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Powers and alternative laws. (English) Zbl 1174.20343

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 \(\mathbb{N}^+\times\mathbb{N}^+\) \[ (r, s)\mapsto(2r, s-r),\quad(r-s,2s),\quad(r/2,s+r/2),\quad(r+s/2,s/2), \] 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 \(n\leq 5\). We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses.

MSC:

20N02 Sets with a single binary operation (groupoids)
20N05 Loops, quasigroups
37E99 Low-dimensional dynamical systems
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