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, as conjectured in [4], we show that the probability is not zero in the case of sets of two elements and calculate the exact value. We explore some generalisations on deficient sets and their likelihoods.

11 pages, 1 figure. Under review at Algebra Universalis