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Flat ultrafilters follow-up

There are a couple of reasons to write a quick follow-up post to the notes for my talk on flat ultrafilters.

Small addenda included

I updated the notes a little. Nothing much, some bad pictures and some comments.

Video recording is online

I uploaded my talk to vimeo – find it embedded below or look it up on vimeo.

Flat Ultrafilters (2011/09/21 University of Michigan Logic Seminar) from Peter Krautzberger on Vimeo.

Flat ultrafilters and the Katetov order

There’s one thing worth stressing though. Remember when I wrote that it’s a little cheating to say that a flat ultrafilter is simply an ultrafilter on $[\omega]^{<\omega}$ that extends a certain filter?

Instead of cheating, I should have talked about the Katetov order. Usually, the Katetov order is defined for ideals, but let me give the formulation for filters.

Definition If $F,G$ are filters on $X$. Then $F\leq_K G$ if there exists a map $f:X \to X$ such that (the filter generated by) $f[G] \supseteq F$.

For ultrafilters, this is of course the same as the Rudin-Keisler order. But for filters it is more general. If you confuse it with near coherence of filters, remember that near coherence asks only for coherence (the union generates a filter), not inclusion of the image (also, near coherence allows only finite-to-one functions).

In any case, we can now formulate flatness very easily. The key is the important filter that Farah, Philips and Steprans had identified in their construction for flat ultrafilters.

The flatness filter Let’s define the flatness filter $F$ on $[\omega]^{<\omega}$ to be the filter generated by

for all $n \in \omega$ as well as

for $f\in \omega^\omega$ increasing with $f(0)>0$ – where $(s_i)_{i \in |s|}$ is the natural enumeration of $s$.

Then it’s easy to formulate flatness:

Flat ultrafilter An ultrafilter $p$ (on some countable set) is flat if $p\geq_K F$.

Why is this true? The work with the original definition (simplifying the flatness scale) gave us something along the lines of “$p$ is isomorphic to an ultrafilter that extends $F$” – but together with the observation that Rudin-Keisler successors of flat ultrafilters are flat, we get the above.

Now this reformulation is not just the result of trying to understand (and simplify) the original definition. It also makes a few observations absolutely immediate. Let me jot down a few without proof

  • Since $F \geq_K Fr \otimes Fr$ (where $Fr$ is the Fréchet filter on $\omega$), no flat ultrafilter can be a P-point.
  • Similarly, $F\geq \otimes_n Fr$ and so this must hold for every flat ultrafilter.
  • In particular, this tells that there are infinitely many skies in the ultrapower of $\omega$ of a flat ultrafilter.

And so forth. But that’s for another post if I get around to it.