ALGEBRAIC COMBINATORICS

no. 3
Maximally nonassociative quasigroups via quadratic orthomorphisms
Drápal, Aleš1; Wanless, Ian M.2
1 Department of Mathematics Charles University Sokolovská 83 186 75 Praha 8, Czech Republic
2 School of Mathematics Monash University Clayton Vic 3800, Australia
Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 501-515.

A quasigroup Q is called maximally nonassociative if for x,y,z∈Q we have that x·(y·z)=(x·y)·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=2p 1 or n=2p 1 p 2 for primes p 1 ,p 2 with p 1 ≤p 2 <2p 1 .

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

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.165
Classification: 20N05, 11T22, 05D99, 05E16
Keywords: Quasigroup, maximally nonassociative, quadratic orthomorphism, idempotent.
Drápal, Aleš 1; Wanless, Ian M. 2

1 Department of Mathematics Charles University Sokolovská 83 186 75 Praha 8, Czech Republic
2 School of Mathematics Monash University Clayton Vic 3800, Australia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@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/}
}
TY  - JOUR
AU  - Drápal, Aleš
AU  - Wanless, Ian M.
TI  - Maximally nonassociative quasigroups via quadratic orthomorphisms
JO  - Algebraic Combinatorics
PY  - 2021
SP  - 501
EP  - 515
VL  - 4
IS  - 3
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.165/
DO  - 10.5802/alco.165
LA  - en
ID  - ALCO_2021__4_3_501_0
ER  - 
%0 Journal Article
%A Drápal, Aleš
%A Wanless, Ian M.
%T Maximally nonassociative quasigroups via quadratic orthomorphisms
%J Algebraic Combinatorics
%D 2021
%P 501-515
%V 4
%N 3
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.165/
%R 10.5802/alco.165
%G en
%F ALCO_2021__4_3_501_0
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 | DOI | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

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

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

Cited by Sources: