非阿基米德乌里松通用度量空间上的代数结构
Algebraic structures on non-Archimedean Urysohn universal metric spaces
AI总结：
研究乌里松通用超度量空间上的值域结构，引入\(p\)-adic Levi–Civita 域等，证明具有无限剩余域的完备值域对可分超度量空间通用，还给出可分情况下值域与乌里松空间等距及特定剩余域时全 Hahn 型值域成为乌里松通用超度量空间的条件。
AI中文摘要：
我们研究乌里松通用超度量空间上的值域结构。引入\(p\)-adic Levi–Civita 域作为\(p\)-adic Hahn 域的子域，并将它们与普通 Levi–Civita 域一起处理。对于包含\(\mathbb{Z}\)的\(\mathbb{R}\)的子群\(G\)和可数无限完美域\(k\)，相应 Levi–Civita 值域与\(R\)-乌里松通用超度量空间等距，其中\(R = \{0\} \cup \{ \eta^{-g} \mid g \in G\}\)。这些空间允许扩展规定素值域的域结构，包括平凡赋值的\(\mathbb{Q}\)和\(p\)-adic 域\(\mathbb{Q}_{p}\)。我们还证明了具有无限剩余域的完备值域是晕的，因此对于具有相应距离集的可分超度量空间是通用的。在可分情况下，这样的值域本身与相应的乌里松空间等距。最后，对于可数无限完美剩余域，相应的全 Hahn 型值域恰好当其值群与\(\mathbb{Z}\)序同构时是乌里松通用超度量空间。
英文摘要：
We investigate valued-field structures on Urysohn universal ultrametric spaces. We introduce $p$-adic Levi--Civita fields as subfields of $p$-adic Hahn fields and treat them together with ordinary Levi--Civita fields. For a subgroup $G$ of $\mathbb{R}$ containing $\mathbb{Z}$ and a countably infinite perfect field $k$, the corresponding Levi--Civita valued field is isometric to the $R$-Urysohn universal ultrametric space, where $R=\{0\}\cup\{η^{-g}\mid g\in G\}$. Thus these spaces admit field structures extending prescribed prime valued fields, including $\mathbb{Q}$ with the trivial valuation and the $p$-adic fields $\mathbb{Q}_{p}$. We also prove that complete valued fields with infinite residue fields are haloed, and hence universal for separable ultrametric spaces with corresponding distance sets. In the separable case, such a valued field is itself isometric to the corresponding Urysohn space. Examples include $\mathbb{C}_{p}$, the completion of the maximal unramified extension of $\mathbb{Q}_{p}$, Laurent series fields, and completions of their algebraic closures. Finally, for a countably infinite perfect residue field, the corresponding full Hahn-type valued field is a Urysohn universal ultrametric space exactly when its value group is order-isomorphic to $\mathbb{Z}$.
