| Title:
|
Nonassociative triples in involutory loops and in loops of small order (English) |
| Author:
|
Drápal, Aleš |
| Author:
|
Hora, Jan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
459-479 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
quasigroup |
| Keyword:
|
loop |
| Keyword:
|
prolongation |
| Keyword:
|
involutory loop |
| Keyword:
|
associative triple |
| Keyword:
|
maximally nonassociative |
| MSC:
|
05B15 |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 07332722 |
| idMR:
|
MR4230953 |
| DOI:
|
10.14712/1213-7243.2020.037 |
| . |
| Date available:
|
2021-02-25T12:37:40Z |
| Last updated:
|
2023-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148658 |
| . |
| Reference:
|
[1] Dickson L. E.: On finite algebras.Nachr. Ges. Wiss. Göttingen (1905), 358–393. |
| Reference:
|
[2] Drápal A., Lisoněk P.: Maximal nonassociativity via nearfields.Finite Fields Appl. 62 (2020), 101610, 27 pages. MR 4032787 |
| Reference:
|
[3] Drápal A., Valent V.: Extreme nonassociativity in order nine and beyond.J. Combin. Des. 28 (2020), no. 1, 33–48. MR 4033745, 10.1002/jcd.21679 |
| Reference:
|
[4] Drápal A., Wanless I. M.: Maximally nonassociative quasigroups via quadratic orthomorphisms.accepted in Algebr. Comb., available at arXiv:1912.07040v1 [math.CO] (2019), 13 pages. |
| Reference:
|
[5] Evans A. B.: Orthogonal Latin Squares Based on Groups.Developments in Mathematics, 57, Springer, Cham, 2018. MR 3837138, 10.1007/978-3-319-94430-2_2 |
| Reference:
|
[6] Kepka T.: A note on associative triples of elements in cancellation groupoids.Comment. Math. Univ. Carolin. 21 (1980), no. 3, 479–487. MR 0590128 |
| Reference:
|
[7] Wanless I. M.: Diagonally cyclic Latin squares.European J. Combin. 25 (2004), no. 3, 393–413. MR 2036476, 10.1016/j.ejc.2003.09.014 |
| Reference:
|
[8] Wanless I. M.: Atomic Latin squares based on cyclotomic orthomorphisms.Electron. J. Combin. 12 (2005), Research Paper 22, 23 pages. MR 2134185, 10.37236/1919 |
| . |
