A note on the probability of a groupoid having deficient sets
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913442778054656 |
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| author | Cardó, Carles |
| author_facet | Cardó, Carles |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the probability of a groupoid having deficient sets Cardó, Carles Group Theory Combinatorics Rings and Algebras 08A30, 60B99 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. |
| title | A note on the probability of a groupoid having deficient sets |
| topic | Group Theory Combinatorics Rings and Algebras 08A30, 60B99 |
| url | https://arxiv.org/abs/2407.16766 |