The isomorphism problem for reduced finitary power monoids

Abstract

Let $H$ be a multiplicatively written monoid with identity $1_H$ and let ${\mathcal{P}}_{\text{fin},1}(H)$ denote the reduced finitary power monoid of $H$, that is, the monoid consisting of all finite subsets of $H$ containing $1_H$ with set multiplication as operation. Building on work of Tringali and Yan, we give a complete description of pairs of commutative and cancellative monoids $H,K$ for which ${\mathcal{P}}_{\text{fin},1}(H)$ and ${\mathcal{P}}_{\text{fin},1}(K)$ are isomorphic.