Higher power polyadic group rings

Steven Duplij

arxiv: 2510.14029 · v2 · pith:KQPXG6XXnew · submitted 2025-10-15 · 🧮 math.RA · cs.IT· hep-th· math-ph· math.GR· math.IT· math.MP

Higher power polyadic group rings

Steven Duplij This is my paper

Pith reviewed 2026-05-21 20:29 UTC · model grok-4.3

classification 🧮 math.RA cs.IThep-thmath-phmath.GRmath.ITmath.MP

keywords polyadic group ringshigher arityquantization conditionsarity freedom principlepolyadic ringspolyadic groupsaugmentation ideal

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The pith

Polyadic group rings are formed by pairing a nonderived higher-arity ring with a nonderived higher-arity group whose arities must obey quantization conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs polyadic group rings as a direct generalization of ordinary group rings to settings in which both the coefficient ring and the underlying group carry operations of arity greater than two. It defines the corresponding multiary addition and multiplication on the set of functions from the group to the ring and then isolates the precise numerical relations that the arities must satisfy so that the resulting structure is consistent. These relations, called quantization conditions, are controlled by an arity freedom principle that leaves some arity choices free while fixing others. The construction preserves familiar features such as a zero element, an identity, total associativity, and a generalized augmentation ideal. The framework is illustrated by explicit examples and is presented as a foundation for further algebraic work.

Core claim

The central claim is that the polyadic group ring R^{[m_r, n_r]}[G^{[n_g]}] is well-defined precisely when the arities satisfy quantization conditions derived from the arity freedom principle; these conditions interrelate the ring arities m_r, n_r with the group arity n_g and extend to operations of still higher polyadic power while guaranteeing the existence of zero and identity elements together with total associativity.

What carries the argument

The quantization conditions on the arities m_r, n_r and n_g, obtained from the arity freedom principle so that the multiary ring and group operations combine consistently on the function space.

If this is right

Total associativity of the polyadic operations holds exactly when the quantization conditions are met.
A zero element and a multiplicative identity exist in the polyadic group ring under the same conditions.
The classical augmentation map and augmentation ideal extend directly to the polyadic setting.
The constructions remain valid when all operations are raised to still higher polyadic powers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same quantization technique may apply to other polyadic algebraic objects such as polyadic modules or polyadic Hopf algebras.
Concrete finite examples for small allowed arity tuples could be used to test whether new cryptographic or coding schemes become feasible.
The arity freedom principle suggests that similar numerical constraints will appear whenever one combines two independent multiary structures into a single composite algebra.

Load-bearing premise

Nonderived polyadic rings and groups with independently selectable arities exist and can be combined whenever the derived quantization conditions hold.

What would settle it

An explicit choice of arities obeying the quantization conditions for which the defined addition or multiplication fails to be associative or well-defined, or conversely a choice violating the conditions for which the operations nevertheless close and associate.

read the original abstract

This paper introduces and systematically develops the theory of polyadic group rings, a higher arity generalization of classical group rings $\mathcal{R}[\mathsf{G}]$. We construct the fundamental operations of these structures, defining the $\mathbf{m}_{r}$-ary addition and $\mathbf{n}_{r} $-ary multiplication for a polyadic group ring $\mathrm{R}^{[\mathbf{m} _{r},\mathbf{n}_{r}]}=\mathcal{R}^{[m_{r},n_{r}]}[\mathsf{G}^{[n_{g}]}]$ built from a nonderived $(m_{r},n_{r})$-ring and a nonderived $n_{g}$-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. This work establishes a new foundation in ring theory with potential applications in cryptography and coding theory, as evidenced by recent schemes utilizing polyadic structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines polyadic group rings from non-derived higher-arity pieces and derives arity quantization conditions, but assumes those base structures exist without supplying general existence results.

read the letter

Colleague, the main takeaway is that this paper builds polyadic group rings R^[m_r,n_r][G^[n_g]] from non-derived (m_r,n_r)-rings and n_g-ary groups, then derives quantization conditions that link the arities through an arity freedom principle, while generalizing the augmentation map and ideal. It also checks basic properties such as total associativity, zero elements, and identities, and includes some explicit examples that show how the setup works and reduces to ordinary group rings in the binary case. That systematic development and the bridge back to classical theory are the parts that feel most concrete and useful. The examples help make the definitions less abstract. The softer spot is the reliance on the existence of those non-derived base objects for arbitrary arities that satisfy the conditions. The framework treats them as independent building blocks that can be chosen freely once the quantization rules hold, yet there is no general existence theorem or wide family of concrete constructions supplied. Without that, many parameter choices could yield empty or trivial structures that collapse to derived cases, so the conditions are necessary but the actual content of the theory may be thinner than it first appears. The derivations of the properties are stated at a high level in the abstract, so a close check of the full arguments would be needed to confirm there are no gaps. This is for readers already working in polyadic or n-ary algebra who want to see group rings extended in this direction, or who might explore the mentioned links to cryptography and coding. A specialist in higher-arity structures would get the most out of the new constructions and the arity interrelations. It deserves a serious referee. The framework is developed enough and the generalizations are explicit enough to warrant review, with the main questions being existence of the base objects and verification of the key derivations.

