A conjecture by Bienvenu and Geroldinger on power monoids

Abstract

Let $S$ be a numerical monoid, i.e., a submonoid of the additive monoid $(\mathbb{N},+)$ of non-negative integers such that $\mathbb{N}\setminus S$ is finite. Endowed with the operation of set addition, the family of all finite subsets of $S$ containing $0$ is itself a monoid, which we denote by ${\mathcal{P}}_{\mathrm{fin},0}(S)$. We show that, if $S_1$ and $S_2$ are numerical monoids and ${\mathcal{P}}_{\mathrm{fin},0}(S_1)$ is isomorphic to ${\mathcal{P}}_{\mathrm{fin},0}(S_2)$, then $S_1=S_2$. (In fact, we establish a more general result, in which $S_1$ and $S_2$ are allowed to be subsets of the non-negative rational numbers that contain zero and are closed under addition.) This proves a conjecture of Bienvenu and Geroldinger.

6 pages, no figures. Final version to appear in Proc. Amer. Math. Soc

Cited by in corpus (3)

• On the arithmetic of power monoids
• On the automorphisms of the power semigroups of a numerical semigroup
• Torsion groups and the Bienvenu--Geroldinger conjecture