Firstly note that, as ideals are downwards-closed, any \preceq _S-ideal subsets of \mathcal {P}^\ast _f(\mathbb {N}_\bot ) will contain all finite subsets of \mathbb {N}_\bot that contain \bot . Let us call those sets trivial. So, a set in \mathcal {P}^\ast _f(\mathbb {N}_\bot ) is non-trivial if it does not contain \bot and an ideal subset of \mathcal {P}^\ast _f(\mathbb {N}_\bot ) is non-trivial if it contains a non-trivial set. Observe that for non-trivial sets X and Y, X \preceq _S Y iff X \supseteq Y. Thus, we can identify an ideal I \in \mathsf {Id}({\mathcal {P}_f^\ast (\mathbb {N}_\bot )}) with the down-closure of the intersection of its non-trivial elements. The smaller this set is, the larger the ideal I. Hence, and confusingly, the non-trivial ideals in the powerdomain (ordered by subset inclusion) correspond to finite subsets of \mathbb {N} ordered by superset inclusion. As all trivial sets are identified, we just need to throw in the unique trivial ideal, and we can see that the Smyth powerdomain of \mathbb {N} is isomorphic to the domain of sets \{ \mathbb {N} \} \cup \mathcal {P}_f^\ast (\mathbb {N}) ordered by subset inclusion.

Note that all sets that contain \bot are considered equivalent by the Smyth construction \approx _S. Thus, this models demonic non-determinism: Of the three programs in Remark [dt-0063], the Smyth semantics considers 2 and 3 to be the same. Thus, the possibility of divergence \bot is no different from inevitable divergence.