A conjecture by Bienvenu and Geroldinger on power monoids
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913773487390720 |
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| author | Tringali, Salvatore Yan, Weihao |
| author_facet | Tringali, Salvatore Yan, Weihao |
| contents | Let $S$ be a numerical monoid, i.e., a submonoid of the additive monoid $(\mathbb N, +)$ of non-negative integers such that $\mathbb N \setminus S$ is finite. Endowed with the operation of set addition, the family of all finite subsets of $S$ containing $0$ is itself a monoid, which we denote by $\mathcal P_{{\rm fin}, 0}(S)$.
We show that, if $S_1$ and $S_2$ are numerical monoids and $\mathcal P_{{\rm fin}, 0}(S_1)$ is isomorphic to $\mathcal P_{{\rm fin}, 0}(S_2)$, then $S_1 = S_2$. (In fact, we establish a more general result, in which $S_1$ and $S_2$ are allowed to be subsets of the non-negative rational numbers that contain zero and are closed under addition.) This proves a conjecture of Bienvenu and Geroldinger. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_17713 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A conjecture by Bienvenu and Geroldinger on power monoids Tringali, Salvatore Yan, Weihao Combinatorics Number Theory Primary 11B13, 11B30, 20M13 Let $S$ be a numerical monoid, i.e., a submonoid of the additive monoid $(\mathbb N, +)$ of non-negative integers such that $\mathbb N \setminus S$ is finite. Endowed with the operation of set addition, the family of all finite subsets of $S$ containing $0$ is itself a monoid, which we denote by $\mathcal P_{{\rm fin}, 0}(S)$. We show that, if $S_1$ and $S_2$ are numerical monoids and $\mathcal P_{{\rm fin}, 0}(S_1)$ is isomorphic to $\mathcal P_{{\rm fin}, 0}(S_2)$, then $S_1 = S_2$. (In fact, we establish a more general result, in which $S_1$ and $S_2$ are allowed to be subsets of the non-negative rational numbers that contain zero and are closed under addition.) This proves a conjecture of Bienvenu and Geroldinger. |
| title | A conjecture by Bienvenu and Geroldinger on power monoids |
| topic | Combinatorics Number Theory Primary 11B13, 11B30, 20M13 |
| url | https://arxiv.org/abs/2310.17713 |