Commuting Pairs in Quasigroups

TL;DR

This paper investigates the structure of quasigroups by analyzing the set of commuting pairs, establishing which proportions of commuting pairs are possible, and characterizing the spectrum of quasigroup orders for given commuting proportions.

Contribution

It characterizes the possible numbers of commuting pairs in quasigroups of a given order and determines the spectrum of orders for specified commuting pair proportions.

Findings

Every rational in (0,1] can be realized as the proportion of commuting pairs in some quasigroup.

For each positive integer n, the set of achievable numbers of commuting pairs is characterized.

The spectrum of orders for quasigroups with a given commuting proportion q is determined.

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 $q$.