2026/07/05 by Yuxu Chen · 1 voice
Mathematics · #math.GN #math.PR
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 \(x≤ t\), \(y≤ t\), and \(x∥ y\). A useful point is that any dcpo P without lower forks is continuous. For normalized valuations the least element remains necessary, and we prove \beginaligned \Vone(P) is RB \Longleftrightarrow \Vone(P) is a pointed bc-domain \Longleftrightarrow P has a least element and contains no lower fork. \endaligned For subprobability and extended valuations, the analogous classifications hold without the pointedness assumption on \(P\).