On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups

Abstract

Let $q$ be an odd prime power and suppose that $a,b\in {\mathbb{F}}_q$ are such that $ab$ and $(1-a)(1-b)$ are nonzero squares. Let $Q_{a,b}=({\mathbb{F}}_q,\ast )$ be the quasigroup in which the operation is defined by $u\ast v=u+a(v-u)$ if $v-u$ is a square, and $u\ast v=u+b(v-u)$ is $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x\ast (y\ast z)=(x\ast y)\ast z$ $\iff$ $x=y=z$. Denote by $\sigma (q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $\alpha \approx 0.02908$ and $\beta \approx 0.01259$ such that if $q\equiv 1\mathrm{mod}4$, then $\lim \sigma (q)/q^2=\alpha$, and if $q\equiv 3\mathrm{mod}4$, then $\lim \sigma (q)/q^2=\beta$.

References in corpus (1)

• Maximally nonassociative quasigroups via quadratic orthomorphisms

Cited by in corpus (3)

• Maximally nonassociative quasigroups via quadratic orthomorphisms
• Isomorphisms of quadratic quasigroups
• Cycles of quadratic Latin squares and anti-perfect $1$-factorisations