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<p>A subset <i>X</i> of a groupoid is said to be deficient if <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_906_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">\(|X \cdot X|\le |X|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo>·</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>. It is well-known that the probability that a random groupoid has a deficient <i>t</i>-element set with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_906_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(t\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

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A note on the probability of a groupoid having deficient sets

  • 期刊论文

  • 发表日期：

    2025年9月22日

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  • Carles Cardó

摘要

A subset X of a groupoid is said to be deficient if \(|X \cdot X|\le |X|\) | X · X | ≤ | X | . It is well-known that the probability that a random groupoid has a deficient t-element set with \(t\ge 3\) t ≥ 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.