Geodesic
Nonassociative triples in involutory loops and in loops of small order
Commentationes Mathematicae Universitatis Carolinae, Tome 61 (2020) no. 4, pp. 459-479.

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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.
DOI : 10.14712/1213-7243.2020.037
Classification : 05B15, 20N05
Keywords: quasigroup; loop; prolongation; involutory loop; associative triple; maximally nonassociative 
@article{10_14712_1213_7243_2020_037,
     author = {Dr\'apal, Ale\v{s} and Hora, Jan},
     title = {Nonassociative triples in involutory loops and in loops of small order},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {459--479},
     publisher = {mathdoc},
     volume = {61},
     number = {4},
     year = {2020},
     doi = {10.14712/1213-7243.2020.037},
     mrnumber = {4230953},
     zbl = {07332722},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2020.037/}
}
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
SP  - 459
EP  - 479
VL  - 61
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2020.037/
DO  - 10.14712/1213-7243.2020.037
LA  - en
ID  - 10_14712_1213_7243_2020_037
ER  - 
%0 Journal Article
%A Drápal, Aleš
%A Hora, Jan
%T Nonassociative triples in involutory loops and in loops of small order
%J Commentationes Mathematicae Universitatis Carolinae
%D 2020
%P 459-479
%V 61
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2020.037/
%R 10.14712/1213-7243.2020.037
%G en
%F 10_14712_1213_7243_2020_037
Drápal, Aleš; Hora, Jan. Nonassociative triples in involutory loops and in loops of small order. Commentationes Mathematicae Universitatis Carolinae, Tome 61 (2020) no. 4, pp. 459-479. doi : 10.14712/1213-7243.2020.037. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2020.037/

Cité par Sources :