Peter Krautzberger · on the web

Idempotent Ultrafilters, an introduction (Michigan Logic Seminar Nov 09, 2011)

Because of a power outage at the department my talk announced for October 29th was postponed by a week.

[Idempotent Ultrafilters: An Introduction (University of Michigan Logic Seminar 2011-11-08)](http://vimeo.com/32109926) from [Peter Krautzberger](http://vimeo.com/pkrautzberger) on [Vimeo](http://vimeo.com).

Here are transcripts of my notes (as well as the originals at the end).

Hindman's Theorem

Hindman's Theorem If \(\mathbb{N} = A_ 0 \dot\cup A_ 1\), then \(\exists j \exists (x_ i)_ {i\in \omega}\) such that \[FS(x_ i) \subseteq A_ j.\]

Imagine you'd like to prove this with an ultrafilter: \[p \in \beta \mathbb{N} \Rightarrow \exists j A_ j=:A \in p.\]

What do we need? We will build \((x_ i)\) inductively!

Pick \(x_ 0 \in A\) -- we can't really choose better than that (except maybe by shrinking the set first).

If we're looking for our result, we need

In other words, we need \(x_ 0 \in \\{ x: -x+A \in p \\}\) to begin with, i.e., \(\\{ x: -x+A \in p \\} \in p\) -- for any \(A\in p\)!

Galvin in 1970: \(p \in \beta \mathbb{N}\) is almost left-translation invariant iff \(\forall A\in p: \\{ x : -x +A\in p\\} \in p\).

Is this enough?

But to continue the process, we need more!

We need \(x_ 2\) such that:

What does this mean? \(-(x_ 0 +x_ 1) + A = -x_ 1 + (-x_ 0 +A)\) by associativity.

Ah! But we have seen this before!

We needed \(x_ 1 \in \\{ x: -x + (-x_ 0 +A) \in p\\}\), so we needed \(\\{ x: -x + (-x_ 0 +A) \in p\\}\in p\)!

But that's ok!! \(-x_ 0 + A \in p\) & \(\forall B\in p: \\{x : -x+B \in p \\} \in p\)!

How do we get to the end?

Question: Do "almost left-translation invariant" ultrafilters exist?

Glazer, ~1975: Yes of course! These are the idempotent ultrafilters! We know these exist since Ellis 1958.

What does this mean?

Now remember: what did Galvin need?

\[(\forall A \in p) \{ x: -x+A \in p\}\in p\]

I.e., \(A\in p \Rightarrow A^{-p} \in p \Rightarrow A \in p+p\), so \(p \subseteq p+p\)
I.e. \(p+p = p\) (since ufs)

Ellis 1958 \((X,\cdot)\) compact, Hausdorff, right-topological semigroup \(\Rightarrow \exists x\in X: x\cdot x =x\).

Proof.

Part 1 Idempotent Ultrafilters
Part 1 Idempotent Ultrafilters, an introduction (Michigan Logic Seminar Nov 09, 2011)
Part 2 Idempotent Ultrafilters
Part 2 Idempotent Ultrafilters, an introduction (Michigan Logic Seminar Nov 09, 2011)
Part 3 Idempotent Ultrafilters
Part 3 Idempotent Ultrafilters, an introduction (Michigan Logic Seminar Nov 09, 2011)
Part 4 Idempotent Ultrafilters
Part 4 Idempotent Ultrafilters, an introduction (Michigan Logic Seminar Nov 09, 2011)