Two Quasigroup Elements Can Commute With Any Positive Rational Probability

Abstract

SummaryA quasigroup is a set with a binary operation in which both left and right division are unique. Equivalently, every row and column in a quasigroup table is a permutation of its elements. The commuting probability of a quasigroup is the probability that two of its elements, chosen at random, will commute. In this paper, we show that a quasigroup may have any rational number in (0,1] as a commuting probability. AcknowledgmentsThe author would like to thank their advisor, Vadim Ponomarenko, for helping and supporting them throughout the process of writing this article.