Powers and alternative laws

Nicholas Ormes; Petr Vojtěchovský

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 1, page 25-40
  • ISSN: 0010-2628

Abstract

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A groupoid is alternative if it satisfies the alternative laws x ( x y ) = ( x x ) y and x ( y y ) = ( x y ) y . These laws induce four partial maps on ℕ + × ℕ + ( r , s ) ↦ ( 2 r , s - r ) , ( r - s , 2 s ) , ( r / 2 , s + r / 2 ) , ( 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 ≤ 5 . We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses.

How to cite

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Ormes, Nicholas, and Vojtěchovský, Petr. "Powers and alternative laws." Commentationes Mathematicae Universitatis Carolinae 48.1 (2007): 25-40. <http://eudml.org/doc/250223>.

@article{Ormes2007,
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 $\mathbb \{N\}^+ \times \mathbb \{N\}^+$\[ (r,\,s)\mapsto (2r,\,s-r),\quad (r-s,\,2s),\quad (r/2,\,s+r/2),\quad (r+s/2,\,s/2), \] 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 $n\le 5$. We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses.},
author = {Ormes, Nicholas, Vojtěchovský, Petr},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {alternative laws; alternative groupoid; powers; dynamical system; alternative loop; two-sided inverse; alternative laws; alternative groupoids; powers; dynamical systems; alternative loops; two-sided inverses},
language = {eng},
number = {1},
pages = {25-40},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Powers and alternative laws},
url = {http://eudml.org/doc/250223},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Ormes, Nicholas
AU - Vojtěchovský, Petr
TI - Powers and alternative laws
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 1
SP - 25
EP - 40
AB - 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 $n\le 5$. We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses.
LA - eng
KW - alternative laws; alternative groupoid; powers; dynamical system; alternative loop; two-sided inverse; alternative laws; alternative groupoids; powers; dynamical systems; alternative loops; two-sided inverses
UR - http://eudml.org/doc/250223
ER -

References

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  1. Burton D.M., Elementary Number Theory, third edition, Wm. C. Brown Publishers, 1994. Zbl1084.11001MR0990017
  2. Dehornoy P., The structure group for the associative identity, J. Pure Appl. Algebra 111 (1996), 59-82. (1996) MR1394345
  3. Dehornoy P., Braids and Self-Distributivity, Progress in Mathematics 192, Birkhäuser, Basel, 2000. Zbl0958.20033MR1778150
  4. Dehornoy P., The fine structure of LD-equivalence, Adv. Math. 155 (2000), 264-316. (2000) Zbl0974.20048MR1794713
  5. Pflugfelder H.O., Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics 7, Heldermann, Berlin, 1990. Zbl0715.20043MR1125767
  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. Zbl0980.05001MR1207813
  8. Wirsching G.J., The dynamical system generated by the 3 n + 1 function, Lecture Notes in Mathematics 1681, Springer, Berlin, 1998. Zbl0892.11002MR1612686

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