Example. Hoare powerdomain of \mathbb {N}_\bot [dt-0062]

Let us find the Hoare powerdomain of \mathbb {N}_\bot . Note that \mathbb {N}_\bot is a flat domain so all elements are compact, i.e. \mathsf {K}(\mathbb {N}_\bot ) = \mathbb {N}_\bot . Let us find examine the \preceq _H-ideal subsets of \mathcal {P}_f^\ast (\mathbb {N}_\bot ). If v and u both contain \bot , then v \preceq _H u iff v \subseteq u. As ideal subsets are downwards closed, we can identify an ideal I \in \mathsf {Id}({\mathcal {P}_f^\ast (\mathbb {N}_\bot )}) with its union \bigcup I. The ideals of \mathcal {P}^\ast _f(\mathbb {N}_\bot ) are then isomorphic to domain \mathcal {P}(\mathbb {N}) of all subsets of \mathbb {N} under subset inclusion.