) “in components” takes the form

$$\begin{array}{cc}\hfill {\mathsf{\mu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]}\left[\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_1\right),\overrightarrow{\mathsf{g}}\right),\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_2\right),\overrightarrow{\mathsf{g}}\right)\right]& =\underset{i}{\mathsf{\Sigma }}\underset{j}{\mathsf{\Sigma }}{\mathsf{\mu }}_R^{\left[2\right]}\left[r_{{\mathsf{g}}_i}\left({\alpha }_1\right),r_{{\mathsf{g}}_j}\left({\alpha }_2\right)\right]·{\mathsf{g}}_{k\left(i.j\right)},\hfill \\ \hfill r_{{\mathsf{g}}_{i,j,k\left(i,j\right)}}\left({\alpha }_{1,2}\right)& \in R,{\mathsf{g}}_i\in G.\hfill \end{array}$$

(23)

We can resolve (22) with respect to ${\mathsf{g}}_j$, reorder indices and present (21) as follows

$${\mathsf{\mu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]}\left[\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_1\right),\overrightarrow{\mathsf{g}}\right),\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_2\right),\overrightarrow{\mathsf{g}}\right)\right]=\underset{k}{\mathsf{\Sigma }}{r}_{{\mathsf{g}}_k}^{\prime }\left({\alpha }_1,{\alpha }_2\right)·{\mathsf{g}}_k,$$

(24)

where

$$r_{{\mathsf{g}}_k}^{\prime }\left({\alpha }_1,{\alpha }_2\right)=\underset{i}{\mathsf{\Sigma }}{\mathsf{\mu }}_R^{\left[2\right]}\left[r_{{\mathsf{g}}_i}\left({\alpha }_1\right),r_{{\mathsf{g}}_i^{-1}{\mathsf{g}}_k}\left({\alpha }_2\right)\right],$$

(25)

which is the component/function version of (18).

The set of sums (19) $\left\{\mathbf{r}\right\}=\mathrm{R}$ together with associative addition (20) and multiplication (21) being distributive is the standard ring $⟨\mathrm{R}\mid {\mathit{v}}_{\mathrm{R}}^{\left[\mathbf{2}\right]},{\mathsf{\mu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]}⟩$. However, the group ring $\mathcal{R}\left[\mathsf{G}\right]$ is more than a ring, because by definition it has the additional module-like operation, the “scalar” multiplication (15). Since the “scalars” are from the initial ring $\mathcal{R}$, it is an internal ring product, because every ring is a module by itself. But for the group ring $\mathcal{R}\left[\mathsf{G}\right]$ (15) is the unnoticed new operation (1-place action or $\mathcal{R}$-module) ${\mathsf{\rho }}_{\mathrm{R}}^{\left[1\right]}:R\times \mathrm{R}\to \mathrm{R}$, which is in the “polyadic” notation becomes (for k-place actions, see [10])

$${\mathsf{\rho }}_{\mathrm{R}}^{\left[\mathbf{1}\right]}\left(\lambda \mid \mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left(\alpha \right),\overrightarrow{\mathsf{g}}\right)\right)=\underset{i}{\mathsf{\Sigma }}{\mathsf{\mu }}_R^{\left[2\right]}\left[\lambda ,r_{{\mathsf{g}}_i}\left(\alpha \right)\right]·{\mathsf{g}}_i,\lambda ,r_{{\mathsf{g}}_i}\left(\alpha \right)\in R,{\mathsf{g}}_i\in G.$$

(26)

In this way, we could think that the formal “polyadic” definition of the group ring, as 2 set and 3 operation algebraic structure $\mathbf{R}=⟨\mathrm{R},R,G\mid {\mathit{\nu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]},{\mathsf{\mu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]},{\mathsf{\rho }}_{\mathrm{R}}^{\left[\mathbf{1}\right]}⟩$, $\lambda \in R$. Nevertheless, after adding the operations (addition ${\nu }_R^{\left[2\right]}$ and multiplication ${\mathsf{\mu }}_R^{\left[2\right]}$) in $\mathcal{R}$, we have (by classification of [10]).

Definition 3.

The binary group ring $\mathcal{R}\left[\mathsf{G}\right]$ is the 3 set and 6 operation algebra-like structure

$$\mathbf{R}=⟨\mathrm{R},R,G\mid {\mathit{\nu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]},{\mathsf{\mu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]},{\mathsf{\rho }}_{\mathrm{R}}^{\left[\mathbf{1}\right]}\mid {\nu }_R^{\left[2\right]},{\mathsf{\mu }}_R^{\left[2\right]}\mid {\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{2}\right]}⟩,$$

(27)

where the first 3 operations are defined in (20), (21) and (26), respectively.

4. Polyadic Group Rings

Now we generalize the group ring concept to higher arity case and introduce a novel algebraic structure, the polyadic group ring. By doing so, we take advantage of the “arity freedom principle” [10]: in any algebraic structure, initial arities of all its operations can be taken arbitrary, then the structural constraints appear from the general dependences leading to “quantization rules” which forbidden certain combination of arities. In this way, polyadic structures can have exotic properties, for instance n-ary groups without identity or with many identities, polyadic fields without zero or/and without unit, and so on [10], which leads to revision even standard theorems and statements.

In the abstract setting from beginning of the previous section, the initial polyadic algebraic structures now carry arbitrary arities ${\mathcal{A}}^{\left[arit{y}_{\mathcal{A}}\right]}\left(1\right)$ and ${\mathcal{B}}^{\left[arit{y}_{\mathcal{B}}\right]}\left(1\right)$. Then the resulting polyadic algebra-like structure ${\mathcal{C}}^{\left[arit{y}_{\mathcal{C}}\right]}\left(2\right)={\mathcal{A}}^{\left[arit{y}_{\mathcal{A}}\right]}\left(1\right)\left[{\mathcal{B}}^{\left[arit{y}_{\mathcal{B}}\right]}\left(1\right)\right]$ will possess the specific arity $arit{y}_{\mathcal{C}}$ which is determined by the operations framework.

So instead of the binary ring $\mathcal{R}$ as $\mathcal{A}\left(1\right)$ and group $\mathsf{G}$ as $\mathcal{B}\left(1\right)$, we consider $\left(n_r,m_r\right)$-ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$ as ${\mathcal{A}}^{\left[arit{y}_{\mathcal{A}}\right]}\left(1\right)$ and $n_g$-ary group ${\mathsf{G}}^{\left[n_g\right]}$ as ${\mathcal{B}}^{\left[arit{y}_{\mathcal{B}}\right]}\left(1\right)$, where

$${\mathcal{R}}^{\left[m_r,n_r\right]}=⟨R\mid {\nu }_R^{\left[m_r\right]},{\mathsf{\mu }}_R^{\left[n_r\right]}⟩,$$

(28)

with totally associative $m_r$-ary addition ${\nu }_R^{\left[m_r\right]}:R^{\times m_r}\to R$ and $n_r$-ary multiplication ${\mathsf{\mu }}_R^{\left[n_r\right]}:R^{\times n_r}\to R$, which satisfy polyadic distributivity.

If polyadic zero $z_R$ and polyadic identity (or unity) $e_R$ in ${\mathcal{R}}^{\left[m_r,n_r\right]}$ exist, they satisfy additive neutrality and multiplicative absorption, and multiplicative neutrality, respectively

$$\begin{array}{cc}\hfill {\nu }_R^{\left[m_r\right]}\left[r,\stackrel{m_r-1}{\overbrace{z_R,\dots ,z_R}}\right]& =r,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\mathsf{\mu }}_R^{\left[m_r\right]}\left[r_1,r_2,\dots ,r_{m_r-1},z_R\right]& =z_R\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\mathsf{\mu }}_R^{\left[n_r\right]}\left[r,\stackrel{n_r-1}{\overbrace{e_R,\dots ,e_R}}\right]& =r,r,r_i,z_R,e_R\in R,\hfill \end{array}$$

(31)

where $z_R$ and $e_R$ can be on any places. The polyadic zero $z_R$ in ${\mathcal{R}}^{\left[m_r,n_r\right]}$ is not necessary for $⟨R\mid {\nu }_R^{\left[m_r\right]}⟩$ to be a $m_r$-ary additive group for $m_r\ge 3$ (which is impossible for binary groups), but the additive querelement is important [26]. Nevertheless, one can adjoin the extraneous polyadic zero ${\dot{z}}_R\notin R$ externally by extending the underlying set R of the initial ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$ as follows

$$\dot{R}=R\cup \left\{{\dot{z}}_R\right\},$$

(32)

where ${\dot{z}}_R$ satisfies the needed standard relations (29) and (30).

