Nonassociative triples in involutory loops and in loops of small order
Commentationes Mathematicae Universitatis Carolinae (2020)
- Volume: 61, Issue: 4, page 459-479
- ISSN: 0010-2628
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topDrápal, Aleš, and Hora, Jan. "Nonassociative triples in involutory loops and in loops of small order." Commentationes Mathematicae Universitatis Carolinae 61.4 (2020): 459-479. <http://eudml.org/doc/297272>.
@article{Drápal2020,
abstract = {A loop of order $n$ possesses at least $3n^2-3n+1$ associative triples. However, no loop of order $n>1$ that achieves this bound seems to be known. If the loop is involutory, then it possesses at least $3n^2-2n$ associative triples. Involutory loops with $3n^2-2n$ associative triples can be obtained by prolongation of certain maximally nonassociative quasigroups whenever $n-1$ is a prime greater than or equal to $13$ or $n-1=p^\{2k\}$, $p$ an odd prime. For orders $n\le 9$ the minimum number of associative triples is reported for both general and involutory loops, and the structure of the corresponding loops is described.},
author = {Drápal, Aleš, Hora, Jan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasigroup; loop; prolongation; involutory loop; associative triple; maximally nonassociative},
language = {eng},
number = {4},
pages = {459-479},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Nonassociative triples in involutory loops and in loops of small order},
url = {http://eudml.org/doc/297272},
volume = {61},
year = {2020},
}
TY - JOUR
AU - Drápal, Aleš
AU - Hora, Jan
TI - Nonassociative triples in involutory loops and in loops of small order
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 4
SP - 459
EP - 479
AB - A loop of order $n$ possesses at least $3n^2-3n+1$ associative triples. However, no loop of order $n>1$ that achieves this bound seems to be known. If the loop is involutory, then it possesses at least $3n^2-2n$ associative triples. Involutory loops with $3n^2-2n$ associative triples can be obtained by prolongation of certain maximally nonassociative quasigroups whenever $n-1$ is a prime greater than or equal to $13$ or $n-1=p^{2k}$, $p$ an odd prime. For orders $n\le 9$ the minimum number of associative triples is reported for both general and involutory loops, and the structure of the corresponding loops is described.
LA - eng
KW - quasigroup; loop; prolongation; involutory loop; associative triple; maximally nonassociative
UR - http://eudml.org/doc/297272
ER -
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