Scott T. Chapman, Joshua Jang, JASON MAO, Skyler Mao
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引用次数: 0
摘要
假设 M 是普伊塞克斯单元,即由非负有理数组成的单元(在标准加法下)。在本文中,我们将从相关贝蒂图的角度研究原子普伊塞克斯单元中的因式分解。M$ 中 $b \ 的贝蒂图是其顶点为 b 的因式的图,因式之间的边至少共享一个原子。如果与 b 关联的贝蒂图是断开的,那么我们称 b 为 M 的贝蒂元。我们明确计算了一大类 Puiseux monoids(某些有理数无限序列的原子化)的贝蒂元集合。原子化过程对研究 Puiseux 单元的算术非常有用,最近的文献也在积极考虑这个问题。这引出了一个论点,即对于每一个正整数 n,都存在一个原子 Puiseux 单元,它恰好有 n 个贝蒂元。
Let M be a Puiseux monoid, that is, a monoid consisting of nonnegative rationals (under standard addition). In this paper, we study factorisations in atomic Puiseux monoids through the lens of their associated Betti graphs. The Betti graph of
$b \in M$
is the graph whose vertices are the factorisations of b with edges between factorisations that share at least one atom. If the Betti graph associated to b is disconnected, then we call b a Betti element of M. We explicitly compute the set of Betti elements for a large class of Puiseux monoids (the atomisations of certain infinite sequences of rationals). The process of atomisation is quite useful in studying the arithmetic of Puiseux monoids, and it has been actively considered in recent literature. This leads to an argument that for every positive integer n, there exists an atomic Puiseux monoid with exactly n Betti elements.
期刊介绍:
Bulletin of the Australian Mathematical Society aims at quick publication of original research in all branches of mathematics. Papers are accepted only after peer review but editorial decisions on acceptance or otherwise are taken quickly, normally within a month of receipt of the paper. The Bulletin concentrates on presenting new and interesting results in a clear and attractive way.
Published Bi-monthly
Published for the Australian Mathematical Society