The neutral element $e_R$ being a polyadic identity, has nothing with the multiplicative invertibility in the ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$. Nevertheless, some elements of ${\mathcal{R}}^{\left[m_r,n_r\right]}$ can be invertible, which means that for them there exists a polyadic analog of multiplicative inverse, the querelement $\overline{r}$ defined by [7]

$${\mathsf{\mu }}_R^{\left[n_r\right]}\left[\overline{r},\stackrel{n_r-1}{\overbrace{r,\dots ,r}}\right]=r,\overline{r},r\in R,$$

(33)

where $\overline{r}$ can be on any place and $n_r\ge 3$. By analogy with the binary ring, a polyadic ring without unity can be called a polyadic rng, a non-unital polyadic ring or pseudo-ring. The simplest $\left(n_r,m_r\right)$-rng example is $2\mathbb{Z}$.

We denote in ${\mathcal{R}}^{\left[m_r,n_r\right]}$ the subset of multiplicatively invertible elements (sometimes called units) by ${\mathtt{U}}_R\subset R$ which, in the binary case, is called a unit group ${\mathcal{U}}^{\left[n_u\right]}\left({\mathcal{R}}^{\left[m_r,n_r\right]}\right)$. In the polyadic case the set ${\mathcal{U}}^{\left[n_u\right]}$ should be the ${\mathrm{n}}_u$-ary group with ${\mathrm{n}}_u=n_r$.

The $n_g$-ary group is

$${\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}=⟨G\mid {\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]},\overline{\left(\right)}⟩,$$

(34)

with ${\mathsf{n}}_g$-ary multiplication ${\mathsf{n}}_g:G^{\times {\mathsf{n}}_g}\to G$, and each element $\mathsf{g}$ has the analog of inverse, its querelement $\overline{\mathsf{g}}$ obeying

$${\mathsf{\mu }}_{\mathsf{G}}\left[\overline{\mathsf{g}},\stackrel{{\mathsf{n}}_g-1}{\overbrace{\mathsf{g},\dots ,\mathsf{g}}}\right]=\mathsf{g},\overline{\mathsf{g}},\mathsf{g}\in G,$$

(35)

where $\overline{\mathsf{g}}$ can be on any place.

If in ${\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$ the polyadic identity ${\mathsf{e}}_{\mathsf{G}}$ exists (which is not necessary for ${\mathsf{n}}_g\ge 3$), it satisfies multiplicative neutrality

$${\mathsf{\mu }}_{\mathsf{G}}\left[\mathsf{g},\stackrel{{\mathsf{n}}_g-1}{\overbrace{{\mathsf{e}}_{\mathsf{G}},\dots ,{\mathsf{e}}_{\mathsf{G}}}}\right]=\mathsf{g},{\mathsf{e}}_{\mathsf{G}},\mathsf{g}\in G,$$

(36)

where $\mathsf{g}$ can be on any place. For more details and definitions, see [10].

Let us construct from $\left(m_r,n_r\right)$-ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$ (28) and $n_g$-ary group ${\mathsf{G}}^{\left[n_g\right]}$ (34) the polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ with the same underlying set of the formal sums (19) $\mathrm{R}=\left\{\mathbf{r}\right\}$

$${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}={\mathcal{R}}^{\left[n_r,m_r\right]}\left[{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right],$$

(37)

but now obeying ${\mathit{m}}_r$-ary addition ${\mathit{\nu }}^{\left[{\mathit{m}}_r\right]}:{\mathrm{R}}^{\times {\mathit{m}}_r}\to \mathrm{R}$ and ${\mathit{n}}_r$-ary multiplication ${\mathsf{\mu }}^{\left[{\mathit{n}}_r\right]}:{\mathrm{R}}^{\times {\mathit{n}}_r}\to \mathrm{R}$. Using the “arity freedom principle” [10] and analogy with the binary case (27) we have

Definition 4.

The polyadic group ring is the 3 set and 6 operation polyadic algebra-like structure

$${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}=⟨\mathrm{R},R,G\mid {\mathit{\nu }}_{\mathrm{R}}^{\left[{\mathit{m}}_r\right]},{\mathsf{\mu }}_{\mathrm{R}}^{\left[{\mathit{n}}_r\right]},{\mathsf{\rho }}_{\mathrm{R}}^{\left[{\mathbf{k}}_{\rho }\right]}\mid {\nu }_R^{\left[m_r\right]},{\mathsf{\mu }}_R^{\left[n_r\right]}\mid {\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}⟩,$$

(38)

where ${\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$ is ${\mathsf{n}}_g$-ary multiplication of the initial group ${\mathsf{G}}^{\left[n_g\right]}$ (with the underlying set G), ${\nu }_R^{\left[m_r\right]}$ and ${\mathsf{\mu }}_R^{\left[n_r\right]}$ are $m_r$-ary addition and $n_r$-ary multiplication of the initial polyadic ring ${\mathcal{R}}^{\left[n_r,m_r\right]}$ (with the underlying set R), ${\mathit{\nu }}_{\mathrm{R}}^{\left[{\mathit{m}}_r\right]}$, ${\mathsf{\mu }}_{\mathrm{R}}^{\left[{\mathit{n}}_r\right]}$ and ${\mathsf{\rho }}_{\mathrm{R}}^{\left[{\mathbf{k}}_{\rho }\right]}$ are ${\mathit{\nu }}_{\mathrm{R}}^{\left[{\mathit{m}}_r\right]}$-ary addition, ${\mathsf{\mu }}_{\mathrm{R}}^{\left[{\mathit{n}}_r\right]}$-ary multiplication and ${\mathit{k}}_{\rho }$-ary multiplication on scalar of the polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ (with the underlying set R).

Now we generalize the binary operations ${\mathit{\nu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]}$ (20), ${\mathsf{\mu }}_{\mathrm{R}}^{\left[\mathbf{2}\right]}$ (21) and ${\mathsf{\rho }}_{\mathrm{R}}^{\left[\mathbf{1}\right]}$ (26) to higher arity setting, implying that the arities of initial $\left(m_r,n_r\right)$-ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$ (28) and $n_g$-ary group ${\mathsf{G}}^{\left[n_g\right]}$ (34) are given.

Definition 5.

The ${\mathit{m}}_r$-ary addition ${\mathit{\nu }}^{\left[{\mathit{m}}_r\right]}$ can be defined by analogy with (20) left-“componentwise” by

$$\begin{array}{cc}\hfill {\mathit{\nu }}_{\mathrm{R}}^{\left[{\mathit{m}}_r\right]}\left[\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_1\right),\overrightarrow{\mathsf{g}}\right),\dots ,\mathbf{r}\left({\overrightarrow{r}}_{\tilde{\mathsf{g}}}\left({\alpha }_{{\mathit{m}}_r}\right),\overrightarrow{\mathsf{g}}\right)\right]& =\underset{i}{\mathsf{\Sigma }}{\nu }_R^{\left[m_r\right]}\left[r_{{\mathsf{g}}_i}\left({\alpha }_1\right),\dots ,r_{{\mathsf{g}}_i}\left({\alpha }_{m_r}\right)\right]·{\mathsf{g}}_i,\hfill \\ \hfill r_{{\mathsf{g}}_i}\left({\alpha }_{1,\dots ,{\mathit{m}}_r}\right)& \in R,{\mathsf{g}}_i\in G,\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_{1,\dots ,{\mathit{m}}_r}\right),\overrightarrow{\mathsf{g}}\right)\in \mathrm{R}.\hfill \end{array}$$

(39)

Proposition 1.

The arity of addition in the polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ coincides with the arity of addition in the initial polyadic ring ${\mathcal{R}}^{\left[n_r,m_r\right]}$ that is

$${\mathit{m}}_r=m_r,$$

(40)

if in both sides of (39) there is one polyadic operation (addition in ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ and addition in ${\mathcal{R}}^{\left[m_r,n_r\right]}$).

Proof. 

The statement (40) directly follows from the construction (39). □

Remark 1.

