主 讲 人 ：Andreas Reinhart    

活动时间：2025-09-21 18:00:00

地      点 ：数学科学学院D203报告厅(Zoom: https://us06web.zoom.us/j/86763384947?pwd=qXhOzOcHvaaiw2jqADI8iqNavdgm14.1, ID: 86763384947; passcode: 612593)

主办单位：数学科学学院

讲座内容：

Let $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$ be the family of all non-empty finite subsets of $\mathbb{N}$ that contain $0$, where $\mathbb{N}$ is the set of non-negative integers. Equipped with the binary operation of set addition\[(A,B) \mapsto A+B := \{a+b: a \in A, \, b \in B \},\] this family forms a commutative monoid, called the reduced finitary power monoid of $\mathbb{N}$.

The identity element is the singleton $\{0\}$. A non-identity element $A \in \mathcal{P}_{{\rm fin},0}(\mathbb{N})$ is called an atom if it cannot be expressed as the sum of two elements of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$, both distinct from $\{0\}$. For each $B \in \mathcal{P}_{{\rm fin},0}(\mathbb{N})$, we denote by $\mathsf{L}(B)$ the \textsf{length set} of $B$, that is, the set of all integers $n \ge 0$ such that $B$ can be written as the sum of $n$ atoms of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$. The collection\[\mathcal{L}(\mathcal{P}_{{\rm fin},0}(\mathbb{N})) := \{\mathsf{L}(B): B \in \mathcal{P}_{{\rm fin},0}(\mathbb{N}) \}\] is called the system of length sets of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$.

It is not difficult to verify that every length set of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$, except for $\{0\}$ and $\{1\}$, is a non-empty finite subset $L$ of $\mathbb{N}$ whose minimum is larger than or equal to $2$. Fan and Tringali (2018) conjectured that the converse also holds, namely, every such set $L$ is indeed a length set of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$.

In this talk, we present several results supporting this conjecture. In particular, we show that $\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ is \textsf{fully elastic}, that is, for every rational number $r \ge 1$, there exists a set $C \in \mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$, distinct from $\{0\}$, such that $\max \mathsf{L}(C) = r \, \min \mathsf{L}(C)$.

主讲人介绍：

Andreas Reinhart is currently a postdoctoral researcher at the University of Graz. He earned his Ph.D. under the supervision of Professor Franz Halter-Koch in 2010 and completed his habilitation at the University of Graz in 2020. His research focuses on various topics in commutative ring theory, factorization theory, multiplicative ideal theory, and algebraic number theory. He has published approximately 25 research papers in a range of academic journals.