A Puiseux monoid is a submonoid of $(\mathbb{Q},+)$ consisting of nonnegative rational numbers. Although the operation of addition is continuous with respect to the standard topology, the set of irreducibles of a Puiseux monoid is, in general, difficult to describe. In this paper, we use topological density to understand how much a Puiseux monoid, as well as its set of irreducibles, spread through $\mathbb{R}_{\ge 0}$. First, we separate Puiseux monoids according to their density in $\mathbb{R}_{\ge 0}$, and we characterize monoids in each of these classes in terms of generating sets and sets of irreducibles. Then we study the density of the difference group, the root closure, and the conductor semigroup of a Puiseux monoid. Finally, we prove that every Puiseux monoid generated by a strictly increasing sequence of rationals is nowhere dense in $\mathbb{R}_{\ge 0}$ and has empty conductor.

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Puiseux幺半群的原子性和密度

Puiseux 幺半群是 $(\mathbb{Q},+)$ 的子幺半群，由非负有理数组成。尽管加法运算相对于标准拓扑是连续的，但 Puiseux 幺半群的不可约集合通常难以描述。在本文中，我们使用拓扑密度来了解 Puiseux 幺半群及其不可约集合通过 $\mathbb{R}_{\ge 0}$ 传播的程度。首先，我们根据它们在 $\mathbb{R}_{\ge 0}$ 中的密度分离 Puiseux 幺半群，并且我们根据生成集和不可约集来表征这些类中的每一个类中的幺半群。然后我们研究了 Puiseux 幺半群的差分群、根闭合和导体半群的密度。最后，