A conjecture by Bienvenu and Geroldinger on power monoids

TL;DR

This paper proves a conjecture that the monoid of finite subsets containing zero of a numerical monoid uniquely determines the monoid itself, extending to certain subsets of non-negative rationals.

Contribution

It establishes that isomorphic power monoids imply the original monoids are identical, confirming a conjecture by Bienvenu and Geroldinger.

Findings

Isomorphic power monoids imply identical numerical monoids.

Extension of the result to subsets of non-negative rationals.

Confirms a conjecture in the structure theory of numerical 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 $\mathcal P_{{\rm fin}, 0}(S)$. We show that, if $S_1$ and $S_2$ are numerical monoids and $\mathcal P_{{\rm fin}, 0}(S_1)$ is isomorphic to $\mathcal P_{{\rm fin}, 0}(S_2)$, 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.