In the polyadic framework and from “arity freedom principle” [10], it follows that number of operations in both sides of (39) can be different, such that the arities of addition in the initial ring and the group ring may also differ, but the total number of ring elements in brackets should remain the same.

Remark 2.

Denote the number of $m_r$-ary additions in the initial $\left(m_r,n_r\right)$-ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$ by ${\mathcal{\ell }}_m$, being actually the polyadic power, and their composition by ${\left({\nu }_R^{\left[m_r\right]}\right)}^{\circ {\mathcal{\ell }}_m}$, where the total number of arguments is not arbitrary, as in the binary case, but “quantized” becoming

$${\mathcal{\ell }}_m\left(m_r-1\right)+1,$$

(41)

which directly follows from the general form of admissible length of words appearing in composition of higher arity operations (2).

Definition 6.

The higher (polyadic) power ${\mathit{m}}_r$-ary addition in the polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ can be defined by analogy with (39) left-“componentwise” by

$$\begin{array}{cc}\hfill & {\mathit{\nu }}_{\mathrm{R}}^{\left[{\mathit{m}}_r\right]}\left[\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_1\right),\overrightarrow{\mathsf{g}}\right),\dots ,\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_{{\mathit{m}}_r}\right),\overrightarrow{\mathsf{g}}\right)\right]\hfill \\ \hfill & =\underset{i}{\mathsf{\Sigma }}{\left({\nu }_R^{\left[m_r\right]}\right)}^{\circ {\mathcal{\ell }}_m}\left[r_{{\mathsf{g}}_i}\left({\alpha }_1\right),\dots ,r_{{\mathsf{g}}_i}\left({\alpha }_{{\mathcal{\ell }}_m\left(m_r-1\right)+1}\right)\right]·{\mathsf{g}}_i,\hfill \\ \hfill & r_{{\mathsf{g}}_i}\left({\alpha }_{1,\dots ,{\mathcal{\ell }}_m\left(m_r-1\right)+1}\right)\in R,{\mathsf{g}}_i\in G,\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_{1,\dots ,{\mathit{m}}_r}\right),\overrightarrow{\mathsf{g}}\right)\in \mathrm{R}.\hfill \end{array}$$

(42)

Therefore, we have

Theorem 1.

The arity of addition ${\mathit{m}}_r$ in the polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ which possesses a higher polyadic power of $m_r$-ary addition is

$${\mathit{m}}_r={\mathcal{\ell }}_m\left(m_r-1\right)+1.$$

(43)

Proof. 

The statement (43) follows from the construction (42), Remarks 1, 2 and the “quantization” condition (41). □

The multiplication in ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ can be defined similarly to the binary case in functional notation (21)

Definition 7.

In the polyadic group ring ${\mathcal{R}}^{\left[m_r,n_r\right]}\left[{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right]$ (37) the multiplication ${\mathsf{\mu }}_{\mathrm{R}}^{\left[{\mathit{n}}_r\right]}:{\mathrm{R}}^{\times {\mathit{n}}_r}\to \mathrm{R}$ can be defined by the both-“componentwise” convolution-like operation

$$\begin{array}{cc}\hfill & {\mathsf{\mu }}_{\mathrm{R}}^{\left[{\mathit{n}}_r\right]}\left[\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_1\right),\overrightarrow{\mathsf{g}}\right),\dots ,\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_{{\mathit{n}}_r}\right),\overrightarrow{\mathsf{g}}\right)\right]\hfill \\ \hfill & =\underset{i_1}{\mathsf{\Sigma }}\dots \underset{i_{n_r}}{\mathsf{\Sigma }}\underset{j_1}{\mathsf{\Sigma }}\dots \underset{j_{{\mathsf{n}}_g}}{\mathsf{\Sigma }}{\mathsf{\mu }}_R^{\left[n_r\right]}\left[r_{{\mathsf{g}}_{i_1}}\left({\alpha }_1\right),\dots ,r_{{\mathsf{g}}_{i_{n_r}}}\left({\alpha }_{n_r}\right)\right]·{\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\left[{\mathsf{g}}_{j_1},\dots ,{\mathsf{g}}_{j_{{\mathsf{n}}_g}}\right],\hfill \\ \hfill & r_{{\mathsf{g}}_i}\left({\alpha }_j\right)\in R,{\mathsf{g}}_i\in G,\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_j\right),\overrightarrow{\mathsf{g}}\right)\in \mathrm{R},\hfill \end{array}$$

(44)

if polyadic power of all operations is 1.

Proposition 2.

The arities of multiplications defined by (44) in ${\mathcal{R}}^{\left[n_r,m_r\right]}$, ${\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$ and ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ coincide

$${\mathit{n}}_r=n_r={\mathsf{n}}_g.$$

(45)

Proof. 

The statement (45) follows directly from both-“componentwise” convolution (44). □

The construction of the multiplication is more elaborate and interesting, if we take into account Remarks 1 and 2 and use the “arity freedom principle” [10] (initial arities are taken arbitrary). Indeed, let us denote polyadic powers of ${\mathsf{\mu }}_R^{\left[n_r\right]}$ and ${\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$ by ${\mathcal{\ell }}_n$ and ${\mathcal{\ell }}_g$, correspondingly, then we have

Definition 8.

The polyadic group ring with higher polyadic power of multiplications is defined by

$$\begin{array}{cc}\hfill & {\mathsf{\mu }}_{\mathrm{R}}^{\left[{\mathit{n}}_r\right]}\left[\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_1\right),\overrightarrow{\mathsf{g}}\right),\dots ,\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_{{\mathit{n}}_r}\right),\overrightarrow{\mathsf{g}}\right)\right]\hfill \\ \hfill & =\underset{i_1}{\mathsf{\Sigma }}\dots \underset{{\mathcal{\ell }}_n\left(n_r-1\right)+1}{\mathsf{\Sigma }}\underset{j_1}{\mathsf{\Sigma }}\dots \underset{{\mathcal{\ell }}_g\left({\mathsf{n}}_g-1\right)+1}{\mathsf{\Sigma }}{\left({\mathsf{\mu }}_R^{\left[n_r\right]}\right)}^{\circ {\mathcal{\ell }}_n}\left[r_{{\mathsf{g}}_{i_1}}\left({\alpha }_1\right),\dots ,r_{{\mathsf{g}}_{i_{{\mathcal{\ell }}_n\left(n_r-1\right)+1}}}\left({\alpha }_{{\mathcal{\ell }}_n\left(n_r-1\right)+1}\right)\right]\hfill \\ \hfill & ·{\left({\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right)}^{{\mathcal{\ell }}_{{\mathsf{n}}_g}}\left[{\mathsf{g}}_{j_1},\dots ,{\mathsf{g}}_{j_{{\mathcal{\ell }}_g\left({\mathsf{n}}_g-1\right)+1}}\right],r_{{\mathsf{g}}_i}\left({\alpha }_j\right)\in R,{\mathsf{g}}_i\in G,\mathbf{r}\left({\overrightarrow{r}}_{\overrightarrow{\mathsf{g}}}\left({\alpha }_j\right),\overrightarrow{\mathsf{g}}\right)\in \mathrm{R}.\hfill \end{array}$$

(46)

The “quantization” conditions for multiplications in ${\mathcal{R}}^{\left[n_r,m_r\right]}$ and ${\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$, analogous to those for additions in (41), now arise from the equality of the total number of arguments in the Equation (46).

$${\mathcal{\ell }}_n\left(n_r-1\right)+1={\mathcal{\ell }}_g\left({\mathsf{n}}_g-1\right)+1.$$

(47)

Because each $r_{{\mathsf{g}}_i}\left({\alpha }_j\right)\in R$ corresponds to one ${\mathsf{g}}_i\in G$ in (46), their amount (or word lengths, see the general word size (2)) should be the same, while they enter into the different operations: $n_r$-ary ring multiplication with polyadic power ${\mathcal{\ell }}_n$ and $n_g$-ary group multiplication with polyadic power ${\mathcal{\ell }}_g$ (see (3)).

Thus, we have

Theorem 2.

The arity of multiplication ${\mathit{n}}_r$ in the polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ (with higher polyadic powers of multiplications in the initial polyadic ring ${\mathsf{\mu }}_R^{\left[n_r\right]}$ and the ${\mathsf{n}}_g$-ary group ${\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$) is

$${\mathit{n}}_r={\mathcal{\ell }}_n\left(n_r-1\right)+1={\mathcal{\ell }}_g\left({\mathsf{n}}_g-1\right)+1.$$

(48)

Proof. 

