$G_δ$半滤子和$ω^*$
$G_δ$ semifilters and $ω^*$
摘要：
偏序$([ω]^ω,\subseteq^*)$上的超滤子是$ω$上的自由超滤子，它们构成了空间$ω^*$，即$ω$的Stone-Cech剩余。如果$U$是该偏序的上集（即，半滤子），则$U$上的超滤子通过Stone对偶对应于$ω^*$的闭子集。此外，如果$U$足够“简单”（更准确地说，$G_δ$作为$2^ω$的子集），我们表明，$U$是类似于$[ω]^ω$在几个方面。第一，$\mathfrak{p}_U = \mathfrak{t}_U = \mathfrak{p}$（这推广了Malliaris和Shelah的结果）。其次，如果是$\mathfrak{d} = \mathfrak{c}$，则$U$上存在也是$P$ -过滤器的超过滤器（这扩展了Ketonen的结果）。第三，在$U$上存在弱$P$ -滤子的超滤子（这扩展了Kunen的一个结果）。通过选择适当的$U$，这些相似性定理在动力学，代数和组合学中找到应用。最值得注意的是，我们将证明$(ω^*,+)$包含极小左理想，也是弱$P$ -集。
英文摘要：
The ultrafilters on the partial order $([ω]^ω,\subseteq^*)$ are the free ultrafilters on $ω$, which constitute the space $ω^*$, the Stone-Cech remainder of $ω$. If $U$ is an upperset of this partial order (i.e., a semifilter), then the ultrafilters on $U$ correspond to closed subsets of $ω^*$ via Stone duality. If, in addition, $U$ is sufficiently "simple" (more precisely, $G_δ$ as a subset of $2^ω$), we show that $U$ is similar to $[ω]^ω$ in several ways. First, $\mathfrak{p}_U = \mathfrak{t}_U = \mathfrak{p}$ (this extends a result of Malliaris and Shelah). Second, if $\mathfrak{d} = \mathfrak{c}$ then there are ultrafilters on $U$ that are also $P$-filters (this extends a result of Ketonen). Third, there are ultrafilters on $U$ that are weak $P$-filters (this extends a result of Kunen). By choosing appropriate $U$, these similarity theorems find applications in dynamics, algebra, and combinatorics. Most notably, we will prove that $(ω^*,+)$ contains minimal left ideals that are also weak $P$-sets.
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