 Graph model of lambda calculus

Let $P\omega =\mathcal{P}(\mathbb{N})$ ordered by subset. Encode a finite subset $u$ and output $n$ by a natural number $⟨u,n⟩$. Define $a\cdot b=\{n:\exists u\subseteq b\text{ finite},⟨u,n⟩\in a\}$ and $e(f)=\{⟨u,n⟩:n\in f(u)\}$. Every Scott continuous map satisfies $f(b)={\bigcup }_{u\subseteq b\text{ finite}}f(u)$, proving that application is the projection of a retraction. Lambda abstraction has graph $\{⟨u,n⟩:n\in [[M{]]}_{\rho [x:=u]}\}$.