On the automorphisms of numerical power monoids

Abstract

Let $H$ be a numerical monoid, that is, a cofinite submonoid of $\mathbb{N}$ (the non-negative integers under addition). Denote by ${\mathcal{P}}_{\text{fin},0}(H)$ the monoid obtained by endowing the family of all finite subsets of $H$ containing $0$ with the operation of setwise addition induced by $H$ on its power set. Tringali and Yan [JCTA, 2025] have recently established that ${\mathcal{P}}_{\text{fin},0}(\mathbb{N})$ has a unique non-trivial automorphism, and conjectured that the automorphism group of ${\mathcal{P}}_{\text{fin},0}(H)$ is trivial whenever $H\ne \mathbb{N}$. We prove this conjecture and, as a byproduct, give a new proof of the Tringali--Yan theorem.

arXiv comment11 pages, no figures