Referee Report

1 major / 2 minor

Summary. The paper introduces the theory of polyadic group rings as a higher-arity generalization of classical group rings. It constructs the m_r-ary addition and n_r-ary multiplication operations for the structure R^{[m_r, n_r]}[G^{[n_g]}] from a nonderived (m_r, n_r)-ring and a nonderived n_g-ary group. A central result is the derivation of quantization conditions that interrelate these arities according to the arity freedom principle, which is extended to higher polyadic powers. The paper establishes algebraic properties such as total associativity, the existence of a zero element and identity, and generalizes the polyadic augmentation map and augmentation ideal. Explicit examples are provided to illustrate the framework.

Significance. If the central constructions and derivations hold, this work extends classical group ring theory to higher arities and provides a systematic framework via the arity freedom principle and quantization conditions. The generalization of the augmentation map and ideal creates a direct bridge to the classical case, while the explicit examples help ground the abstract constructions. Potential applications in cryptography and coding theory are noted but not developed in detail.

major comments (1)

[Section on constructions and quantization conditions] The derivation of the quantization conditions (governed by the arity freedom principle) presupposes the existence of nonderived (m_r, n_r)-rings and nonderived n_g-ary groups for arbitrary arity parameters satisfying those conditions. No general existence theorem or family of constructions is supplied to show that such nonderived structures exist and remain nonderived after forming the polyadic group ring for general choices of m_r, n_r, n_g. This assumption is load-bearing for the central claim that the framework yields a non-vacuous theory beyond derived cases.

minor comments (2)

[Introduction] The notation for the polyadic group ring R^{[m_r, n_r]}[G^{[n_g]}] and the distinction between derived and nonderived structures could be introduced with a short preliminary subsection to aid readability.
[Examples] A table or summary listing the specific arity triples (m_r, n_r, n_g) used in the explicit examples would make the illustrations easier to compare.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comment regarding the existence of nonderived structures point by point below. We believe our response clarifies the scope of the work while strengthening the presentation.

read point-by-point responses

Referee: [Section on constructions and quantization conditions] The derivation of the quantization conditions (governed by the arity freedom principle) presupposes the existence of nonderived (m_r, n_r)-rings and nonderived n_g-ary groups for arbitrary arity parameters satisfying those conditions. No general existence theorem or family of constructions is supplied to show that such nonderived structures exist and remain nonderived after forming the polyadic group ring for general choices of m_r, n_r, n_g. This assumption is load-bearing for the central claim that the framework yields a non-vacuous theory beyond derived cases.

Authors: We appreciate the referee's observation on this foundational aspect. The manuscript develops the polyadic group ring construction starting from given nonderived (m_r, n_r)-rings and n_g-ary groups that satisfy the derived quantization conditions. The focus is on how these operations extend to the group ring and the resulting algebraic properties, including total associativity and the generalized augmentation map. While we do not provide a general existence theorem for nonderived structures at arbitrary arities (which would constitute a separate substantial contribution to polyadic algebra), the paper includes explicit examples of nonderived base structures for specific arity triples satisfying the conditions, and we verify in these cases that the constructed polyadic group ring preserves the nonderived character where applicable. To strengthen the manuscript, we will revise the relevant section to explicitly state the assumptions on the base structures and include a brief discussion of known families of nonderived polyadic rings and groups from the literature, along with additional concrete examples. This addresses the concern by clarifying that the theory applies whenever such structures exist, and demonstrates non-vacuity through examples rather than claiming universality. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations follow from explicit definitions of nonderived polyadic structures

full rationale

The paper defines polyadic group rings from independent nonderived (m_r, n_r)-rings and n_g-ary groups, then derives quantization conditions on arities from the requirements of the operations (associativity, zero/identity elements). These steps are constructive and self-contained within the new framework; no reduction of a claimed prediction to a fitted parameter or self-citation chain is exhibited. The arity freedom principle is presented as a governing observation emerging from the constructions rather than an input that forces the output by definition. Existence of the base structures is presupposed as building blocks, but this is an assumption about the domain, not a circularity in the derivation itself.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 3 invented entities

