AI 中文总结
本文围绕数值幂幺半群的自同构展开。研究了\(\mathcal{P}_{\text{fin},0}(H)\)，Tringali和Yan猜想当\(H\neq\mathbb{N}\)时其自同构群平凡，本文证明了该猜想并给出Tringali - Yan定理新证明。
AI 中文摘要
设\(H\)是一个数值幺半群，即\(\mathbb{N}\)（加法下的非负整数）的余有限子幺半群。用\(\mathcal{P}_{\text{fin},0}(H)\)表示通过赋予\(H\)中所有包含\(0\)的有限子集族以由\(H\)在其幂集上诱导的逐点加法运算而得到的幺半群。Tringali和Yan最近证明\(\mathcal{P}_{\text{fin},0}(\mathbb{N})\)有唯一非平凡自同构，并猜想当\(H\neq\mathbb{N}\)时\(\mathcal{P}_{\text{fin},0}(H)\)的自同构群是平凡的。本文证明了该猜想，并给出Tringali - Yan定理的新证明。
英文摘要
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.
Comments11 pages, no figures