The statement (48) follows from the construction (46) and the “quantization” condition (47). □

Obviously, if all polyadic powers are equal to one ${\mathcal{\ell }}_n={\mathcal{\ell }}_g=1$, then all operations share the same arity (45).

Definition 9.

The polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}={\mathcal{R}}^{\left[m_r,n_r\right]}\left[{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right]$ (37) which has initial multiplications ${\mathsf{\mu }}_R^{\left[n_r\right]}$ and ${\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$ of higher polyadic powers (46), is called a higher power polyadic group ring.

5. Properties

Here we consider basic properties of the polyadic group rings, which are in the higher arity case can be unusual and exotic.

In the binary case, the associativity of addition of the group ring $\mathcal{R}\left[\mathsf{G}\right]$ trivially follows from the associativity of addition in the initial ring $\mathcal{R}$ because of the the left-“componentwise” addition (14). The same conclusion is valid for polyadic additions ${\mathit{\nu }}_R^{\left[{\mathit{m}}_r\right]}$ and ${\nu }_R^{\left[m_r\right]}$ with unit polyadic power (39).

Proposition 3.

In case of higher polyadic powers (42) ${\mathcal{\ell }}_m>1$ the total associativity of addition in the polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ follows from the associativity of the addition in the initial ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$, if the strong inequality takes place

$${\mathit{m}}_r>m_r.$$

(49)

Proof. 

It follows from the “quantization” condition (41) and the obvious fact that in the same structure lower arity parentheses are inserted into higher arity ones, which is informally can be treated as “larger brackets can be constructed from smaller brackets”. □

The connection between associativities of multiplications is more complicated to prove.

Theorem 3.

If the multiplication in the initial ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$ is totally polyadic associative, then the polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ is totally associative multiplicatively, when all the arities of multiplication are equal

$${\mathit{n}}_r=n_r={\mathsf{n}}_g.$$

(50)

Proof. 

Using (44) in the notation (19), we compute ${\mathit{n}}_r$ terms in the total associativity

$$\begin{array}{}\mathrm{(51)}& \hfill \mathbf{r}\left({\beta }_1\right)& =\mathsf{\Sigma }{\mathsf{\mu }}_{\mathrm{R}}^{\left[n_r\right]}\left[{\mathsf{\mu }}_{\mathrm{R}}^{\left[n_r\right]}\left[\mathbf{r}\left({\alpha }_1\right),\dots ,\mathbf{r}\left({\alpha }_{n_r}\right)\right],\mathbf{r}\left({\alpha }_{n_r+1}\right),\dots ,\mathbf{r}\left({\alpha }_{2n_r-1}\right)\right],\hfill \mathrm{(52)}& \hfill \mathbf{r}\left({\beta }_2\right)& =\mathsf{\Sigma }{\mathsf{\mu }}_{\mathrm{R}}^{\left[n_r\right]}\left[\mathbf{r}\left({\alpha }_1\right),{\mathsf{\mu }}_{\mathrm{R}}^{\left[n_r\right]}\left[\mathbf{r}\left({\alpha }_2\right),\dots ,\mathbf{r}\left({\alpha }_{n_r+1}\right)\right],\mathbf{r}\left({\alpha }_{n_r+2}\right),\dots ,\mathbf{r}\left({\alpha }_{2n_r-1}\right)\right],\hfill & & ⋮\hfill \mathrm{(53)}& \hfill \mathbf{r}\left({\beta }_{n_r}\right)& =\mathsf{\Sigma }{\mathsf{\mu }}_{\mathrm{R}}^{\left[n_r\right]}\left[\mathbf{r}\left({\alpha }_1\right),\dots ,\mathbf{r}\left({\alpha }_{n_r-1}\right),{\mu }_R^{\left[n_r\right]}\left[\mathbf{r}\left({\alpha }_{n_r}\right),\dots ,\mathbf{r}\left({\alpha }_{2n_r-1}\right)\right]\right],\hfill \end{array}$$

where $\mathsf{\Sigma }$ denotes the sum by all corresponding internal indices. Then we take into account the both-“componentwise” convolution and total associativity of ${\mathsf{n}}_g$-ary group ${\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$ to obtain

$$\begin{array}{cc}\hfill \mathbf{r}\left({\beta }_1\right)& =\mathsf{\Sigma }{\mathsf{\mu }}_R^{\left[n_r\right]}\left[{\mathsf{\mu }}_R^{\left[n_r\right]}\left[r_{{\mathsf{g}}_{i_1}}\left({\alpha }_1\right),\dots ,r_{{\mathsf{g}}_{i_{n_r}}}\left({\alpha }_{n_r}\right)\right],r_{{\mathsf{g}}_{i_{n_r}+1}}\left({\alpha }_{n_r+1}\right),\dots ,r_{{\mathsf{g}}_{2i_{n_r}-1}}\left({\alpha }_{2n_r-1}\right)\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ·{\left({\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right)}^{\circ 2}\left[{\mathsf{g}}_{j_1},\dots ,{\mathsf{g}}_{2j_{{\mathsf{n}}_g}+1}\right],\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill \mathbf{r}\left({\beta }_2\right)& =\mathsf{\Sigma }{\mathsf{\mu }}_R^{\left[n_r\right]}\left[r_{{\mathsf{g}}_{i_1}}\left({\alpha }_1\right),{\mathsf{\mu }}_R^{\left[n_r\right]}\left[r_{{\mathsf{g}}_{i_2}}\left({\alpha }_2\right)\dots ,r_{{\mathsf{g}}_{i_{n_r}+1}}\left({\alpha }_{n_r+1}\right)\right],r_{{\mathsf{g}}_{i_{n_r}+2}}\left({\alpha }_{n_r+2}\right),\dots ,r_{{\mathsf{g}}_{2i_{n_r}-1}}\left({\alpha }_{2n_r-1}\right)\right]\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill & ·{\left({\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right)}^{\circ 2}\left[{\mathsf{g}}_{j_1},\dots ,{\mathsf{g}}_{2j_{{\mathsf{n}}_g}+1}\right],\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & ⋮\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill \mathbf{r}\left({\beta }_{{\mathit{n}}_r}\right)& =\mathsf{\Sigma }{\mathsf{\mu }}_R^{\left[n_r\right]}\left[r_{{\mathsf{g}}_{i_1}}\left({\alpha }_1\right),\dots ,r_{{\mathsf{g}}_{i_{n_r-1}}}\left({\alpha }_{n_r-1}\right),{\mathsf{\mu }}_R^{\left[n_r\right]}\left[r_{{\mathsf{g}}_{i_{n_r}}}\left({\alpha }_{n_r}\right),\dots ,r_{{\mathsf{g}}_{2i_{n_r}-1}}\left({\alpha }_{2n_r-1}\right)\right]\right]\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill & ·{\left({\mathsf{\mu }}_{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right)}^{\circ 2}\left[{\mathsf{g}}_{j_1},\dots ,{\mathsf{g}}_{2j_{{\mathsf{n}}_g}+1}\right],\hfill \end{array}$$

(59)

where the (group dependence) terms on r.h.s. of the formal product $\left(·\right)$ coincide after suitable rename of the summation indices. All the terms on the l.h.s. are equal due to the total polyadic associativity in the initial ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$. Thus, the polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}={\mathcal{R}}^{\left[m_r,n_r\right]}\left[{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right]$ (37) is multiplicatively totally associative for equal arities (50). □

Theorem 4.

In case of higher polyadic powers (42) ${\mathcal{\ell }}_n>1$ and/or ${\mathcal{\ell }}_g>1$ the total associativity of multiplication in the polyadic group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}={\mathcal{R}}^{\left[m_r,n_r\right]}\left[{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right]$ follows from the associativity of the addition in the initial ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$, if the strong inequalities take place

$${\mathit{n}}_r>n_r\cup {\mathit{n}}_r>n_g.$$

(60)

Proof. 

It follows from the “quantization” condition (47) and the informal consequence “larger brackets can be constructed from smaller brackets”. □

The polyadic distributivity in ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ is governed by polyadic distributivity of the initial ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$, because the addition is present only in the l.h.s. of the polyadic group ring elements (13).

