Nicholas Ormes, Petr Vojt\v {e}chovsk\'y
Powers and alternative laws

Comment.Math.Univ.Carolin. 48,1 (2007) 25-40.

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 $\Bbb N^+ \times \Bbb N^+$ $$ (r, s)\mapsto (2r, s-r),\quad (r-s, 2s),\quad (r/2, s+r/2),\quad (r+s/2, s/2), $$ 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.

Keywords: alternative laws, alternative groupoid, powers, dynamical system, alternative loop, two-sided inverse
AMS Subject Classification: Primary 20N02; Secondary 20N05, 37E99