Two forcing notions $\Bbb P$ and $\Bbb Q$ could be defined to be forcing equivalent if the associated complete Boolean algebras are isomorphic (so, the CBA's formed by considering the regular opens of the separative quotients of the forcing notions).
I believe that an equivalent definition can be given by the existence of a $\Bbb Q$-name $\dot{\cal G}$ and a $\Bbb P$-name $\dot{\cal H}$ such that
- $\Bbb Q$ forces that $\dot{\cal G}$ names a $\Bbb P$-generic filter,
- $\Bbb P$ forces that $\dot{\cal H}$ names a $\Bbb Q$-generic filter,
- If $G\subset\Bbb P$ is $\Bbb P$-generic, then $G=\dot{\cal G}{}^{\cal H}$, where $\mathcal H=\dot{\mathcal H}{}^{G}$,
- If $H\subset\Bbb Q$ is $\Bbb Q$-generic, then $H=\dot{\cal H}{}^{\cal G}$, where $\mathcal G=\dot{\mathcal G}{}^{H}$.
This is for instance Shelah's definition of forcing equivalent (Def. 5.2 in Proper and Improper Forcing). One can use the above names $\dot{\cal G}$ and $\dot{\cal H}$ to define an isomorphism between the complete Boolean algebras, by $\iota:\Bbb P\to\Bbb Q$ defined as $p\mapsto ||p\in\dot{\cal G}||$.
Let us call $\Bbb P$ and $\Bbb Q$ almost equivalent if there exist a $\Bbb Q$-name $\dot{\cal G}$ and a $\Bbb P$-name $\dot{\cal H}$ such that
- $\Bbb Q$ forces that $\dot{\cal G}$ names a $\Bbb P$-generic filter,
- $\Bbb P$ forces that $\dot{\cal H}$ names a $\Bbb Q$-generic filter,
- If $\dot G$ is the canonical $\Bbb P$-name for the $\Bbb P$-generic filter, then $\Bbb P$ forces that $\mathbf V[\dot G]=\mathbf V[\dot{\mathcal H}]$
- If $\dot H$ is the canonical $\Bbb Q$-name for the $\Bbb Q$-generic filter, then $\Bbb Q$ forces that $\mathbf V[\dot{\mathcal G}]=\mathbf V[\dot H]$
Here $\dot G$ and $\dot H$ are respectively the cannonical $\Bbb P$ and $\Bbb Q$-names for their generic filters. It is clear that forcing equivalent implies almost equivalent.
The reverse does not hold. For instance, we can consider the forcing notion $\Bbb P$ to be Cohen forcing, and $\Bbb Q=(\omega_1\times\Bbb P)\cup\{1\}$, where $1$ is the top element and $(\alpha',p')\leq (\alpha,p)$ iff $\alpha=\alpha'$ and $p'\leq p$ (so $\Bbb Q$ is the lottery sum of $\aleph_1$-many copies of Cohen forcing).
Since $\Bbb P$ is c.c.c. and $\Bbb Q$ is not, they cannot be forcing equivalent. On the other hand, the names $\dot{\cal G}=\{(p,(\alpha,p))\mid (\alpha,p)\in\Bbb Q\}$ and $\dot{\cal H}=\{((0,p),p)\mid p\in\Bbb P\}$ show that $\Bbb P$ and $\Bbb Q$ are almost equivalent.
My question is if exceptions such as above are the only exceptions.
More precisely, if $\Bbb P$ and $\Bbb Q$ are almost equivalent, does this imply that $\Bbb P$ and $\Bbb Q$ are forcing equivalent to lottery sums of forcings from $\{\Bbb P_i\mid i\in I\}$ and $\{\Bbb Q_j\mid j\in J\}$ respectively, where each $\Bbb P_i$ is forcing equivalent to some $\Bbb Q_j$ , and vice versa?
