Def: T\subseteq S dick <=> \{ x^{-1}T : x\in S \} hat eDe
Prop: Aeq: (a) T dick; (b) \forall e \subseteq_ e S \exists y \in S: ( y \in \bigcap_ {x\in e} x^{-1} T ) \Longleftrightarrow \stackrel{\stackrel{\text{punktweise}}{\downarrow}}{e \cdot y} \subseteq T)
(c) ex. L \subseteq \widehat{T} (min) LID.
[proof]
a <=> c: x\in S, p\in \beta S: x^{-1} T \in p \stackrel{\text{Skript}}{\Leftrightarrow} T\in x\cdot p \Leftrightarrow x\cdot p \in \widehat{T}
also T \text{ dick} \Leftrightarrow \exists p \in \beta S: S\cdot p \subseteq \widehat{T} \stackrel{\text{stetig & closed}}{\Longleftrightarrow} \exists p \in \beta S: \beta S \cdot p \subseteq \widehat{T} \Leftrightarrow \text{Beh.}
Bem.A dick => A zentral.
[A dick => \exists L \text{ min. LID} \subseteq \widehat{A} \Rightarrow \exists \epsilon: E_ \min \cap L \subseteq \widehat{A} \Rightarrow A \in \epsilon.
Bem. (a) A \subseteq (\omega, +) dick <=> A enthaelt beliebig lange Intervalle
(b) dick \stackrel{\not \Rightarrow}{\not \Leftarrow} synd.
A := \bigcup_ {2 \in \omega} I_ {2n}, B = \bigcup_ {n\in \omega} I_ {2n+1} \rightarrow A, B dick, nicht syndetisch.
A = 2 \cdot \mathbb{N} synd, A nicht dick.
(c) A \subseteq \text{ pws} \Leftrightarrow \exists g \subseteq_ e S { t^{-1} \bigcup_ {x \in g} x^{-1}A : t \in S} \text{ eDE} \Leftrightarrow \exists g \subseteq_ e : \bigcup_ {x \in g} x^{-1}A \text{ dick.}
(d) A dick (pws, synd) => x^{-1} A dick (pws, synd)
A synd => x\cdot A synd.
Translation
Chapter 2 Thick subsets of S
Definition: T\subseteq S thick <=> \{ x^{-1}T : x\in S \} has the finite intersection property (FIP)
Proposition: TFAE
(a) T thick;
(b) \forall e \subseteq S \text{ finite } \exists y \in S: ( y \in \bigcap_ {x\in e} x^{-1} T) \Longleftrightarrow \stackrel{\stackrel{\text{pointwise}}{\downarrow}}{e \cdot y} \subseteq T)
(c) \exists L \subseteq \widehat{T} (minimal) left ideal (LID).
[proof of a <=> c]
note: x\in S, p\in \beta S: x^{-1} T \in p \Leftrightarrow T\in x\cdot p \Leftrightarrow x\cdot p \in \widehat{T}
therefore: T \text{ thick} \Leftrightarrow \exists p \in \beta S: S\cdot p \subseteq \widehat{T} \stackrel{\text{continuous & closed}}{\Longleftrightarrow} \exists p \in \beta S: \beta S \cdot p \subseteq \widehat{T} \Leftrightarrow \text{ the claim}.
RemarkA thick => A central.
proof. A central => \exists L \text{ min. LID} \subseteq \widehat{A} \Rightarrow \exists \epsilon: E_ \min \cap L \subseteq \widehat{A} \Rightarrow A \in \epsilon.
Remark.
(a) A \subseteq (\omega, +) thick <=> A contains arbitrarily long intervals
Consider a partition \omega = \bigcup_ {n \in \omega} I_ n, \vert I_ n\vert = n+1 with $0\in I_ 0 << I_ 1 << \ldots $
A := \bigcup_ {2 \in \omega} I_ {2n}, B = \bigcup_ {n\in \omega} I_ {2n+1} \rightarrow A, B thick, not syndetic.
A = 2 \cdot \mathbb{N} syndetic, A not thick.
(c) A \subseteq \text{ piecewise syndetic (pws)} \Leftrightarrow \exists g \subseteq S \text{ finite } { t^{-1} \bigcup_ {x \in g} x^{-1}A : t \in S} \text{ eDE} \Leftrightarrow \exists g \subseteq S \text{ finite} : \bigcup_ {x \in g} x^{-1}A \text{ thick.}
(d) A thick (pws, synd) => x^{-1} A thick (pws, synd)
A synd => x\cdot A synd.
Notes
Same lecture, new chapter. This is the first of two pages on the basics of thick sets.
"Thick" is an odd notion. It always seems a little made up to me, something stated after the fact (after asking "what does a set look like that covers a left ideal?"). On the other hand, for \omega, I can imagine that the notion "a set that contains arbitrarily long intervals" might actually come up independently of ultrafilters. However, I don't know the history of the notion, so I'm probably wrong here (if you know anything about this, please leave a comment).
A technical note. I realized that using the section heading "partial translation" was a bit misleading; as would be "augmented/corrected translation". In fact, I do both -- leave some things out (negligible comments etc), clear up the layout, and add corrections (e.g. \vert I_ n\vert = n+1 instead of n). So I will just call it "translation" from now on.
