A note on the probability of a groupoid having deficient sets

TL;DR

This paper investigates the probability of deficient subsets in groupoids, showing that for sets of size two the probability is non-zero and providing exact calculations, extending understanding of deficient set likelihoods.

Contribution

It proves that the probability of a two-element deficient set in a groupoid is non-zero and calculates its exact value, addressing a conjecture and exploring generalizations.

Findings

Probability of two-element deficient sets is non-zero.

Exact probability value for two-element deficient sets.

Generalizations on deficient sets and their likelihoods.

Abstract

A subset $X$ of a groupoid is said to be deficient if $|X \cdot X|\leq |X|$. It is well-known that the probability that a random groupoid has a deficient $t$-element set with $t\geq 3$ 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.