The isomorphism problem for reduced finitary power monoids

TL;DR

This paper characterizes when the reduced finitary power monoids of two commutative, cancellative monoids are isomorphic, extending previous work and providing a complete classification of such pairs.

Contribution

It offers a complete description of isomorphism conditions between reduced finitary power monoids of commutative, cancellative monoids, building on prior research.

Findings

Provides a full classification of isomorphic reduced finitary power monoids

Extends previous work by Tringali and Yan on power monoids

Clarifies the structure of monoid isomorphisms in this context

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.