Power monoids and the Bienvenu-Geroldinger problem for torsion groups

Abstract: Let $M$ be a (multiplicatively written) monoid with identity element $1_M$. Endowed with the operation of setwise multiplication induced by $M$, the collection of finite subsets of $M$ containing $1_M$ forms a monoid in its own right, denoted by $\mathcal{P}_{\mathrm{fin},1}(M)$ and called the reduced finitary power monoid of $M$. It is natural to ask whether, for all $H$ and $K$ in a given class of monoids, $\mathcal{P}_{\mathrm{fin},1}(H)$ is isomorphic to $\mathcal{P}_{\mathrm{fin},1}(K)$ if and only if $H$ is isomorphic to $K$. Originating from a conjecture of Bienvenu and Geroldinger recently settled by Yan and myself, the problem --- together with its numerous variants and ramifications --- has non-trivial connections to additive number theory and related fields. In this talk, I will present a positive solution for the class of torsion groups.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)