By the same reason, if the initial ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$ has the polyadic zero $z_R$, then the group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ has the polyadic zero ${\mathbf{z}}_{\mathrm{R}}$ of the form

$${\mathbf{z}}_{\mathrm{R}}=z_R·\underset{i}{\mathsf{\Sigma }}{\mathsf{g}}_i,z_R\in R,{\mathsf{g}}_i\in G,{\mathbf{z}}_{\mathrm{R}}\in \mathrm{R},$$

(61)

such that no group elements appear with nonzero coefficients, since the finite support, and such element is unique. Also, the zero ${\mathbf{z}}_{\mathrm{R}}$ in ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ is the additive polyadic identity and is multiplicatively absorbing (for ${\mathcal{R}}^{\left[m_r,n_r\right]}$ see (29) and (30))

$$\begin{array}{cc}\hfill {\mathit{\nu }}_{\mathrm{R}}^{\left[{\mathit{m}}_r\right]}\left[\mathbf{r}\left(\alpha \right),\stackrel{m_r-1}{\overbrace{{\mathbf{z}}_R,\dots ,{\mathbf{z}}_R}}\right]& =\mathbf{r}\left(\alpha \right),\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill {\mathsf{\mu }}_R^{\left[{\mathit{n}}_r\right]}\left[\mathbf{r}\left({\alpha }_1\right),\mathbf{r}\left({\alpha }_2\right),\dots ,\mathbf{r}\left({\alpha }_{n_r-1}\right),{\mathbf{z}}_{\mathrm{R}}\right]& ={\mathbf{z}}_{\mathrm{R}},\mathbf{r}\left(\alpha \right),\mathbf{r}\left({\alpha }_i\right),{\mathbf{z}}_{\mathrm{R}}\in \mathrm{R}.\hfill \end{array}$$

(63)

In the polyadic case, the identity of multiplication is only a neutral element (36) and has no connection with invertibility (see, e.g., [10]). If the initial $\left(m_r,n_r\right)$-ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$ has the polyadic identity $e_R$ (31) and the identity of the $n_g$-ary group ${\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$ (34) is ${\mathsf{e}}_{\mathsf{G}}$, then the trivial polyadic identity ${\mathbf{e}}_{\mathrm{R}}$ of the group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ is

$${\mathbf{e}}_{\mathrm{R}}=e_R·{\mathsf{e}}_{\mathsf{G}},e_R\in R,{\mathsf{e}}_{\mathsf{G}}\in G,{\mathbf{e}}_{\mathrm{R}}\in \mathrm{R},$$

(64)

such that in the sum (19) one coefficient from ${\mathcal{R}}^{\left[m_r,n_r\right]}$ at the group identity ${\mathsf{e}}_{\mathsf{G}}$ is $e_R$, while others are equal to $z_R$ and therefore are not written here.

The invertibility properties of polyadic structures are governed not by neutral elements, but by querelements [10]. So for elements from multiplicative ${\mathrm{n}}_u$-ary unit group ${\mathcal{U}}^{\left[{\mathrm{n}}_u\right]}\left[{\mathcal{R}}^{\left[m_r,n_r\right]}\right]$ (the subset ${\mathtt{U}}_R\subset R$ having the querelement $\overline{r}$), and the $n_g$-ary group ${\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$ (34) having the querelement $\overline{\mathsf{g}}$ (35) for each $\mathsf{g}\in \mathsf{G}$, we can formulate

Definition 10.

In the group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ the querelement $\overline{\mathbf{r}}\left(\alpha \right)$ for some elements $\mathbf{r}\left(\alpha \right)\in \mathrm{R}$ is defined by

$$\overline{\mathbf{r}}\left(\alpha \right)=\underset{i}{\mathsf{\Sigma }}{\overline{r}}_{{\mathsf{g}}_i}\left(\alpha \right)·{\overline{\mathsf{g}}}_i,\overline{r}\in \mathtt{U},{\overline{\mathsf{g}}}_i,{\mathsf{g}}_i\in G,\overline{\mathbf{r}}\in \mathrm{R},$$

(65)

where in the simplest case $\overline{r}$ and $\overline{\mathsf{g}}$ are defined in (33) and (35), respectively, if all arities coincide (50).

In the nontrivial approach, the r.h.s. can contain also more general elements for which one should solve system of equations in each concrete case. We denote the subset of multiplicatively invertible elements in the group ring ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ (sometimes called group units) by ${\mathtt{U}}_R\in R$ which should form the ${\mathit{n}}_u$-ary group ${\mathcal{U}}^{\left[{\mathit{n}}_u\right]}\left({\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}\right)$, or a polyadic unit group.

Definition 11.

The polyadic augmentation map $\mathsf{\epsilon }:{\mathcal{R}}^{\left[m_r,n_r\right]}\left[{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right]\to {\mathcal{R}}^{\left[m_r,n_r\right]}$ can be defined for αth element (19) of ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}$ as follows

$$\underset{i=1}{{\mathsf{\Sigma }}^{i_{max}}}{r}_{{\mathsf{g}}_i}\left(\alpha \right)·{\mathsf{g}}_i\mapsto {\left({\nu }_R^{\left[m_r\right]}\right)}^{\circ {\mathcal{\ell }}_i}\left[r_{{\mathsf{g}}_1}\left(\alpha \right),\dots ,r_{{\mathsf{g}}_{{\mathcal{\ell }}_i\left(m_r-1\right)+1}}\left(\alpha \right)\right],r_{{\mathsf{g}}_i}\left(\alpha \right)\in {\mathtt{U}}_R,{\mathsf{g}}_i\in G.$$

(66)

Remark 3.

Note that on the l.h.s. of (66) we have the formal sum Σ by i, if finite, then till $i_{max}$, while on the r.h.s. the sum becomes $m_r$-ary addition in the initial $\left(m_r,n_r\right)$-ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$, therefore we have the “quantization” condition for the polyadic augmentation

$$i_{max}=i_{max}\left({\mathcal{\ell }}_i,m_r\right)={\mathcal{\ell }}_i\left(m_r-1\right)+1,$$

(67)

where ${\mathcal{\ell }}_i$ is the polyadic power of the initial ring ${\mathcal{R}}^{\left[m_r,n_r\right]}$ addition.

Definition 12.

The kernel of the polyadic augmentation map is called a polyadic augmentation ideal and is defined by setting coefficients sum in (66) equal to the polyadic zero (61), as follows

$$ker\epsilon =⟨R\mid {\left({\nu }_R^{\left[m_r\right]}\right)}^{\circ {\mathcal{\ell }}_i}\left[r_{{\mathsf{g}}_1}\left(\alpha \right),\dots ,r_{{\mathsf{g}}_{{\mathcal{\ell }}_i\left(m_r-1\right)+1}}\left(\alpha \right)\right]={\mathbf{z}}_{\mathrm{R}}⟩,r_{{\mathsf{g}}_i}\left(\alpha \right)\in {\mathtt{U}}_R,{\mathsf{g}}_i\in G.$$

(68)

The polyadic augmentation map $\mathsf{\epsilon }$ preserves addition and multiplication without changing the arities and maps the corresponding identities in ${\mathcal{R}}^{\left[m_r,n_r\right]}\left[{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right]$ and ${\mathcal{R}}^{\left[m_r,n_r\right]}$.

6. Examples

Let us present simple but nontrivial examples of the polyadic group rings ${\mathbf{R}}^{\left[{\mathit{m}}_r,{\mathit{n}}_r\right]}={\mathcal{R}}^{\left[m_r,n_r\right]}\left[{\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}\right]$ and list their main properties. First, we present in detail the example of the polyadic power equal to one, then briefly show higher polyadic powers in addition and multiplication separately.

Example 1.

We take for the initial ring the commutative nonderived $\left(2,3\right)$-ring with the underlying set $R=\mathrm{j}\mathbb{Z}$ (${\mathrm{j}}^2=-1$), operations are in $\mathbb{C}$. Now ${\nu }_R^{\left[2\right]}$ and ${\mathsf{\mu }}_R^{\left[3\right]}$ are usual addition and product. Note that $⟨\mathrm{j}\mathbb{Z}\mid {\nu }_R^{\left[2\right]}⟩$ is a binary group with respect to addition, and $⟨\mathrm{j}\mathbb{Z}\mid {\mathsf{\mu }}_R^{\left[3\right]}⟩$ is not a ternary group, but only a ternary semigroup, because there is no multiplicative querelement for each $r\in R$. Obviously, that ${\mathcal{R}}^{\left[2,3\right]}$ is unitless. Note that polyadic distributivity follows from the binary distributivity in $\mathbb{Z}$. Therefore,

$${\mathcal{R}}^{\left[2,3\right]}=⟨\mathrm{j}\mathbb{Z}\mid {\nu }_R^{\left[2\right]},{\mathsf{\mu }}_R^{\left[3\right]},z_R⟩$$

(69)

is a commutative nonderived $\left(2,3\right)$-ring without multiplicative neutral element and $z_R=\mathrm{j}0=0$.

