发表机构
Nanchang Institute of Technology; Guilin University of Technology（南昌工程学院; 桂林理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究可数交连续 dcpo 的 Scott 积与 sobriety，构造反例否定两个问题，并证明 L-dcpo 在 Scott 平方为普通乘积拓扑时是 Scott sober，同时刻画 well-filteredness。
AI 中文摘要
我们研究可数交连续 directed-complete 偏序集（dcpo）的 Scott 积相容性与 sobriety。我们构造了一个可数交连续 dcpo $\mathcal R$，其 Scott 空间是 well-filtered 但不是 sober，尽管每个有限幂上的 Scott 拓扑是因子 Scott 拓扑的乘积。$\mathcal R$ 的理想具有唯一的有限参数描述。我们还构造了一个具有最大元的可数交连续 dcpo $\mathbb{H}$，其 Scott 空间是 coherent 且 well-filtered，但其 Scott 平方的拓扑严格细于普通乘积拓扑。这些构造否定了关于可数交连续 dcpo 的两个问题。在正面方面，我们证明了一个交连续 $L$-dcpo（即主理想是完全格的 dcpo）是 Scott sober，当且仅当其 Scott 平方具有普通乘积拓扑时。这肯定了 Jia 关于 $L$-dcpo 的 core-compactness 问题，无需任何可数性假设。最后，我们用闭 Rudin 集的有界性刻画了 $L$-dcpo 的 well-filteredness，并提出了相关的开放问题。
英文摘要
We study Scott-product compatibility and sobriety for countable meet-continuous directed-complete partial orders (dcpos). We construct a countable meet-continuous dcpo $\mathcal R$ whose Scott space is well-filtered but not sober, although the Scott topology on every finite power is the product of the factor Scott topologies. The ideals of $\mathcal R$ admit a unique finite-parameter description. We also construct a countable meet-continuous dcpo $\mathbb{H}$ with a greatest element whose Scott space is coherent and well-filtered, but whose Scott square has a topology strictly finer than the ordinary product topology. These constructions answer negatively two questions about countable meet-continuous dcpos. On the positive side, we prove that a meet-continuous $L$-dcpo, meaning a dcpo whose principal ideals are complete lattices, is Scott sober whenever its Scott square carries the ordinary product topology. This answers Jia's core-compactness question affirmatively for $L$-dcpos without any countability assumption. Finally, we characterize well-filteredness of $L$-dcpos by boundedness of closed Rudin sets and formulate related open questions.
Comments17 pages
