A counterexample to an isomorphism problem for power monoids

Abstract

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 ${\mathbb{Z}}^2$.