In my previous question https://mathoverflow.net/questions/370411/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.