# Preprint 'On strongly summable ultrafilters'

I have just uploaded a preprint titled On strongly summable ultrafilters to the arXiv. Let me give a short account of what it’s about.

In the preprint I extend a theorem orginally due to Neil Hindman and Dona Strauss. The theorem shows that certain ultrafilters can only be written as sums in the (so-called) trivial fashion. On the one hand, this property is quite unique and I find it algebraically fascinating. On the other, the existence of the ultrafilters in question is independent of ZFC, so set theoretic interests are immediate. Let’s start with the ultrafilters.

## Strongly summable ultrafilters

Among the idempotent ultrafilters on *FS-set* is.

For a sequence

Then **strongly summable** if it has a base of FS-sets.

By the Galvin-Glazer Theorem, any set in an idempotent ultrafilter contains an FS-set. The difference to arbitrary idempotent ultrafilters is that strongly summables have such a set in the ultrafilter itself! So in measure theoretic terms, not only does every measure 1 set contain an FS-set, but one of measure 1. As a comparison, the very important *minimal* idempotents can never be strongly summable; in fact strongly summable ultrafilters are at the other end of the spectrum — they are what is called strongly right maximal idempotents.

In other words, what selective ultrafilters are for Ramsey’s Theorem, strongly summable ultrafilters are for Hindman’s Theorem. Unlike idempotent ultrafilters strongly summable ultrafilters might not exist, since their existence implies the existence of P-points in

## Union Ultrafilters

There is an equivalent notion, union ultrafilters, see here. In the preprint, as always, it’s hard to speak about the one without the other.

Denote the non-empty, finite subsets of

The **FU-set (generated by )** is the set of all finite unions, i.e.,

A **union ultrafilter** on

### Trivial sums

As mentioned, the main result in the preprint is about writing ultrafilters as sums. Any idempotent ultrafilter can be written as a sum in many ways, most trivially as

If

This is simply due to the fact that the integers still commute with everything in

Let’s say that an (idempotent) ultrafilter **the trivial sums property** if this is the only way to write is as a sum, i.e.,

This property fascinates me for many reasons. One easy but equally fascinating consequence is that the maximal group of such an idempotent

In contrast, minimal idempotent ultrafilters have huge maximal groups. In fact, those always include a copy of the free group on

### Using union ultrafilters

I gave a simplified version of the main theorem here. For this consider the very natural map

**Theorem** If

In fact, the theorem in the preprint does a bit more. You see, such an **disjoint binary support** — simply because that’s what happens to pairwise disjoint sets under

Now if you take any other divisible sequence

Then we can formulate the full theorem from the preprint.

**Theorem** If

## About the result

The original theorem by Neil Hindman and Dona Strauss [doi] (also to be found in their book, chapter 12) showed that the trivial sums property holds for strongly summable ultrafilters with special properties. These included the restriction that the strongly summable ultrafilters must have a base of FS-sets coming from divisible sequences. In particular, these strongly summables will contain the FS-set for a sequence with disjoint support for a divisible sequence (just for the sequence itself), i.e., the result in the preprint entails the original result.

To see that it is a little bit more general, you need to know another concept, *additive isomorphism*. To keep it short, let’s just state it for strongly summable ultrafilters.

Two strongly summable ultrafilters **additively isomorphic** if there are

In their original paper [doi], Neil Hindman and Dona Strauss give a beautiful example of a strongly summable ultrafilter that does not concentrate on FS-sets from divisible sequences (this is done using unordered union ultrafilters) — and no additively isomorphic copy does. However, every strongly summable is additively isomorphic to one as in the result from the preprint. Hence the result is a little bit stronger.

(Un)fortunately, the trivial sums property is not transferred via an additive isomorphism, so one open question is whether all strongly summable ultrafilters have it. Although I included some evidence in the preprint towards believing that all strongly summables have it, I hope that it is not the case, simply because I hope that a counterexample would be a new kind of strongly summable ultrafilter. Of course, the big question is whether any other idempotents have this property! In particular, if there might exist such examples in ZFC.

## About the proof

The proof in the preprint follows the same strategy as the proof of the original theorem by Neil Hindman and Dona Strauss. So the main task (or rather the initial observation that got me thinking about their proof again) was to overcome some of the restrictions on the strongly summable ultrafilters. Besides the ones already mentioned, there was another technical condition in the original result, a strange combinatorial property which I prefer to think about in terms of union ultrafilters.

An ultrafilter **special** if for every infinite

In other words,

**Theorem** Every union ultrafilter is special.

The proof is not very complicated, it involves a standard parity argument that often appears when arguing with union ultrafilters (and then you just have to look carefully to see that you’re done).

But now this strange and difficult extra condition from the original result by Neil Hindman and Dona Strauss is suddenly available for free.

The final result is derived in essentially two steps. There is a bit of technicality involved, but I think nothing really complicated (to read).

Fixing some divisible sequence **-support**. Then it roughly proceeds as follows.

- Show trivial sums for the ultrafilters that contain the set of multiples for each member of
. - This essentially reflects from a simple observation: if two elements with
-support sufficiently far apart have their sum in an FS-set with disjoint -support, then they must both be in the FS-set already.

- This essentially reflects from a simple observation: if two elements with
- Finally show that any sum equal to the strongly summable is essentially only possible by integer translates of ultrafilters that contain these sets of multiples.

In this tricky final part the specialness condition comes into play. It is used to create an FS-set that has a lot of ‘holes’ in the

Well, of course that’s only a vague description of the proof, but the preprint should have enough details.

## Questions

The most important questions I have already asked. Does every strongly summable have the trivial sums property and are their other ultrafilters with it, maybe even in ZFC? A result by Neil Hindman and Dona Strauss shows that all strongly summable ultrafilters have a maximal group isomorphic to the integers. It is still unkown if any other ultrafilters carry this property. So I think there is still a lot to be learned about this property.