发表机构
School of Mathematics, Sichuan University（四川大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文刻画了$C_\sigma$-唯一dcpo的内在性质，并据此解决了三个开放问题：幂保持唯一性、二元积不保持唯一性及sober可数frame的反例。
AI 中文摘要
一个dcpo $D$ 被称为 $C_{\sigma}$-唯一的，如果对于每个dcpo $Q$，$\Gamma(D)\cong \Gamma(Q)$ 蕴含 $D\cong Q$，其中 $\Gamma(D)$ 表示 $D$ 的Scott闭子集格。我们给出了 $C_{\sigma}$-唯一性在Skula密度和Scott闭包方面的内在刻画，精确确定了何时一个真子dcpo能保持Scott闭集格。基于此刻画，我们回答了三个开放问题：(1) Johnstone dcpo的每个幂 $\J^I$ 都是 $C_{\sigma}$-唯一的；(2) $C_{\sigma}$-唯一性不被二元积保持，即使两个因子及其积都是sober的；(3) 一个sober的可数完全格不必是 $C_{\sigma}$-唯一的，即使它是一个frame。
英文摘要
A dcpo \(D\) is called \(C_σ\)-unique if for every dcpo \(Q\), \( Γ(D)\cong Γ(Q) \) implies \( D\cong Q, \) where \(Γ(D)\) denotes the lattice of Scott-closed subsets of \(D\). We give an intrinsic characterization of \(C_σ\)-uniqueness in terms of Skula-density and Scott closure, identifying precisely when a proper sub-dcpo can preserve the lattice of Scott-closed sets. Based on this characterization, we answer three open problems: (1) every power $\J^I$ of Johnstone's dcpo is $C_σ$-unique; (2) $C_σ$-uniqueness is not preserved by binary products, even when both factors and their product are sober; and (3) a sober countable complete lattice need not be $C_σ$-unique, even when it is a frame.
