Torsion groups and the Bienvenu--Geroldinger conjecture
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915844828692480 |
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| author | Tringali, Salvatore Yan, Weihao |
| author_facet | Tringali, Salvatore Yan, Weihao |
| contents | Equipped with the operation of setwise multiplication induced by a (multiplicatively written) monoid $H$ on its parts, the collection of all finite subsets of $H$ containing the identity element is itself a monoid, denoted by $\mathcal P_{\textrm{fin}, 1}(H)$ and called the reduced finitary power monoid of $H$.
One is naturally led to ask whether, for all $H$ and $K$ in a given class of monoids, $\mathcal P_{\textrm{fin},1}(H)$ and $\mathcal P_{\textrm{fin},1}(K)$ are isomorphic if and only if $H$ and $K$ are. The problem originates from a conjecture of Bienvenu and Geroldinger that was recently settled by the authors. Here, we provide a positive answer to the problem in the case where $H$ and $K$ are cancellative monoids, one of which is torsion. In particular, the answer is in the affirmative when $H$ and $K$ are torsion groups. Whether the conclusion extends to arbitrary groups remains open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19592 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Torsion groups and the Bienvenu--Geroldinger conjecture Tringali, Salvatore Yan, Weihao Group Theory Combinatorics Rings and Algebras Equipped with the operation of setwise multiplication induced by a (multiplicatively written) monoid $H$ on its parts, the collection of all finite subsets of $H$ containing the identity element is itself a monoid, denoted by $\mathcal P_{\textrm{fin}, 1}(H)$ and called the reduced finitary power monoid of $H$. One is naturally led to ask whether, for all $H$ and $K$ in a given class of monoids, $\mathcal P_{\textrm{fin},1}(H)$ and $\mathcal P_{\textrm{fin},1}(K)$ are isomorphic if and only if $H$ and $K$ are. The problem originates from a conjecture of Bienvenu and Geroldinger that was recently settled by the authors. Here, we provide a positive answer to the problem in the case where $H$ and $K$ are cancellative monoids, one of which is torsion. In particular, the answer is in the affirmative when $H$ and $K$ are torsion groups. Whether the conclusion extends to arbitrary groups remains open. |
| title | Torsion groups and the Bienvenu--Geroldinger conjecture |
| topic | Group Theory Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2601.19592 |