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
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.
are filters on . Then if there exists a map such that (the filter generated by) .
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
on to be the filter generated by
as well as
increasing with -- where is the natural enumeration of .
Then it's easy to formulate flatness:
Flat ultrafilter An ultrafilter
(on some countable set) is flat if .
Why is this true? The work with the original definition (simplifying the flatness scale) gave us something along the lines of "
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
(where is the Fréchet filter on ), no flat ultrafilter can be a P-point.
and so this must hold for every flat ultrafilter.
- In particular, this tells that there are infinitely many skies in the ultrapower of
of a flat ultrafilter.
And so forth. But that's for another post if I get around to it.