The central constructions rest on the postulated existence of nonderived multi-ary rings and groups together with the newly introduced arity freedom principle; the arities themselves function as free parameters chosen to satisfy the derived conditions.

free parameters (1)

m_r, n_r, n_g
Arity parameters for the ring addition, ring multiplication, and group operation; chosen as part of the definition and constrained by quantization conditions.

axioms (2)

domain assumption Existence of nonderived (m_r, n_r)-rings and nonderived n_g-ary groups
The polyadic group ring is built directly from these structures; invoked in the definition of R^[m_r,n_r][G^[n_g]].
ad hoc to paper Arity freedom principle governs quantization conditions
The principle is introduced to relate the arities and is not derived from prior literature in the abstract.

invented entities (3)

polyadic group ring R^[m_r,n_r][G^[n_g]] no independent evidence
purpose: Higher-arity generalization of classical group rings
New composite structure defined by combining the multi-ary ring and group with m_r-ary addition and n_r-ary multiplication.
quantization conditions no independent evidence
purpose: Relations that must hold between arities for consistency
Derived relations introduced to make the polyadic operations well-defined.
arity freedom principle no independent evidence
purpose: Governing principle for arity interrelations and higher powers
New principle postulated to organize the quantization conditions.

pith-pipeline@v0.9.0 · 5763 in / 1765 out tokens · 87200 ms · 2026-05-21T20:29:12.756428+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean and polyadic operation modules polyadic word-length quantization (L_admiss) echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the length w_μ(n) of a word in a composition of n-ary multiplications is not arbitrary but quantized, assuming only the admissible values... L_admiss(n, ℓ_μ) = ℓ_μ(n−1)+1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

[1]

BERLEKAMP, E. R. (1968).Algebraic Coding Theory. New York: McGraw-Hill. [Cited on page2]

work page 1968
[2]

BOVDI, A. A. (1974).Group Rings. Uzhgorod: Uzgorod. Univ. [Cited on pages2, 5, and 6]

work page 1974
[3]

CURTIS, C. W.ANDI. REINER(1962).Representation theory of finite groups and associative algebras. Providence: AMS. [Cited on page6] D ¨ORNTE, W. (1929). Unterschungen ¨uber einen verallgemeinerten Gruppenbegriff.Math. Z.29, 1–19. [Cited on pages2, 4, and 8]

work page 1962
[4]

DUPLIJ, S. (2017). Polyadic integer numbers and finitepm, nq-fields.p-Adic Numbers, Ultrametric Analysis and Appl.9 (4), 257–281. arXiv: math.RA/1707.00719. [Cited on page8]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[5]

(2022).Polyadic Algebraic Structures

DUPLIJ, S. (2022).Polyadic Algebraic Structures. London-Bristol: IOP Publishing. [Cited on pages2, 3, 4, 7, 9, 10, 12, and 13]

work page 2022
[6]

GUO(2025)

DUPLIJ, S.ANDQ. GUO(2025). Polyadic encryption,preprintUniv. M ¨unster, CIT, M¨unster, 9 p., arXiv: cs.CR/2507.05683. [Cited on pages3 and 17]

work page arXiv 2025
[7]

HOLM(2018).Algebras and Representation Theory

ERDMANN, K.ANDT. HOLM(2018).Algebras and Representation Theory. Cham: Springer. [Cited on page6]

work page 2018
[8]

KIRILLOV, A. A. (1976).Elements of the Theory of Representations. Berlin: Springer-Verlag. [Cited on page6]

work page 1976
[9]

LEESON, J. J.ANDA. T. BUTSON(1980). On the general theory ofpm, nqrings.Algebra Univers.11, 42–76. [Cited on pages2 and 4]

work page 1980
[10]

MENEZES, A. J., P. C.VANOORSCHOT,ANDS. A. VANSTONE(1997).Handbook of Applied Cryptography. Boca Raton: CRC Press. [Cited on page2]

work page 1997
[11]

MILIES, C. P.ANDS. K. SEHGAL(2002).An Introduction to Group Rings. Dordrecht: Kluwer. [Cited on pages5 and 6]

work page 2002
[12]

OPPENHEIM, A. V. (1978).Applications of Digital Signal Processing. Englewood Cliffs, N.J.: Prentice-Hall. [Cited on page3]

work page 1978
[13]

