Positional numeral systems over polyadic rings

TL;DR

This paper develops positional numeral systems over polyadic rings with multiple arguments for addition and multiplication, revealing quantized word lengths and invariants, and explores their potential for advanced arithmetic and coding schemes.

Contribution

It introduces the first systematic construction of positional systems over polyadic rings, establishing existence, bounds, and invariants for finite expansions.

Findings

Every commutative (m,n)-ring admits a base-p expansion respecting word length constraints.

The minimum number of digits is at least the arity of addition m.

For m,n ≥ 3, only a subset of elements have finite expansions characterized by invariants.

Abstract

We construct positional numeral systems that work natively over nonderived polyadic $\left( m,n\right) $-rings whose addition takes $m$ arguments and multiplication takes $n$. In such rings, the length of an admissible additive word and a multiplicative tower are not arbitrary (as in the binary case), but "quantized". Our main contributions are the following. Existence: every commutative $\left( m,n\right) $-ring admits a base-$p$ place-value expansion that respects the word length constraint in terms of numbers of operation compositions $\ell_{mult}=\ell_{add}(m-1)+1$. Lower bound: the minimum number of digits is greater than or equal to the arity of addition $m$. Representability gap: for $m,n\geq3$ only a proper subset of ring elements possess finite expansions, characterized by congruence-class arity shape invariants $I^{(m)}$ and $J^{(n)}$. Mixed-base "polyadic clocks": allowing a different base at each position enlarges the design space quadratically in the digit count. Catalogues: explicit tables for the integer rings $\mathbb{Z}_{4,3}$ and $\mathbb{Z}_{6,5}$ illustrate how ordinary integers lift to distinct polyadic variables. These results lay the groundwork for faster arity-aware arithmetic, exotic coding schemes, and hardware that exploits operations beyond the binary pair.