Let ${\mathcal{C}}_3=⟨\left\{e,a,a^2\right\}\mid a^3=a^0=e⟩$ be the cyclic group of order 3 with the identity e and one generator a. We take for the initial polyadic group ${\mathsf{G}}^{\left[{\mathsf{n}}_g\right]}$ the finite set of $2\times 2$ antidiagonal symbolic matrices $G=adiag\left({\mathcal{C}}_3,{\mathcal{C}}_3\right)$, which is closed with respect to triple matrix multiplication. The element of G can be presented in the form

$${\mathsf{g}}_i=g\left(m,n\right)=\left(\begin{array}{cc}0& a^m\\ a^n& 0\end{array}\right),$$

(70)

where $m,n\in \mathbb{Z}mod3$, such that the cardinality $\left|G\right|=9$, and the manifest form of the elements (70) are

$$\begin{array}{ccc}\hfill {\mathsf{g}}_1=\left(\begin{array}{cc}0& e\\ e& 0\end{array}\right),& {\mathsf{g}}_2=\left(\begin{array}{cc}0& a\\ e& 0\end{array}\right),& {\mathsf{g}}_3=\left(\begin{array}{cc}0& a^2\\ e& 0\end{array}\right),\hfill \\ \hfill {\mathsf{g}}_4=\left(\begin{array}{cc}0& e\\ a& 0\end{array}\left(\begin{array}{cc}0& a^2\\ e& 0\end{array}\right)\right),& {\mathsf{g}}_5=\left(\begin{array}{cc}0& a\\ a& 0\end{array}\right),& {\mathsf{g}}_6=\left(\begin{array}{cc}0& a^2\\ a& 0\end{array}\right),\hfill \\ \hfill {\mathsf{g}}_7=\left(\begin{array}{cc}0& e\\ a^2& 0\end{array}\right),& {\mathsf{g}}_8=\left(\begin{array}{cc}0& a\\ a^2& 0\end{array}\right),& {\mathsf{g}}_9=\left(\begin{array}{cc}0& a^2\\ a^2& 0\end{array}\right).\hfill \end{array}$$

(71)

The ternary multiplication ${\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}$ is nonderived (any even product gives a diagonal matrix that is out of the set $G=\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}$), ternary noncommutative and has the form

$${\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\left[{\mathsf{g}}_i,{\mathsf{g}}_j,{\mathsf{g}}_k\right]=g\left(m_1,n_1\right)g\left(m_2,n_2\right)g\left(m_3,n_3\right)=g\left(m_1+n_2+m_3,n_1+m_2+n_3\right).$$

(72)

The ternary identity ${\mathsf{e}}_{\mathsf{G}}$ (36) is defined by

$${\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\left[{\mathsf{e}}_{\mathsf{G}},{\mathsf{e}}_{\mathsf{G}},\mathsf{g}\right]=\mathsf{g}.$$

(73)

Using (70) and (72) we obtain the identity in matrix form

$${\mathsf{e}}_{\mathsf{G}}=e\left(t\right)=\left(\begin{array}{cc}0& a^t\\ a^{3-t}& 0\end{array}\right),t=\mathbb{Z}mod3.$$

(74)

So the ternary group ${\mathsf{G}}^{\left[\mathsf{3}\right]}$ has 3 identities

$${\mathsf{e}}_{\mathsf{G},1}=\mathrm{e}\left(0\right)=\left(\begin{array}{cc}0& e\\ e& 0\end{array}\right)={\mathsf{g}}_1,{\mathsf{e}}_{\mathsf{G},2}=\mathrm{e}\left(1\right)=\left(\begin{array}{cc}0& a\\ a^2& 0\end{array}\right)={\mathsf{g}}_9,{\mathsf{e}}_{\mathsf{G},3}=\mathrm{e}\left(2\right)=\left(\begin{array}{cc}0& a^2\\ a& 0\end{array}\right)={\mathsf{g}}_6.$$

(75)

Each element in ${\mathsf{G}}^{\left[\mathsf{3}\right]}$ has its querelement $\overline{\mathsf{g}}\left(\mathsf{g}\right)$ or $\overline{g}\left(m,n\right)=\overline{g\left(m,n\right)}$ defined by

$${\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\left[\mathsf{g},\mathsf{g},\overline{\mathsf{g}}\right]=\mathsf{g},g\left(m,n\right)g\left(m,n\right)\overline{g}\left(m,n\right)=g\left(m,n\right),$$

(76)

and in the matrix form

$${\overline{\mathsf{g}}}_i=\overline{g}\left(m,n\right)=\left(\begin{array}{cc}0& a^{3-n}\\ a^{3-m}& 0\end{array}\right),m,n=\mathbb{Z}mod3.$$

(77)

Note that each element of ${\mathsf{G}}^{\left[\mathsf{3}\right]}$ is polyadic multiplicative idempotent (4) because from (72) it follows

$${\mathsf{g}}^{⟨3⟩}={\left({\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\right)}^{\circ 3}\left[{\mathsf{g}}^7\right]=\mathsf{g},\forall \mathsf{g}\in G.$$

(78)

So during “polyadization” ${\mathcal{C}}_3\to adiag\left({\mathcal{C}}_3,{\mathcal{C}}_3\right)$ the binary cyclic group of order 3 becomes the ternary group of idempotents of polyadic power 3.

Thus, we have that

$${\mathsf{G}}^{\left[\mathsf{3}\right]}=⟨\left\{{\mathsf{g}}_i\right\}\mid {\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]},{\mathsf{e}}_{\mathsf{G},i},\overline{\left(\right)}⟩,$$

(79)

where the queroperation $\overline{\left(\right)}$ is defined in (77), becomes the ternary group having 3 identities (75).

The polyadic group $\left(\mathbf{2},\mathbf{3}\right)$-ring ${\mathcal{R}}^{\left[2,3\right]}\left[{\mathsf{G}}^{\left[\mathsf{3}\right]}\right]={\mathbf{R}}^{\left[\mathbf{2},\mathbf{3}\right]}$, as the set R consists of the finite formal sums (13), and the αth element of R has the form

$$\mathbf{r}\left(\alpha \right)=\underset{i}{\mathsf{\Sigma }}{r}_i\left(\alpha \right)·{\mathsf{g}}_i,r_i\left(\alpha \right)\in R,{\mathsf{g}}_i\in G,\mathbf{r}\left(\alpha \right)\in \mathrm{R},$$

(80)

where $r_i\left(\alpha \right)\in j\mathbb{Z}$ and ${\mathsf{g}}_i$ are defined in (70). Manifestly, we obtain

$$\mathbf{r}\left(\alpha \right)=\underset{i}{\mathsf{\Sigma }}j{k}_i\left(\alpha \right)·\left(\begin{array}{cc}0& a^{m_i}\\ a^{n_i}& 0\end{array}\right),m_i,n_i=\mathbb{Z}mod3,k_i\left(\alpha \right)\in \mathbb{Z}.$$

(81)

There is no identity in the polyadic group ring ${\mathcal{R}}^{\left[2,3\right]}\left[{\mathsf{G}}^{\left[\mathsf{3}\right]}\right]$, because $\left(2,3\right)$-ring ${\mathcal{R}}^{\left[2,3\right]}$ is unitless, but zero is

$${\mathbf{z}}_{\mathrm{R}}=z_R·\mathsf{g},z_R=\mathrm{j}0=0\in \mathbb{Z},\mathsf{g}\in {\mathsf{G}}^{\left[\mathsf{3}\right]}.$$

(82)

The commutative binary addition ${\mathit{\nu }}_R^{\left[\mathbf{2}\right]}$ in ${\mathbf{R}}^{\left[\mathbf{2},\mathbf{3}\right]}$ is simply “left”-componentwise (42) and reduces to the binary addition ${\nu }_{\mathrm{R}}^{\left[2\right]}$ in ${\mathcal{R}}^{\left[2,3\right]}$. The ternary multiplication ${\mathsf{\mu }}_{\mathrm{R}}^{\left[\mathbf{3}\right]}$ in the polyadic group $\left(\mathbf{2},\mathbf{3}\right)$-ring ${\mathbf{R}}^{\left[2,\mathbf{3}\right]}$ is noncommutative and needs gathering of the similar coefficients before each product of the ternary group ${\mathsf{G}}^{\left[\mathsf{3}\right]}$ (see (17) and (44)).

