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arXiv · 2407.16766

A note on the probability of a groupoid having deficient sets

Carles Cardó
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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.

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Carles Cardó. 2024-07-23. A note on the probability of a groupoid having deficient sets. https://arxiv.org/abs/2407.16766

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