有限可嵌入的极大超滤子
Ultrafilters maximal for finite embeddability
摘要：
在[1]中，作者证明了组合数论中出现的预序的一些基本性质，即自然数集合之间的有限嵌入性，并将其推广到超滤子，这与自然数的离散空间的斯通-切赫紧化的代数和拓扑结构有关。在本文中，我们继续研究这些前序。特别地，我们证明了存在有限可嵌入的极大超滤子，并且证明了这种超滤子的集合是半群$(\bN,\oplus)$中极小双边理想的闭包，即$\overline{K(\bN,\oplus)}$ .因此，我们很容易得到超滤子在$\overline{K(\bN,\oplus)}$中的许多组合性质。我们也给出了我们的主要结果的另一种证明的基础上的非标准模型的算术。
英文摘要：
In [1] the authors showed some basic properties of a pre-order that arose in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and they presented its generalization to ultrafilters, which is related to the algebraical and topological structure of the Stone-Čech compactification of the discrete space of natural numbers. In this present paper we continue the study of these pre-orders. In particular, we prove that there exist ultrafilters maximal for finite embeddability, and we show that the set of such ultrafilters is the closure of the minimal bilateral ideal in the semigroup $(\bN,\oplus)$, namely $\overline{K(\bN,\oplus)}$. As a consequence, we easily derive many combinatorial properties of ultrafilters in $\overline{K(\bN,\oplus)}$. We also give an alternative proof of our main result based on nonstandard models of arithmetic.
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