# ALGEBRAIC COMBINATORICS

Maximally nonassociative quasigroups via quadratic orthomorphisms
Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 501-515.

A quasigroup $Q$ is called maximally nonassociative if for $x,y,z\in Q$ we have that $x·\left(y·z\right)=\left(x·y\right)·z$ only if $x=y=z$. We show that, with finitely many exceptions, there exists a maximally nonassociative quasigroup of order $n$ whenever $n$ is not of the form $n=2{p}_{1}$ or $n=2{p}_{1}{p}_{2}$ for primes ${p}_{1},{p}_{2}$ with ${p}_{1}\le {p}_{2}<2{p}_{1}$.

Supplementary Materials:
Supplementary materials for this article are supplied as separate files:

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.165
Classification: 20N05,  11T22,  05D99,  05E16
Keywords: Quasigroup, maximally nonassociative, quadratic orthomorphism, idempotent.
@article{ALCO_2021__4_3_501_0,
author = {Dr\'apal, Ale\v{s} and Wanless, Ian M.},
title = {Maximally nonassociative quasigroups via quadratic orthomorphisms},
journal = {Algebraic Combinatorics},
pages = {501--515},
publisher = {MathOA foundation},
volume = {4},
number = {3},
year = {2021},
doi = {10.5802/alco.165},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.165/}
}
Drápal, Aleš; Wanless, Ian M. Maximally nonassociative quasigroups via quadratic orthomorphisms. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 501-515. doi : 10.5802/alco.165. https://alco.centre-mersenne.org/articles/10.5802/alco.165/

[1] Drápal, Aleš; Lisoněk, Petr Maximal nonassociativity via nearfields, Finite Fields Appl., Volume 62 (2020), p. 101610, 27pp | Article | MR 4032787 | Zbl 07174350

[2] Drápal, Aleš; Valent, Viliam Few associative triples, isotopisms and groups, Des. Codes Cryptogr., Volume 86 (2018), pp. 555-568 | Article | MR 3767788 | Zbl 1434.20051

[3] Drápal, Aleš; Valent, Viliam High nonassociativity in order 8 and an associative index estimate, J. Combin. Des., Volume 27 (2019), pp. 205-228 | Article | MR 3915173 | Zbl 1434.20052

[4] Drápal, Aleš; Valent, Viliam Extreme nonassociativity in order nine and beyond, J. Combin. Des., Volume 28 (2020), pp. 33-48 | Article | MR 4033745

[5] Drápal, Aleš; Wanless, Ian M. On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups (2020) (https://arxiv.org/abs/2005.11674)

[6] Evans, Anthony B. Orthogonal Latin squares based on groups, Developments in Mathematics, 57, Springer, Cham, 2018, xv+537 pages | Article | MR 3837138 | Zbl 1404.05002

[7] Grošek, Otokar; Horák, Peter On quasigroups with few associative triples, Des. Codes Cryptogr., Volume 64 (2012), pp. 221-227 | Article | MR 2914413 | Zbl 1250.94036

[8] Kepka, Tomáš A note on associative triples of elements in cancellation groupoids, Comment. Math. Univ. Carolin., Volume 21 (1980), pp. 479-487 | MR 590128 | Zbl 0444.20069

[9] Kotzig, Anton; Reischer, Corina Associativity index of finite quasigroups, Glas. Mat. Ser. III, Volume 18 (1983), pp. 243-253 | MR 733164 | Zbl 0528.20055

[10] Lisoněk, Petr Maximal nonassociativity via fields, Des. Codes Cryptogr., Volume 88 (2020), pp. 2521-2530 | Article | MR 4171315 | Zbl 07272713

[11] Rojas-León, Antonio More general exponential and character sums, Handbook of Finite Fields (Mullen, Gary L.; Panario, Daniel, eds.), CRC Press, Boca Raton FL, 2013, pp. 161-169

[12] Wanless, Ian M. Diagonally cyclic Latin squares, European J. Combin., Volume 25 (2004), pp. 393-413 | Article | MR 2036476 | Zbl 1047.05007

[13] Wanless, Ian M. Atomic Latin squares based on cyclotomic orthomorphisms, Electron. J. Combin., Volume 12 (2005), Paper no. R22, 23 pages | MR 2134185 | Zbl 1079.05016

[14] Wanless, Ian M. Author homepage (2020) (http://users.monash.edu.au/~iwanless/data)