Maximally nonassociative quasigroups via quadratic orthomorphisms

TL;DR

This paper proves that for most orders, there exist maximally nonassociative quasigroups, except for specific cases related to prime factorizations, expanding understanding of quasigroup structures.

Contribution

It establishes the existence of maximally nonassociative quasigroups for almost all orders, except for certain prime-related cases, using quadratic orthomorphisms.

Findings

Existence of maximally nonassociative quasigroups for most orders

Exceptions occur when order is of the form 2p1 or 2p1p2 with specific primes

Provides a classification based on prime factorization constraints

Abstract

A quasigroup $Q$ is called maximally nonassociative if for $x,y,z\in Q$ we have that $x\cdot (y\cdot z) = (x\cdot y)\cdot z$ 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 $n=2p_1p_2$ for primes $p_1,p_2$ with $p_1\le p_2<2p_1$.