Commuting Pairs in Quasigroups

Abstract

A quasigroup is a pair $\left(Q,\ast \right)$, where $Q$ is a nonempty set and $\ast$ is a binary operation on $Q$ such that for every $\left(a,b\right)\in Q^2$, there exists a unique $\left(x,y\right)\in Q^2$ such that $a\ast x=b=y\ast a$. Let $\left(Q,\ast \right)$ be a quasigroup. A pair $\left(x,y\right)\in Q^2$ is a commuting pair of $\left(Q,\ast \right)$ if $x\ast y=y\ast x$. Recently, it has been shown that every rational number in the interval $\left(0,1\right]$ can be attained as the proportion of ordered pairs that are commuting in some quasigroup. For every positive integer $n$, we establish the set of all integers $k$ such that there is a quasigroup of order $n$ with exactly $k$ commuting pairs. This allows us to determine, for a given rational $q\in \left(0,1\right]$, the spectrum of positive integers $n$ for which there is a quasigroup of order $n$ whose proportion of commuting pairs is equal to $q$.