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Higher power polyadic group rings

Steven Duplij

TL;DR

The paper develops polyadic group rings as a higher-arity generalization of group rings, establishing the object $\mathrm{R}^{[\bm{m}_{r},\bm{n}_{r}]}=\mathcal{R}^{[m_{r},n_{r}]}[\mathsf{G}^{[n_{g}]}]$ with $\bm{m}_{r}$-ary addition and $\bm{n}_{r}$-ary multiplication. Using the arity freedom principle, it derives quantization conditions linking the arities of the initial ring, the polyadic group, and the resulting ring, including cases with higher polyadic powers via $\ell_{m}$ and $\ell_{n}$. The work proves total associativity criteria, defines a polyadic augmentation map and augmentation ideal, and provides explicit nontrivial examples illustrating the convoluted, nonbinary multiplication. It argues for potential cryptographic and coding-theoretic applications and lays a foundation for future exploration in representation theory and homology of polyadic rings.

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.

Higher power polyadic group rings

TL;DR

The paper develops polyadic group rings as a higher-arity generalization of group rings, establishing the object with -ary addition and -ary multiplication. Using the arity freedom principle, it derives quantization conditions linking the arities of the initial ring, the polyadic group, and the resulting ring, including cases with higher polyadic powers via and . The work proves total associativity criteria, defines a polyadic augmentation map and augmentation ideal, and provides explicit nontrivial examples illustrating the convoluted, nonbinary multiplication. It argues for potential cryptographic and coding-theoretic applications and lays a foundation for future exploration in representation theory and homology of polyadic 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 -ary addition and -ary multiplication for a polyadic group ring built from a nonderived -ring and a nonderived -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.
Paper Structure (7 sections, 6 theorems, 82 equations)

This paper contains 7 sections, 6 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Binary group rings
  4. Polyadic group rings
  5. Properties
  6. Examples
  7. Conclusion

Key Result

Proposition 4.3

The arity of addition in the polyadic group ring $\hbox{\upshape R}^{\left[ \bm{m}_{r},\bm{n}_{r}\right] }$ coincides with the arity of addition in the initial polyadic ring $\mathcal{R}^{\left[ n_{r},m_{r}\right] }$ that is if in both sides of (mm) there is one polyadic operation (addition in $\hbox{\upshape R}^{\left[ \bm{m}_{r},\bm{n}_{r}\right] }$ and addition in $\mathcal{R}^{\left[ m_{

Theorems & Definitions (30)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 4.1
  • Definition 4.2
  • Proposition 4.3
  • proof
  • Remark 4.4
  • Remark 4.5
  • Definition 4.6
  • ...and 20 more