论文标题
线性代数的末端
Ends of spaces via linear algebra
论文作者
论文摘要
我们开发了一种理论，可以被视为粗略理论的前传。我们将空间的末端视为无穷大的额外点。为了讨论无限空间的行为，需要一个接近无限的概念（一种度量）。最简单的方法是列出有限的$ x $的子集（即远离无限），该列表应满足某些基本属性。这样的列表$ s_x $我们在集合$ x $上调用a \ textbf {scale}（请参阅第3节）。为了使用石材二元定理中的想法，我们考虑了sub-boolean代数$ ba_x $ of power set $ 2^x $ $ x $的$ x $，其中包含$ s_x $，并且自然而然地导致了\ textbf {scaped bool bool elgebra} $ textbf {x，s_x，s_x，ba_x $ bue y s $ bue n a $ bo bo bo bo y s y x， $（\ bar x，s_x，\ overline {ba_x}）$是\ textbf {compact in infinity}。 给定一个缩放空间$（x，s_x）$最自然的布尔代数为$（x，s_x，2^x）$，与$ x $的几何形状相距太远。因此，我们需要弄清楚如何将$ 2^x $修剪为较小的子树状元代数$ ba_x $。通常，如何将sub-Boolean代数$ ba_x $修剪为较小的代数。这是使用线性代数的想法完成的。 Namely, we consider a family $\mathcal{F}$ of naturally arising $S_X$-linear operators on $BA_X$ and the smaller sub-Boolean algebra $BA_{\mathcal{F}}$ consists of eigensets of $\mathcal{F}$, an analog of eigenvectors from linear algebra.我们表明，到目前为止的文献中定义的所有末端（Freundenthal End，有限生成的群体的末端，斑点末端，Cornulier端，粗空间的末端）都是这种过程的特殊情况。
We develop a theory that may be considered as a prequel to the coarse theory. We are viewing ends of spaces as extra points at infinity. In order to discuss behaviour of spaces at infinity one needs a concept (a measure) of approaching infinity. The simplest way to do so is to list subsets of $X$ that are bounded (i.e. far from infinity) and that list should satisfy certain basic properties. Such a list $S_X$ we call a \textbf{scale} on a set $X$ (see Section 3). In order to use ideas from the Stone Duality Theorem we consider sub-Boolean algebras $BA_X$ of the power set $2^X$ of $X$ that contain $S_X$ and that leads naturally to the concept of ends of a \textbf{scaled Boolean algebra} $(X,S_X,BA_X)$ which can be attached to $X$ and form a new scaled Boolean algebra $(\bar X,S_X,\overline{BA_X})$ that is \textbf{compact at infinity}. Given a scaled space $(X,S_X)$ the most natural scaled Boolean algebra is $(X,S_X,2^X)$ which can be too far removed from the geometry of $X$. Therefore we need to figure out how to trim $2^X$ to a smaller sub-Boolean algebra $BA_X$. More generally, how to trim a sub-Boolean algebra $BA_X$ to a smaller one. That is done using ideas from linear algebra. Namely, we consider a family $\mathcal{F}$ of naturally arising $S_X$-linear operators on $BA_X$ and the smaller sub-Boolean algebra $BA_{\mathcal{F}}$ consists of eigensets of $\mathcal{F}$, an analog of eigenvectors from linear algebra. We show that all ends defined in literature so far (Freundenthal ends, ends of finitely generated groups, Specker ends, Cornulier ends, ends of coarse spaces) are special cases of such a process.
