Geodesic
Powers and alternative laws
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 1, pp. 25-40 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 20N02, 20N05, 37B10, 37E99
Keywords: alternative laws; alternative groupoid; powers; dynamical system; alternative loop; two-sided inverse 
@article{CMUC_2007_48_1_a2,
     author = {Ormes, Nicholas and Vojt\v{e}chovsk\'y, Petr},
     title = {Powers and alternative laws},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {25--40},
     year = {2007},
     volume = {48},
     number = {1},
     mrnumber = {2338827},
     zbl = {1174.20343},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2007_48_1_a2/}
}
TY  - JOUR
AU  - Ormes, Nicholas
AU  - Vojtěchovský, Petr
TI  - Powers and alternative laws
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 2007
SP  - 25
EP  - 40
VL  - 48
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMUC_2007_48_1_a2/
LA  - en
ID  - CMUC_2007_48_1_a2
ER  - 
%0 Journal Article
%A Ormes, Nicholas
%A Vojtěchovský, Petr
%T Powers and alternative laws
%J Commentationes Mathematicae Universitatis Carolinae
%D 2007
%P 25-40
%V 48
%N 1
%U http://geodesic.mathdoc.fr/item/CMUC_2007_48_1_a2/
%G en
%F CMUC_2007_48_1_a2
Ormes, Nicholas; Vojtěchovský, Petr. Powers and alternative laws. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 1, pp. 25-40. http://geodesic.mathdoc.fr/item/CMUC_2007_48_1_a2/