Powers of elements in Jordan loops

Kyle Pula

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 2, page 291-299
  • ISSN: 0010-2628

Abstract

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A Jordan loop is a commutative loop satisfying the Jordan identity ( x 2 y ) x = x 2 ( y x ) . We establish several identities involving powers in Jordan loops and show that there is no nonassociative Jordan loop of order 9 .

How to cite

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Pula, Kyle. "Powers of elements in Jordan loops." Commentationes Mathematicae Universitatis Carolinae 49.2 (2008): 291-299. <http://eudml.org/doc/250469>.

@article{Pula2008,
abstract = {A Jordan loop is a commutative loop satisfying the Jordan identity $(x^2 y)x = x^2(y x)$. We establish several identities involving powers in Jordan loops and show that there is no nonassociative Jordan loop of order $9$.},
author = {Pula, Kyle},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Jordan loop; Jordan quasigroup; well-defined powers; nonassociative loop; order of a loop; commutative loops; identities; finite nonassociative Jordan loops; Jordan quasigroups; finite Jordan loops},
language = {eng},
number = {2},
pages = {291-299},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Powers of elements in Jordan loops},
url = {http://eudml.org/doc/250469},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Pula, Kyle
TI - Powers of elements in Jordan loops
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 2
SP - 291
EP - 299
AB - A Jordan loop is a commutative loop satisfying the Jordan identity $(x^2 y)x = x^2(y x)$. We establish several identities involving powers in Jordan loops and show that there is no nonassociative Jordan loop of order $9$.
LA - eng
KW - Jordan loop; Jordan quasigroup; well-defined powers; nonassociative loop; order of a loop; commutative loops; identities; finite nonassociative Jordan loops; Jordan quasigroups; finite Jordan loops
UR - http://eudml.org/doc/250469
ER -

References

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  1. Bruck R.H., A Survey of Binary Systems, Ergebnisse der Mathematik und Ihrer Grenzgebiete, New Series, Vol. 20, Springer, Berlin, 1958. Zbl0141.01401MR0093552
  2. Goodaire E.G., Keeping R.G., Jordan loops and loop rings, preprint. MR2376868
  3. Kinyon M.K., Pula J.K., Vojtěchovský P., Admissible Orders of Jordan Loops, J. Combinatorial Designs, to appear. 
  4. McCrimmon K., A Taste of Jordan Algebras, Universitext, Springer, New York, 2004. Zbl1044.17001MR2014924
  5. McCune W.W., Mace4 Reference Manual and Guide, Tech. Memo ANL/MCS-TM-264, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, August 2003; http://www.cs.unm.edu/mccune/mace4/. 
  6. Pflugfelder H.O., Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics 7, Heldermann Verlag, Berlin, 1990. Zbl0715.20043MR1125767

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