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Keywords:
alternative laws; alternative groupoid; powers; dynamical system; alternative loop; two-sided inverse
alternative laws; alternative groupoid; powers; dynamical system; alternative loop; two-sided inverse
Summary:
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 $\Bbb N^+ \times \Bbb N^+$ $$ (r,\,s)\mapsto (2r,\,s-r),\quad (r-s,\,2s),\quad (r/2,\,s+r/2),\quad (r+s/2,\,s/2), $$ 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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