A note on the probability of a groupoid having deficient sets

Abstract

A subset X of a groupoid is said to be deficient if \(|X \cdot X|\le |X|\). It is well-known that the probability that a random groupoid has a deficient t-element set with \(t\ge 3\) is zero. However, it is believed that the probability is not zero for 2-element sets. We prove that it is indeed not null and calculate the exact value. We explore some generalisations on deficient sets and their likelihoods.