关于概率幂域、RB 域和 bc 域的一个注记
A note on probabilistic powerdomains, RB-domains, and bc-domains
AI总结：
研究有限非空偏序集\(F\)的归一化概率幂域\(\Vone(F)\)成为 RB 域的条件，从禁止结构角度将该分类扩展到任意非空dcpo，给出\(\Vone(P)\)是 RB 等的等价条件。
摘要：
对于有限非空偏序集\(F\)，当且仅当\(F\)是有限根树时，归一化概率幂域\(\Vone(F)\)是 RB 域。从禁止结构观点将此分类扩展到任意非空dcpo。主理想链条件由不存在下叉表示。证明了\(\Vone(P)\)是 RB 当且仅当\(\Vone(P)\)是有尖 bc 域当且仅当\(P\)有最小元且无下叉等结论。
英文摘要：
For a finite nonempty poset \(F\), the normalized probabilistic powerdomain \(\Vone(F)\) is an RB-domain exactly when \(F\) is a finite rooted tree. We extend this classification to arbitrary nonempty dcpos from the viewpoint of forbidden structure. The principal-ideal chain condition is expressed by the absence of a lower fork, i.e. a triple \((x,y,t)\) with \(x\leq t\), \(y\leq t\), and \(x\parallel y\). A useful point is that any dcpo $P$ without lower forks is continuous. For normalized valuations the least element remains necessary, and we prove \[ \begin{aligned} \Vone(P)\text{ is RB} \Longleftrightarrow \Vone(P)\text{ is a pointed bc-domain} \Longleftrightarrow P\text{ has a least element and contains no lower fork}. \end{aligned} \] For subprobability and extended valuations, the analogous classifications hold without the pointedness assumption on \(P\).
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。
