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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) from Peter Krautzberger on Vimeo.

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)