I'm reading Jech and following his Boolean algebra models approach to it. I'm wondering if I've got the right idea here.
Let $M \models \mathrm{ZFC}$ and $B \in \mathbf{CompBoolAlg}$. We construct $M^B$ as follows:
- $M^B_0 = \emptyset$
- $M^B_{\alpha+1} = \{ \text{partial functions} \ f : M^B_\alpha \to B \}$
- $M^B_\Lambda = \bigcup_\lambda M^B_\lambda$ for limits $\Lambda$
and $M^B = \bigcup_\alpha M^B_\alpha$.
This gives an embedding of $B$ valued models $$\begin{align} \check{} : M &\to M^B \\ \emptyset &\mapsto \emptyset \\ x &\mapsto \{ \check y \mid y \in x \} \end{align}$$
Now let $G$ be a generic ultrafilter on $B$. Is it the case that $M[G] = \check M / G$? Any help appreciated!