PASSMAN, D. S. (1977).The Algebraic Structure of Group Rings. New York-London-Sydney: John Wiley and Sons. [Cited on pages2, 5, and 6]

work page 1977
[14]

POST, E. L. (1940). Polyadic groups.Trans. Amer. Math. Soc.48, 208–350. [Cited on page2]

work page 1940
[15]

RICHARDSON, T.ANDR. L. URBANKE(2008).Modern Coding Theory. Cambridge: Cambridge University Press. [Cited on page2]

work page 2008
[16]

SEHGAL, S. K. (1978).Topics in Group Rings. New York: Marcel Dekker. [Cited on pages2, 5, and 6]

work page 1978
[17]

ZALESSKII, A. E.ANDA. V. MIKHALEV(1975). Group rings.J. Soviet Math.4(1), 1–78. [Cited on pages5 and 6] – 18 –

work page 1975

[1] [1]

BERLEKAMP, E. R. (1968).Algebraic Coding Theory. New York: McGraw-Hill. [Cited on page2]

work page 1968

[2] [2]

BOVDI, A. A. (1974).Group Rings. Uzhgorod: Uzgorod. Univ. [Cited on pages2, 5, and 6]

work page 1974

[3] [3]

CURTIS, C. W.ANDI. REINER(1962).Representation theory of finite groups and associative algebras. Providence: AMS. [Cited on page6] D ¨ORNTE, W. (1929). Unterschungen ¨uber einen verallgemeinerten Gruppenbegriff.Math. Z.29, 1–19. [Cited on pages2, 4, and 8]

work page 1962

[4] [4]

DUPLIJ, S. (2017). Polyadic integer numbers and finitepm, nq-fields.p-Adic Numbers, Ultrametric Analysis and Appl.9 (4), 257–281. arXiv: math.RA/1707.00719. [Cited on page8]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[5] [5]

(2022).Polyadic Algebraic Structures

DUPLIJ, S. (2022).Polyadic Algebraic Structures. London-Bristol: IOP Publishing. [Cited on pages2, 3, 4, 7, 9, 10, 12, and 13]

work page 2022

[6] [6]

GUO(2025)

DUPLIJ, S.ANDQ. GUO(2025). Polyadic encryption,preprintUniv. M ¨unster, CIT, M¨unster, 9 p., arXiv: cs.CR/2507.05683. [Cited on pages3 and 17]

work page arXiv 2025

[7] [7]

HOLM(2018).Algebras and Representation Theory

ERDMANN, K.ANDT. HOLM(2018).Algebras and Representation Theory. Cham: Springer. [Cited on page6]

work page 2018

[8] [8]

KIRILLOV, A. A. (1976).Elements of the Theory of Representations. Berlin: Springer-Verlag. [Cited on page6]

work page 1976

[9] [9]

LEESON, J. J.ANDA. T. BUTSON(1980). On the general theory ofpm, nqrings.Algebra Univers.11, 42–76. [Cited on pages2 and 4]

work page 1980

[10] [10]

MENEZES, A. J., P. C.VANOORSCHOT,ANDS. A. VANSTONE(1997).Handbook of Applied Cryptography. Boca Raton: CRC Press. [Cited on page2]

work page 1997

[11] [11]

MILIES, C. P.ANDS. K. SEHGAL(2002).An Introduction to Group Rings. Dordrecht: Kluwer. [Cited on pages5 and 6]

work page 2002

[12] [12]

OPPENHEIM, A. V. (1978).Applications of Digital Signal Processing. Englewood Cliffs, N.J.: Prentice-Hall. [Cited on page3]

work page 1978

[13] [13]

PASSMAN, D. S. (1977).The Algebraic Structure of Group Rings. New York-London-Sydney: John Wiley and Sons. [Cited on pages2, 5, and 6]

work page 1977

[14] [14]

POST, E. L. (1940). Polyadic groups.Trans. Amer. Math. Soc.48, 208–350. [Cited on page2]

work page 1940

[15] [15]

RICHARDSON, T.ANDR. L. URBANKE(2008).Modern Coding Theory. Cambridge: Cambridge University Press. [Cited on page2]

work page 2008

[16] [16]

SEHGAL, S. K. (1978).Topics in Group Rings. New York: Marcel Dekker. [Cited on pages2, 5, and 6]

work page 1978

[17] [17]

ZALESSKII, A. E.ANDA. V. MIKHALEV(1975). Group rings.J. Soviet Math.4(1), 1–78. [Cited on pages5 and 6] – 18 –

work page 1975