论文标题
在数值和puiseux monoids的分子上
On the set of molecules of numerical and Puiseux monoids
论文作者
论文摘要
$ \ mathbb {q} _ {\ ge 0} $的添加剂submonoids（也称为puiseux himoids）通常不是唯一的分解单体化（UFMS）。实际上，唯一的独特分解puiseux monoid是由一个元素产生的。但是，即使puiseux monoid不是UFM，它也可能包含非零元素具有一个分解的元素。我们称之为这样的元素分子。 W. Narkiewicz在代数数理论的背景下首先研究了分子。最近，F。Gotti和第一作者在Puiseux Monoid的背景下研究了分子。在这里，我们解决了与具有不同原子行为的PUISEUX MONOID的各种子类分子的大小相关的一些方面。特别是，我们积极回答以下最新的实现猜想：对于\ Mathbb {n} _ {\ ge 2} $中的每个$ m \，存在一个数值单sonoid，其非原子的一组非原子具有心脏性$ m $。
Additive submonoids of $\mathbb{Q}_{\ge 0}$, also known as Puiseux monoids, are not unique factorization monoids (UFMs) in general. Indeed, the only unique factorization Puiseux monoids are those generated by one element. However, even if a Puiseux monoid is not a UFM, it may contain nonzero elements having exactly one factorization. We call such elements molecules. Molecules were first investigated by W. Narkiewicz in the context of algebraic number theory. More recently, F. Gotti and the first author studied molecules in the context of Puiseux monoids. Here we address some aspects related to the size of the sets of molecules of various subclasses of Puiseux monoids with different atomic behaviors. In particular, we positively answer the following recent realization conjecture: for each $m \in \mathbb{N}_{\ge 2}$ there exists a numerical monoid whose set of molecules that are not atoms has cardinality $m$.
