Torsion groups and the Bienvenu--Geroldinger conjecture

TL;DR

This paper investigates the structure of reduced finitary power monoids derived from cancellative monoids, proving that isomorphism of these monoids implies the original monoids are isomorphic in the torsion case, thus advancing the understanding of the Bienvenu--Geroldinger conjecture.

Contribution

The authors prove that for cancellative monoids with one being torsion, the associated power monoids' isomorphism implies the original monoids are isomorphic, confirming a special case of the conjecture.

Findings

Isomorphism of power monoids implies monoid isomorphism for torsion cancellative monoids.

Confirmed the conjecture for the case where one monoid is torsion.

Open question remains for non-torsion or arbitrary groups.

Abstract

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.