Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2509.23818 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
Table des matières:
- Let $H$ be a (multiplicatively written) monoid. The family $\mathcal{P}_{\text{fin},1}(H)$ of finite subsets of $H$ containing the identity element is itself a monoid when endowed with setwise multiplication induced by $H$. Tringali and Yan proved that two monoids $H_1$ and $H_2$ contained in a special class of commutative, cancellative monoids are isomorphic if and only if $\mathcal{P}_{\text{fin},1}(H_1)$ and $\mathcal{P}_{\text{fin},1}(H_2)$ are. Moreover, they raised the question whether the same holds in the general setting of cancellative monoids. We show that if $H_1$ and $H_2$ are (commutative) valuation monoids with trivial unit groups and isomorphic quotient groups, then $\mathcal{P}_{\text{fin},1}(H_1)\simeq\mathcal{P}_{\text{fin},1}(H_2)$. This provides a negative answer to Tringali and Yans question already within the class of valuation submonoids of the additive group $\mathbb{Z}^2$.