Maximal nonassociativity via fields

TL;DR

This paper proves the existence of maximally nonassociative quasigroups for many more orders than previously known, using finite fields and quadratic character sums, with a largely constructive approach.

Contribution

It extends the existence results of maximally nonassociative quasigroups to a broader set of orders using finite field techniques.

Findings

Existence of maximally nonassociative quasigroups for many new orders.

Use of finite fields and Weil bounds to construct such quasigroups.

Results are largely constructive, improving on previous non-constructive proofs.

Abstract

We say that $(x,y,z)\in Q^3$ is an associative triple in a quasigroup $Q(*)$ if $(x*y)*z=x*(y*z)$. Let $a(Q)$ denote the number of associative triples in $Q$. It is easy to show that $a(Q)\ge |Q|$, and we call the quasigroup maximally nonassociative if $a(Q)= |Q|$. It was conjectured that maximally nonassociative quasigroups do not exist when $|Q|>1$. Dr\'apal and Lison\v{e}k recently refuted this conjecture by proving the existence of maximally nonassociative quasigroups for a certain infinite set of orders $|Q|$. In this paper we prove the existence of maximally nonassociative quasigroups for a much larger set of orders $|Q|$. Our main tools are finite fields and the Weil bound on quadratic character sums. Unlike in the previous work, our results are to a large extent constructive.