Characterizing finite posets whose probabilistic powerdomain are RB-domains

We classify the finite posets whose probabilistic powerdomain is an RB-domain.

For a finite nonempty poset \(P\), let \(\Vone(P)\) be the probability powerdomain of $P$, which is the probability simplex ordered by the stochastic order.

We prove that \(\Vone(P)\) is an RB-domain if and only if \(P\) has a least element and the undirected Hasse graph of \(P\) is a tree.

Consequently, the probabilistic powerdomain does not preserve RB-domains; the four-point diamond gives a finite counterexample.

The proof separates two obstructions.

First, if \(P\) has no least element, then the face of probability measures supported on the minimal points must be fixed pointwise by every deflation below the identity.

Secondly, once a least element exists, the Hasse graph is connected, and a cycle in it makes the local stochastic cone non-simplicial.

A Euclidean finite-step cone argument then rules out the finite-valued monotone approximations supplied by the RB property.