Abstract

A Puiseux monoid is an additive submonoid of the nonnegative rational numbers. If M is a Puiseux monoid, then the question of whether each nonunit element of M can be written as a sum of irreducible elements (that is, M is atomic) is surprisingly difficult. For instance, although various techniques have been developed over the past few years to identify subclasses of Puiseux monoids that are atomic, no general characterization of such monoids is known. Here we survey some of the most relevant aspects related to the atomicity of Puiseux monoids. We provide characterizations of when M is finitely generated, factorial, half-factorial, other-half-factorial, Prüfer, seminormal, root-closed, and completely integrally closed. In addition to the atomic property, precise characterizations are also not known for when M satisfies the ACCP, is a BF-monoid, or is an FF-monoid; in each of these cases, we construct classes of Puiseux monoids satisfying these properties.

摘要

Puiseux monoid是非负有理数的加和亚类。如果M是Puiseux单面体，那么M的每个非单元元素是否可以写成不可约元素之和（即M是原子）的问题出奇地困难。例如，尽管在过去几年中已经开发出各种技术来识别原子级的Puiseux monoid的子类，但是尚不知道此类monoid的一般特征。在这里，我们调查了与Puiseux monoid的原子性相关的一些最相关方面。我们提供M何时的特征是有限生成的，阶乘，半阶乘，半半阶乘，Prüfer，半正规，根封闭和完全整体封闭。除了原子性质以外，当M满足ACCP，BF-monoid或FF-monoid时，还不知道精确的表征。在每种情况下，我们都构造满足这些性质的Puiseux单面体的类。