On Sets of Lengths in Monoids of plus-minus weighted Zero-Sum Sequences
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911854636302336 |
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| author | Geroldinger, Alfred Kainrath, Florian |
| author_facet | Geroldinger, Alfred Kainrath, Florian |
| contents | Let $G$ be an additive abelian group. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are $\varepsilon_1, \ldots, \varepsilon_{\ell} \in \{-1, 1\}$ such that $\varepsilon_1 g_1 + \ldots + \varepsilon_{\ell} g_{\ell}=0$. We study sets of lengths in the monoid $\mathcal B_{\pm} (G)$ of plus-minus weighted zero-sum sequences over $G$. If $G$ is finite, then sets of lengths are highly structured. If $G$ is infinite, then every finite, nonempty subset of $\mathbb N_{\ge 2}$ is the set of lengths of some sequence $S \in \mathcal B_{\pm} (G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17258 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Sets of Lengths in Monoids of plus-minus weighted Zero-Sum Sequences Geroldinger, Alfred Kainrath, Florian Commutative Algebra Combinatorics Number Theory 20M13, 11B30, 13A05 Let $G$ be an additive abelian group. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are $\varepsilon_1, \ldots, \varepsilon_{\ell} \in \{-1, 1\}$ such that $\varepsilon_1 g_1 + \ldots + \varepsilon_{\ell} g_{\ell}=0$. We study sets of lengths in the monoid $\mathcal B_{\pm} (G)$ of plus-minus weighted zero-sum sequences over $G$. If $G$ is finite, then sets of lengths are highly structured. If $G$ is infinite, then every finite, nonempty subset of $\mathbb N_{\ge 2}$ is the set of lengths of some sequence $S \in \mathcal B_{\pm} (G)$. |
| title | On Sets of Lengths in Monoids of plus-minus weighted Zero-Sum Sequences |
| topic | Commutative Algebra Combinatorics Number Theory 20M13, 11B30, 13A05 |
| url | https://arxiv.org/abs/2404.17258 |