Set Theoretic Geology II: The structure of the directed partial order of grounds 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 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

*

*When $GROUNDS(M)$ has the structure of a meet-semilattice?

*When is  $GROUNDS(M)$ equipped with a full lattice structure?

*When $GROUNDS(M)$, 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.
 A: Mirco, this is also a fantastic question - the structure of the grounds as a partial order seems to be a very basic aspect of forcing that is not entirely understood.  Once again I don’t have a complete answer, but I can provide some background & a few observations.
Intersection of grounds.  As pointed out in the comments, it is not the case that the intersection of grounds is (necessarily) a ground - the intersection may fail to satisfy ZFC.  However, the intersection does contain a ground (see Directedness)
Directedness. Theorem (Usuba):  The grounds are downward-set-directed (that is, the intersection of any collection of set-many grounds contains a further ground).
This fundamental result resolved a number of open questions in Set Theoretic Geology - not least of which is the lovely fact that the Mantle (the intersection of all grounds) is always a model of ZFC.
Meets.  Since we can get below any set-indexed collection of grounds, it’s natural to ask whether there is a unique, largest such ground below such a collection.  I believe the answer is no (I think the example linked in comments by @gabe-goldberg Intersection of two generic extensions may be a good candidate for a counterexample, but I haven’t thought it through).
The Ground Axiom.  The ground axiom states “there are no grounds except V itself” -- if this is the case, then GROUND(V) is trivial.
Least element.  The Mantle (the intersection of all grounds) is, by Usuba’s result, a model of ZFC.  If GROUND(V) has a least element, it is equal to the Mantle - this will happen exactly when V is a set forcing extension of a model of the Ground Axiom.  If GROUND(V) does not have a least element, it may be the case that moving from the Mantle to V can be accomplished by class forcing.  Finally, it may be the case that V is not even a class forcing extension of the Mantle (by any class forcing definable in the Mantle) -- see http://jdh.hamkins.org/the-universe-need-not-be-a-class-forcing-extension-of-hod/ .
With regards to your final “more related questions”, I believe that if infs or sups exist then they must be ground models, by an argument similar to @asaf-karagila ‘s in the comments - we can use directedness to get below a collection of grounds, so if an inf exists it is an intermediate model of ZFC between a ground and an extension, hence it is also a ground.
None of the above resolves your three Questions - under what circumstances do we get a nice structure on GROUND(V)?  For example, if we start in a model of the Ground Axiom and carry out some forcing, what is the relationship between properties of the forcing we choose and the structure of grounds in the resulting extension?  I really like this line of thinking as an avenue for understanding structural properties of forcing.
