Powers and alternative laws

TL;DR

This paper investigates the structure of alternative groupoids, analyzing a dynamical system derived from their laws, and determines the conditions under which powers are well-defined, revealing connections to number theory and loop theory.

Contribution

It introduces a dynamical system based on alternative laws, characterizes the orbits, and establishes that nth powers are well-defined only for n ≤ 5 in free alternative groupoids.

Findings

nth powers are well-defined only for n ≤ 5

orbits of the dynamical system are fully described

existence of alternative loops without two-sided inverses

Abstract

A groupoid is alternative if it satisfies the alternative laws $x(xy)=(xx)y$ and $x(yy)=(xy)y$. These laws induce four partial maps on $\mathbb{N}^+\times \mathbb{N}^+$, $(r,\,s)\mapsto (2r,\,s-r)$, $(r-s,\,2s)$, $(r/2,\,s+r/2)$, $(r+s/2,\,s/2)$ that taken together form a dynamical system. We describe the orbits of this dynamical system, which allows us to show that $n$th powers in a free alternative groupoid on one generator are well-defined if and only if $n\le 5$. We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses.