Powers and alternative laws

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.

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