A note on probabilistic powerdomains, RB-domains, and bc-domains

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 $P$.