For instance, we have 3 elements of the polyadic group $\left(\mathbf{2},\mathbf{3}\right)$-ring ${\mathbf{R}}^{\left[\mathbf{2},\mathbf{3}\right]}$

$$\begin{array}{cc}\hfill \mathbf{r}\left(1\right)& =5\mathrm{j}·{\mathsf{g}}_5,\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill \mathbf{r}\left(2\right)& =2\mathrm{j}·{\mathsf{g}}_7 \mathsf{\dotplus }\left(-7\mathrm{j}\right)·{\mathsf{g}}_8,\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill \mathbf{r}\left(3\right)& =\left(-4\mathrm{j}\right)·{\mathsf{g}}_2\dotplus 7\mathrm{j}·{\mathsf{g}}_3\dotplus \left(-3\mathrm{j}\right)·{\mathsf{g}}_6,\hfill \end{array}$$

(85)

where $\left(\dotplus \right)$ is the formal sum ${\mathsf{\Sigma }}_i$.

Now we ternary multiply them (46) and formally open the brackets (for brevity, clearness and conciseness we do not write the ternary multiplications manifestly)

$$\begin{array}{cc}\hfill & {\mathbf{r}}_0={\mathsf{\mu }}_{\mathrm{R}}^{\left[\mathbf{3}\right]}\left[\mathbf{r}\left(1\right),\mathbf{r}\left(2\right),\mathbf{r}\left(3\right)\right]=\left(5\mathrm{j}·{\mathsf{g}}_5\right)\left(2\mathrm{j}·{\mathsf{g}}_7\dotplus \left(-7\mathrm{j}\right)·{\mathsf{g}}_8\right)\left(\left(-5\mathrm{j}\right)·{\mathsf{g}}_2\dotplus 7\mathrm{j}·{\mathsf{g}}_3\dotplus \left(-3\mathrm{j}\right)·{\mathsf{g}}_6\right)\hfill \\ \hfill & =\left(5\mathrm{j}·{\mathsf{g}}_5\right)\left(2\mathrm{j}·{\mathsf{g}}_7\right)\left(\left(-4\mathrm{j}\right)·{\mathsf{g}}_2\right)\dotplus \left(5\mathrm{j}·{\mathsf{g}}_5\right)\left(3\mathrm{j}·{\mathsf{g}}_7\right)\left(7\mathrm{j}·{\mathsf{g}}_3\right)\dotplus \left(5\mathrm{j}·{\mathsf{g}}_5\right)\left(2\mathrm{j}·{\mathsf{g}}_7\right)\left(\left(-3\mathrm{j}\right)·{\mathsf{g}}_6\right)\hfill \\ \hfill & \dotplus \left(5\mathrm{j}·{\mathsf{g}}_5\right)\left(\left(-7\mathrm{j}\right)·{\mathsf{g}}_8\right)\left(\left(-4\mathrm{j}\right)·{\mathsf{g}}_2\right)\dotplus \left(5\mathrm{j}·{\mathsf{g}}_5\right)\left(\left(-7\mathrm{j}\right)·{\mathsf{g}}_8\right)\left(7\mathrm{j}·{\mathsf{g}}_3\right)\hfill \\ \hfill & \dotplus \left(5\mathrm{j}·{\mathsf{g}}_5\right)\left(\left(-7\mathrm{j}\right)·{\mathsf{g}}_8\right)\left(\left(-3\mathrm{j}\right)·{\mathsf{g}}_6\right).\hfill \end{array}$$

(86)

Then we use the ternary group multiplication (with manifest ternary multiplication ${\mathsf{\mu }}_R^{\left[3\right]}$ in the initial ring ${\mathcal{R}}^{\left[2,3\right]}$ and ${\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}$ in the ternary multiplication in ${\mathsf{G}}^{\left[\mathsf{3}\right]}$)

$$\begin{array}{cc}\hfill {\mathbf{r}}_0& ={\mathsf{\mu }}_R^{\left[3\right]}\left[5\mathrm{j},2\mathrm{j},\left(-4\mathrm{j}\right)\right]·{\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\left[{\mathsf{g}}_5,{\mathsf{g}}_7,{\mathsf{g}}_2\right]\dotplus {\mathsf{\mu }}_R^{\left[3\right]}\left[5\mathrm{j},2\mathrm{j},7\mathrm{j}\right]·{\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\left[{\mathsf{g}}_5,{\mathsf{g}}_7,{\mathsf{g}}_3\right]\hfill \\ \hfill & \dotplus {\mathsf{\mu }}_R^{\left[3\right]}\left[5\mathrm{j},2\mathrm{j},\left(-3\mathrm{j}\right)\right]·{\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\left[{\mathsf{g}}_5,{\mathsf{g}}_7,{\mathsf{g}}_6\right]\dotplus {\mathsf{\mu }}_R^{\left[3\right]}\left[5\mathrm{j},\left(-7\mathrm{j}\right),\left(-4\mathrm{j}\right)\right]·{\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\left[{\mathsf{g}}_5,{\mathsf{g}}_8,{\mathsf{g}}_2\right]\hfill \\ \hfill & \dotplus {\mathsf{\mu }}_R^{\left[3\right]}\left[5\mathrm{j},\left(-7\mathrm{j}\right),7\mathrm{j}\right]·{\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\left[{\mathsf{g}}_5,{\mathsf{g}}_8,{\mathsf{g}}_3\right]\dotplus {\mathsf{\mu }}_R^{\left[3\right]}\left[5\mathrm{j},\left(-7\mathrm{j}\right),\left(-3\mathrm{j}\right)\right]·{\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\left[{\mathsf{g}}_5,{\mathsf{g}}_8,{\mathsf{g}}_6\right].\hfill \end{array}$$

(87)

Performing the ternary multiplications ${\mathsf{\mu }}_R^{\left[3\right]}$ and ${\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}$ in ${\mathcal{R}}^{\left[5,3\right]}$ we obtain the formal sum

$${\mathbf{r}}_0=40\mathrm{j}·{\mathsf{g}}_5\dotplus \left(-70\mathrm{j}\right)·{\mathsf{g}}_6\dotplus 30\mathrm{j}·{\mathsf{g}}_9\dotplus \left(-140\right)\mathrm{j}·{\mathsf{g}}_9\dotplus 245\mathrm{j}·{\mathsf{g}}_9\dotplus \left(-105\mathrm{j}\right)·{\mathsf{g}}_3.$$

(88)

Finally, we gather coefficients from the initial ring ${\mathcal{R}}^{\left[2,3\right]}$ to get the ternary product of elements (83)–(85) from the polyadic group ring ${\mathbf{R}}^{\left[\mathbf{2},\mathbf{3}\right]}$ (81)

$${\mathbf{r}}_0=\left(-105\mathrm{j}\right)·{\mathsf{g}}_3\dotplus 40\mathrm{j}·{\mathsf{g}}_5\dotplus \left(-70\mathrm{j}\right)·{\mathsf{g}}_6\dotplus 135\mathrm{j}·{\mathsf{g}}_9.$$

(89)

The polyadic augmentation map ε (66) for the elements (83)–(85) and (89) becomes

$$\begin{array}{cc}\hfill \epsilon \left(\mathbf{r}\left(1\right)\right)& =5\mathrm{j},\hfill \end{array}$$

