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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.10209 |
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| _version_ | 1866911105349058560 |
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| author | Reinhart, Andreas |
| author_facet | Reinhart, Andreas |
| contents | The set $\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ of all finite subsets of $\mathbb{N}_0$ containing the zero element is a monoid with set addition as operation. If a set $A\in\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ can be written in the form $A=\sum_{i=1}^{\ell} A_i$ with $\ell\in\mathbb{N}_0$ and indecomposable elements $(A_i)_{i=1}^{\ell}$ of $\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$, then $\ell$ is a factorization length of $A$ and $\mathsf{L}(A)\subseteq\mathbb{N}_0$ denotes the set of all possible factorization lengths of $A$. We show that for each rational number $q\geq 1$, there is some $A\in\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ such that $q=\frac{\max(\mathsf{L}(A))}{\min(\mathsf{L}(A))}$. This supports a Conjecture of Fan and Tringali. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10209 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the system of length sets of power monoids Reinhart, Andreas Commutative Algebra 11B13, 11B30, 20M13 The set $\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ of all finite subsets of $\mathbb{N}_0$ containing the zero element is a monoid with set addition as operation. If a set $A\in\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ can be written in the form $A=\sum_{i=1}^{\ell} A_i$ with $\ell\in\mathbb{N}_0$ and indecomposable elements $(A_i)_{i=1}^{\ell}$ of $\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$, then $\ell$ is a factorization length of $A$ and $\mathsf{L}(A)\subseteq\mathbb{N}_0$ denotes the set of all possible factorization lengths of $A$. We show that for each rational number $q\geq 1$, there is some $A\in\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ such that $q=\frac{\max(\mathsf{L}(A))}{\min(\mathsf{L}(A))}$. This supports a Conjecture of Fan and Tringali. |
| title | On the system of length sets of power monoids |
| topic | Commutative Algebra 11B13, 11B30, 20M13 |
| url | https://arxiv.org/abs/2508.10209 |