In my previous question Set-theoretic geology: controlled erosion? and the great answer by Jonas Reitz, I have learned a few things, starting from the awareness that I understand the fine-grain structure of Set Theoretic Geology even less than I had assumed.

That is, of course, good news: more to learn!

The second thing I have learned is:

if I want to understand more, I have to start from the STRUCTURAL STANDPOINT, ie I have to grasp, given a transitive model M (I could do away with that, by starting from V, but I prefer concrete set models), the structure of the partial order of grounds of $M$.

To be more specific, Let us begin with $GROUNDS(M)$, and take a look at its structure: it is a partial order, and looks like, assuming the Ground Axiom, that it is directed.

So, given two grounds, say $G_1$ and $G_2$, there is a third G which refines both.

Joel's Modal Logic of Forcing is $S4.2$ (please correct me if I am wrong!), which makes sense to me: this logic corresponds exactly to directed partial pre-orders.

But here is where things become quite hazy to me: what about actual meets?

QUESTIONS

1. When $GROUND(M)$ has the structure of a meet-semilattice?
2. When is $GROUND(M)$ equipped with a full lattice structure?
3. When $GROUND(M)$, assuming 1 and 2, is a complete (sups, infs) lattice?

More related questions:

$GROUNDS(M)$ is a subclass of $TM(M)$, ie the class (set) of transitive sub-models of $M$, so it makes sense to loosen the questions above by asking when the infs and sups asked for are not part of the directed order, but still exist in $TM(M)$.

Any answer to any or some of the questions is welcome.