| Title:
|
Powers and alternative laws (English) |
| Author:
|
Ormes, Nicholas |
| Author:
|
Vojtěchovský, Petr |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
25-40 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
alternative laws |
| Keyword:
|
alternative groupoid |
| Keyword:
|
powers |
| Keyword:
|
dynamical system |
| Keyword:
|
alternative loop |
| Keyword:
|
two-sided inverse |
| MSC:
|
20N02 |
| MSC:
|
20N05 |
| MSC:
|
37B10 |
| MSC:
|
37E99 |
| idZBL:
|
Zbl 1174.20343 |
| idMR:
|
MR2338827 |
| . |
| Date available:
|
2009-05-05T17:01:08Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119636 |
| . |
| Reference:
|
[1] Burton D.M.: Elementary Number Theory.third edition, Wm. C. Brown Publishers, 1994. Zbl 1084.11001, MR 0990017 |
| Reference:
|
[2] Dehornoy P.: The structure group for the associative identity.J. Pure Appl. Algebra 111 (1996), 59-82. MR 1394345 |
| Reference:
|
[3] Dehornoy P.: Braids and Self-Distributivity.Progress in Mathematics 192, Birkhäuser, Basel, 2000. Zbl 0958.20033, MR 1778150 |
| Reference:
|
[4] Dehornoy P.: The fine structure of LD-equivalence.Adv. Math. 155 (2000), 264-316. Zbl 0974.20048, MR 1794713 |
| Reference:
|
[5] Pflugfelder H.O.: Quasigroups and Loops: Introduction.Sigma Series in Pure Mathematics 7, Heldermann, Berlin, 1990. Zbl 0715.20043, MR 1125767 |
| Reference:
|
[6] Smith W.D.: Inclusions among diassociativity-related loop properties.preprint. |
| Reference:
|
[7] van Lint J.H., Wilson R.M.: A Course in Combinatorics.Cambridge University Press, Cambridge, 1992. Zbl 0980.05001, MR 1207813 |
| Reference:
|
[8] Wirsching G.J.: The dynamical system generated by the $3n+1$ function.Lecture Notes in Mathematics 1681, Springer, Berlin, 1998. Zbl 0892.11002, MR 1612686 |
| . |
