In my previous question [Set-theoretic geology: controlled erosion?](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 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 $GROUNDS(M)$ has the structure of a meet-semilattice?
2. When is $GROUNDS(M)$ equipped with a full lattice structure?
3. 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.



Set Theoretic Geology II: The structure of the directed partial order of grounds
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](https://mathoverflow.net/questions/297756/intersection-of-two-generic-extensions/297769) 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.


