Maximally nonassociative quasigroups via quadratic orthomorphisms

Abstract

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 .

Cycles of quadratic Latin squares and antiperfect 1‐factorisations

“…We also found that there are $$7{q}^{2}\unicode{x02215}32+O({q}^{3\unicode{x02215}2})$$ quadratic $${N}_{2}$$ Latin squares of order $$q$$. Drápal and Wanless [12] showed that the quadratic Latin squares $${\rm{ {\mathcal L} }}[a,b]$$ and $${\rm{ {\mathcal L} }}[c,d]$$ of order $$q$$ are isomorphic if and only if $$\{c,d\}=\{\theta (a),\theta (b)\}$$ for an automorphism $$\theta $$ of $${{\mathbb{F}}}_{q}$$. It follows that the number of isomorphism classes of $${N}_{2}$$ quadratic Latin squares of order $$q$$ is at least $${\rm{\Theta }}({q}^{2}\unicode{x02215}\mathrm{log}(q))$$.…”

“…The condition $$\{ab,(a-1)(b-1)\}\subseteq {{\rm{ {\mathcal R} }}}_{q}$$ ensures that $${\rm{ {\mathcal L} }}[a,b]$$ is a Latin square [16]. Quadratic Latin squares have previously been used to construct perfect 1‐factorisations [1, 17, 38], mutually orthogonal Latin squares [15, 16], atomic Latin squares [38], Falconer varieties [1] and maximally nonassociative quasigroups [10, 11]. Quadratic Latin squares are the main focus of this paper.…”

“…[ , ]  is a Latin square [16]. Quadratic Latin squares have previously been used to construct perfect 1-factorisations [1,17,38], mutually orthogonal Latin squares [15,16], atomic Latin squares [38], Falconer varieties [1] and maximally nonassociative quasigroups [10,11]. Quadratic Latin squares are the main focus of this paper.…”

Cycles of quadratic Latin squares and anti-perfect $1$-factorisations

“…(iii) b = a/(2a − 1) and (q, a) ∈ {(13, 2), (13,9), (17,5), (17,8), (37,11), (37,27), (41, 23), (41, 26)}.…”

“…Such squares are called quadratic Latin squares. Quadratic Latin squares have previously been used to construct perfect 1-factorisations [34,15,1], mutually orthogonal Latin squares [14,13], atomic Latin squares [34], Falconer varieties [1], and maximally non-associative quasigroups [9,10]. Quadratic Latin squares are the main focus of this paper.…”