从Freudenthal谱定理到单位阿基米德格群的可投影壳，通过极小谱的紧化
From Freudenthal's Spectral Theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
摘要：
我们使用Riesz空间理论中的一个里程碑式的结果-- Freudenthal的1936年谱定理--来规范地表示任何阿基米德格序群$G$，其强单位为在其\emph{minimal}素理想的空间$Z_G$的适当的$G$ -索引零维紧化$w_GZ_G$上的实值连续函数的（非分离）格群。建立这个表示所需要的另外两个成分是$G$在其\emph{maximal}理想的空间$X_G$上的Yosida表示，以及$Z_G$到$X_G$上的著名的连续满射。然后，我们建立我们的主要结果表明，包含最小的扩展这一表示的$G$，分离点的$G$-即，子格子群的${\rm C}\,(Z_G)$产生的图像的$G$随着所有特征函数的clopen（封闭和开放）子集的$Z_G$这是由元素的$G$ -正是经典的可投影外壳的$G$。因此，我们的主要结果揭示了一个基本的可投影壳和最小谱之间的关系，并提供了最直接和明确的可投影壳的建设日期。我们的技术确实需要一个强大的团队。
英文摘要：
We use a landmark result in the theory of Riesz spaces - Freudenthal's 1936 Spectral Theorem - to canonically represent any Archimedean lattice-ordered group $G$ with a strong unit as a (non-separating) lattice-group of real valued continuous functions on an appropriate $G$-indexed zero-dimensional compactification $w_GZ_G$ of its space $Z_G$ of \emph{minimal} prime ideals. The two further ingredients needed to establish this representation are the Yosida representation of $G$ on its space $X_G$ of \emph{maximal} ideals, and the well-known continuous surjection of $Z_G$ onto $X_G$. We then establish our main result by showing that the inclusion-minimal extension of this representation of $G$ that separates the points of $Z_G$ - namely, the sublattice subgroup of ${\rm C}\,(Z_G)$ generated by the image of $G$ along with all characteristic functions of clopen (closed and open) subsets of $Z_G$ which are determined by elements of $G$ - is precisely the classical projectable hull of $G$. Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
