Polyadic rings of $p$-adic integers

TL;DR

This paper introduces a $p$-adic analog of residue classes and explores conditions under which these form $(m,n)$-rings, potentially revealing new symmetries in particle physics at very short spacetime scales.

Contribution

It proposes a novel $p$-adic framework for residue classes and characterizes when they form $(m,n)$-rings, linking algebraic structures to physical symmetries.

Findings

Defined a $p$-adic analog of residue classes

Derived relations for $(m,n)$-ring formation

Suggests implications for particle symmetries at small scales

Abstract

In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become $\left( m,n\right) $-rings. Second, we introduce a possible $p$-adic analog of the residue class modulo a $p$-adic integer. Then, we find the relations which determine, when the representatives form a $\left( m,n\right) $-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.