Torsion groups and the Bienvenu--Geroldinger conjecture
Salvatore Tringali, Weihao Yan
TL;DR
The paper investigates whether the reduced finitary power monoid $\mathcal{P}_{\mathrm{fin},1}(M)$ encodes the isomorphism type of the underlying monoid $M$, addressing a Bienvenu–Geroldinger–style conjecture. It develops a pullback mechanism: any isomorphism between $\mathcal{P}_{\mathrm{fin},1}(H)$ and $\mathcal{P}_{\mathrm{fin},1}(K)$ induces a bijection $g: H\to K$ with $f(\{1_H,x\})=\{1_K,g(x)\}$, and analyzes its structure via order and power-preservation properties. The main contribution proves that when $H$ and $K$ are cancellative and one is torsion (so both are torsion groups), this pullback is actually an isomorphism, yielding $H\cong K$ from $\mathcal{P}_{\mathrm{fin},1}(H)\cong\mathcal{P}_{\mathrm{fin},1}(K)$. This advances the BG-like isomorphism problem for power monoids and clarifies the torsion/cancellative regime, while leaving open the general case for arbitrary groups.
Abstract
Let $M$ be a monoid (written multiplicatively). Equipped with the operation of setwise multiplication induced by $M$ on its parts, the collection of all finite subsets of $M$ containing the identity element is itself a monoid, denoted by $\mathcal P_{{\rm fin}, 1}(M)$ and called the reduced finitary power monoid of $M$. One is naturally led to ask whether, for all $H$ and $K$ in a given class of monoids, $\mathcal P_{\fin,1}(H)$ and $\mathcal P_{\fin,1}(K)$ are isomorphic if and only if $H$ and $K$ are. The problem originates from a conjecture of Bienvenu and Geroldinger [Israel J. Math., 2025] that was recently settled by the authors [Proc. AMS, 2025]. 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.