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alternative laws; alternative groupoid; powers; dynamical system; alternative loop; two-sided inverse
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.
[1] Burton D.M.: Elementary Number Theory. third edition, Wm. C. Brown Publishers, 1994. MR 0990017 | Zbl 1084.11001
[2] Dehornoy P.: The structure group for the associative identity. J. Pure Appl. Algebra 111 (1996), 59-82. MR 1394345
[3] Dehornoy P.: Braids and Self-Distributivity. Progress in Mathematics 192, Birkhäuser, Basel, 2000. MR 1778150 | Zbl 0958.20033
[4] Dehornoy P.: The fine structure of LD-equivalence. Adv. Math. 155 (2000), 264-316. MR 1794713 | Zbl 0974.20048
[5] Pflugfelder H.O.: Quasigroups and Loops: Introduction. Sigma Series in Pure Mathematics 7, Heldermann, Berlin, 1990. MR 1125767 | Zbl 0715.20043
[6] Smith W.D.: Inclusions among diassociativity-related loop properties. preprint.
[7] van Lint J.H., Wilson R.M.: A Course in Combinatorics. Cambridge University Press, Cambridge, 1992. MR 1207813 | Zbl 0980.05001
[8] Wirsching G.J.: The dynamical system generated by the $3n+1$ function. Lecture Notes in Mathematics 1681, Springer, Berlin, 1998. MR 1612686 | Zbl 0892.11002
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