(90)

$$\begin{array}{cc}\hfill \epsilon \left(\mathbf{r}\left(2\right)\right)& =\epsilon \left(2\mathrm{j}·{\mathsf{g}}_7\dotplus \left(-7\mathrm{j}\right)·{\mathsf{g}}_8\right)={\nu }_R^{\left[2\right]}\left[2\mathrm{j},\left(-7\mathrm{j}\right)\right]=-5\mathrm{j},\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill \epsilon \left(\mathbf{r}\left(3\right)\right)& =\epsilon \left(\left(-4\mathrm{j}\right)·{\mathsf{g}}_2\dotplus 7\mathrm{j}·{\mathsf{g}}_3\dotplus \left(-3\mathrm{j}\right)·{\mathsf{g}}_6\right)={\left({\nu }_R^{\left[2\right]}\right)}^{\circ 2}\left[\left(-4\mathrm{j}\right),7\mathrm{j},\left(-3\mathrm{j}\right)\right]=0,\hfill \\ \hfill \epsilon \left({\mathbf{r}}_0\right)& =\epsilon \left(\left(-105\mathrm{j}\right)·{\mathsf{g}}_3\dotplus 40\mathrm{j}·{\mathsf{g}}_5\dotplus \left(-70\mathrm{j}\right)·{\mathsf{g}}_6\dotplus 135\mathrm{j}·{\mathsf{g}}_9\right)\hfill \end{array}$$

(92)

$$\begin{array}{cc}\hfill & ={\left({\nu }_R^{\left[2\right]}\right)}^{\circ 3}\left[\left(-105\mathrm{j}\right),40\mathrm{j},\left(-70\mathrm{j}\right),135\mathrm{j}\right]=0,\hfill \end{array}$$

(93)

where on the r.h.s. there are binary additions in ${\mathcal{R}}^{\left[2,3\right]}$, while on the l.h.s. there are formal sums. According to (68) the elements $\mathbf{r}\left(3\right)$ and ${\mathbf{r}}_0$ are in the kernel.

Let us modify the previous construction for higher polyadic power case.

Example 2.

Now we take for the initial ring the commutative nonderived $\left(2,5\right)$-ring with the underlying set $R={\mathrm{j}}_4\mathbb{Z}$ (${\mathrm{j}}_4^4=-1$), operations are in $\mathbb{C}$. The operations ${\nu }_R^{\left[2\right]}$ and ${\mathsf{\mu }}_R^{\left[5\right]}$ are usual addition and product being the binary addition and nonderived 5-ary multiplication (only product of 5 elements is closed). Thus $⟨{\mathrm{j}}_4\mathbb{Z}\mid {\nu }_R^{\left[2\right]}⟩$ is a binary group with respect to addition, and $⟨{\mathrm{j}}_4\mathbb{Z}\mid {\mathsf{\mu }}_R^{\left[5\right]}⟩$ is not a 5-ary group, but only a 5-ary semigroup, because there is no multiplicative querelement for each $r\in R$ (no division in $\mathbb{Z}$). Obviously, that ${\mathcal{R}}^{\left[2,5\right]}$ is unitless. Note that polyadic distributivity follows from the binary distributivity in $\mathbb{Z}$. Therefore,

$${\mathcal{R}}^{\left[2,5\right]}=⟨{\mathrm{j}}_4\mathbb{Z}\mid {\nu }_R^{\left[2\right]},{\mathsf{\mu }}_R^{\left[5\right]},z_R⟩$$

(94)

is a commutative nonderived $\left(2,5\right)$-ring without multiplicative neutral element and $z_R=\mathrm{j}0=0$.

The ternary group ${\mathsf{G}}^{\left[\mathsf{3}\right]}$ is the same as in the previous example (79). The element of the polyadic group ring has the same form (80), but now the multiplication (46) is of 2nd polyadic power for ${\mathsf{G}}^{\left[\mathsf{3}\right]}$, i.e., ${\mathcal{\ell }}_{\mathsf{g}}=2$,

$$\begin{array}{cc}\hfill {\left({\mathsf{\mu }}_{\mathsf{G}}^{\left[\mathsf{3}\right]}\right)}^{\circ 2}\left[{\mathsf{g}}_{i_1},{\mathsf{g}}_{i_2},{\mathsf{g}}_{i_3},{\mathsf{g}}_{i_4},{\mathsf{g}}_{i_5}\right]& =g\left(m_1,n_1\right)g\left(m_2,n_2\right)g\left(m_3,n_3\right)g\left(m_4,n_4\right)g\left(m_5,n_5\right)\hfill \\ \hfill & =g\left(m_1+n_2+m_3+n_4+m_5,n_1+m_2+n_3+m_4+n_5\right).\hfill \end{array}$$

(95)

The initial ring ${\mathcal{R}}^{\left[2,5\right]}$ is still of 1st multiplicative polyadic power

$${\mathsf{\mu }}_R^{\left[5\right]}\left[{\mathrm{j}}_4{k}_1,{\mathrm{j}}_4{k}_2,{\mathrm{j}}_4{k}_3,{\mathrm{j}}_4{k}_4,{\mathrm{j}}_4{k}_5\right]=\left({\mathrm{j}}_4{k}_1\right)\left({\mathrm{j}}_4{k}_2\right)\left({\mathrm{j}}_4{k}_3\right)\left({\mathrm{j}}_4{k}_4\right)\left({\mathrm{j}}_4{k}_5\right)=-{\mathrm{j}}_4{k}_1{k}_2{k}_3{k}_4{k}_5,k_i\in \mathbb{Z}.$$

(96)

So manifestly the multiplicative arity of the polyadic group ring (48) using the “quantization” condition (47) with arities $n_r=5$ and ${\mathsf{n}}_g=3$, and different polyadic powers ${\mathcal{\ell }}_n=1$ and ${\mathcal{\ell }}_g=2$ becomes

$${\mathit{n}}_r={\mathcal{\ell }}_n\left(n_r-1\right)+1={\mathcal{\ell }}_g\left({\mathsf{n}}_g-1\right)+1=1(5-1)+1=2(3-1)+1=5.$$

(97)

The additive arity is inherited from the initial ring ${\mathcal{R}}^{\left[2,5\right]}$ and is binary ${\mathit{n}}_r=n_r=2$. Thus, the polyadic group ring is actually a nonderived $\left(\mathbf{2},\mathbf{5}\right)$-ring ${\mathbf{R}}^{\left[\mathbf{2},\mathbf{5}\right]}$ with 2nd polyadic power of the ternary group ${\mathsf{G}}^{\left[\mathsf{3}\right]}$ multiplication. The computations similar to the previous example can be done using the 5-ary multiplications (95) and (96), which manifestly shows the existence of higher power polyadic group rings.

7. Conclusions

This article has laid the foundational framework for the theory of polyadic group rings, a novel algebraic structure that generalizes the classical group ring construction $\mathcal{R}\left[\mathsf{G}\right]$ to the higher arity setting. We have formally defined a novel algebraic object, the polyadic group ring ${\mathbf{R}}^{[{\mathit{m}}_r,{\mathit{n}}_r]}={\mathcal{R}}^{[m_r,n_r]}\left[{\mathsf{G}}^{\left[n_g\right]}\right]$, constructing its ${\mathit{m}}_r$-ary addition and ${\mathit{n}}_r$-ary multiplication by systematically generalizing the binary operations. A central achievement of this work is the derivation of the “quantization” conditions that govern the admissible arities, revealing the profound interplay between the arities of the initial ring, the initial polyadic group, and the resulting polyadic group ring, including the novel case of operations with higher polyadic powers.

We have established essential algebraic properties of these structures, proving conditions for total associativity and characterizing the existence of a zero element and identity. Furthermore, the generalization of key tools such as the augmentation map and augmentation ideal provides a bridge for transferring techniques from the classical theory into this new domain. The explicit, non-trivial examples involving nonderived rings and finite polyadic groups serve to concretely illustrate the theory and demonstrate the noncommutative, convoluted nature of the multiplication.

This work opens several avenues for future research. First, further investigation of recently introduced multiary graded polyadic algebras [14], which reveals fundamentally new phenomena not present in the binary case, such as the existence of higher-power gradings. This can be seen as a new direction in mathematical physics—one that uses polyadic algebraic structures—and, more concretely, as a novel approach to particle model building that employs graded algebras. The representation theory of polyadic group rings, their homology, and other homological invariants remain entirely unexplored and represent a natural next step. Furthermore, the recent emergence of applied polyadic structures, such as in the “polyadic encryption” scheme [18], strongly validates the practical potential of this foundational work [19].

Thus, the novel theory of polyadic group rings presented here not only enriches pure algebra but also provides a new mathematical language for the challenges of modern computation and security.