Commuting Pairs in Quasigroups

Abstract

A quasigroup is a pair $(Q, *)$ where $Q$ is a non-empty set and $*$ is a binary operation on $Q$ such that for every $(a, b) \in Q^2$ there exists a unique $(x, y) \in Q^2$ such that $a*x=b=y*a$. Let $(Q, *)$ be a quasigroup. A pair $(x, y) \in Q^2$ is a commuting pair of $(Q, *)$ if $x * y = y * x$. Recently, it has been shown that every rational number in the interval $(0, 1]$ 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 $q \in (0, 1]$, the spectrum of positive integers $n$ for which there is a quasigroup of order $n$ whose proportion of commuting pairs is equal to $q$.