Let $P$ be a forcing notion. Let $B(P)$ be the boolean completion of $P$ and $i : P \rightarrow B(P)$ be the corresponding dense embedding (in $B(P)^{+}$). Let $G$ be $B(P)$-generic over $M$, the transitive ground model satisfying ZFC. 


I know that if $N$ is a transitive model of ZFC such that $M \subset N \subset M[G]$, then $N = M[D \cap G]$ for some complete subalgebra $D$ of $B(P)$. But can we say anything about $(X :=) ran(i) \cap D$ and $(Y :=) i^{-1}[D]$? Is $X$ dense in $D^{+}$? Is $Y$ the range of some complete embedding into $P$? Are there any other interesting properties about them?


Thanks in advance.



On intermediate transitive models for ZFC between M an M[G]
It depends on the particular forcing, and in general, things may
not work out so nicely.


On the one hand, it could be that $P=\mathbb{B}^+$, in which case
for any intermediate model $N$ we have $X=Y=D^+$ and so $X$ is
dense in $D^+$ and everything you want is true.


On the other hand, consider the case where $P$ is the forcing
consisting of conditions $(s,t)$, where $s,t\in 2^{{\lt}\omega}$
are finite binary sequences of the same length $|s|=|t|$. The
Boolean completion is $\mathbb{B}=\text{Add}(\omega,2)$ the
forcing to add two Cohen reals $M[c,d]$. Consider the intermediate
extension $M[c]$ to add just the first one. In this case, $D$ is
the subalgebra consisting of conditions in $\mathbb{B}$ that do
not determine any information about the second real $d$, although
they may decide information about the first real $c$. This does
not interact well with the image of $P$ inside $\mathbb{B}$,
because conditions in the range of $P$ decide an equal number of bits for both $c$ and $d$. In
particular, $D$ contains no members of $\text{ran}(i)$ except for the
trivial condition $1$. So in this case, $X$ is not dense in $D^+$,
and $Y$ has only the trivial condition.


