Bem. (a) dick <=> enthaelt beliebig lange Intervalle

(b) dick synd.

[proof]

Betrachte , ,

1) dick, nicht syndetisch.

2) synd, nicht dick.

(c)

(d) dick (pws, synd) => dick (pws, synd)

synd => synd.

Translation

Chapter 2 Thick subsets of

Definition: thick <=> has the finite intersection property (FIP)

Proposition: TFAE

(a) thick;

(b)

(c) (minimal) left ideal (LID).

[proof of a <=> c]

note:

therefore: .

Remark thick => central.

proof. central => .

Remark.

(a) thick <=> contains arbitrarily long intervals

(b) thick syndetic, thick syndetic.

[proof]

Consider a partition , with

1) thick, not syndetic.

2) syndetic, not thick.

(c)

(d) thick (pws, synd) => thick (pws, synd)

synd => 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 , 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. instead of ). So I will just call it “translation” from now on.