A conjecture by Bienvenu and Geroldinger on power monoids

TL;DR

The paper resolves a rigidity question for reduced finitary power monoids: if the reduced FPMs $\mathcal{P}_{\mathrm{fin},0}(S_1)$ and $\mathcal{P}_{\mathrm{fin},0}(S_2)$ are isomorphic, then the underlying Puiseux monoids $S_1$ and $S_2$ must be isomorphic, strengthening Bienvenu and Geroldinger’s conjecture from numerical monoids to Puiseux monoids. The authors prove this by exploiting Nathanson’s Fundamental Theorem of Additive Combinatorics to show that any isomorphism on reduced FPMs induces a bijective monoid homomorphism $\Phi:S_1\to S_2$ defined via $\Phi(a)=\max\phi(\{0,a\})$, with $\Phi(a+b)=\Phi(a)+\Phi(b)$. The argument begins with analyzing 2-element subsets and extends to general finite subsets, thereby recovering the entire monoid structure from the power monoid. As a consequence, the result implies the conjecture holds for Puiseux monoids, and in particular yields the numerical-monoid case as a corollary, contributing to the broader isomorphism problem for power monoids and advancing the unifying theory of factorization in additive combinatorics.

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.

Theorems & Definitions (12)

• Remark 1.1
• Theorem 2.1: Nathanson's theorem
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5
• proof