Hindman’s Theorem, partial semigroups and some of my most lacking intuitions (part 6)15 Sep 2011
I know, I know, it’s part 6 already. Last time I finally formulated the Central Sets Theorem. This part will just be a small bridge. But at least you’ll finally know why on earth I am writing this, i.e., what big open question I’m actually trying to build some intuitions for.
Central Sets and ultrafilters
Before we can move on in this series I should tell you a little bit about the relationship between ultrafilters and central sets – and finally give you something close to a definition.
If you already believed me that idempotent ultrafilters exist, you might also believe me that there’s a special kind of idempotent ultrafilters, they are called minimal idempotents. The reason for the name “minimal” is a partial order on idempotents coming from ring theory. Yet again, we’ll skip the definition – I would first have to convince you that is actually a semigroup and, again, I don’t want to go there right now.
In any case, there are those idempotent ultrafilters which are minimal idempotents. As I mentioned before, Hillel Furstenberg had introduced the notion of central set via recurrence in dynamical systems. It took a couple of years until in 1990 Vitaly Bergelson and Neil Hindman helped establish that centrality can be framed extremely well in terms of ultrafilters.
Theorem (Bergelson, Hindman) is central iff is in a minimal idempotent ultrafilter.
This was, I think, quite a crazy and beautiful result at the time and its simplicity is still stunning (although, arguably, you won’t agree since I didn’t give you the complicated definition in terms of dynamics).
This leaves us with the following situation:
- central sets and minimal idempotents correspond neatly and
- we have a Ramsey-type theorem (the Central Sets Theorem) that central sets fulfil
This is not as optimal as it could be! If you remember, we were even better off with Hindman’s Theorem:
A set is contained in an idempotent ultrafilter if and only if it “satisfies” Hindman’s Theorem (i.e., includes an FS-set).
In fact, a set is an FS-set if and only if it is contained in an idempotent ultrafilter.
Hindman’s Theorem and idempotent ultrafilters corresponded directly. The unfortunate situation for central sets is that (as far as I know) no version of the Central Sets Theorem is able to accomplish a correspondence of the form
Wanted “A set is included in a minimal idempotent ultrafilter (i.e., is a central set) if and only if it fulfills the following Central Sets Theorem”
This would be the dream, I think. And this is, in a manner of speaking, the whole point of this series. How can we get to this? As you may have guessed, I believe partial semigroups can help shed light on this. So I will return to them in the continuation.