Filters - Model Theory - Cambridge III notes

4 Filters

Definition 4.1 (Filter). Let $J$ be a set. A filter $F$ on $J$ is a non-empty subset of $P (J)$ such that:

$\emptyset \notin F$ .
$\forall A, B \in F, A \cap B \in F$ (“closed under finite intersections”).
$\forall A \in F$ , if $A \subseteq B \subseteq J$ , then $B \in F$ (“closed under super set”).

Example 4.2.

For $J$ infinite,
$F : = {A \subseteq J : J ∖ A is finite}$

is a filter.
For $J$ non-empty, and any $i \in J$ ,

$F = {A \subseteq J : i \in J}$

is a filter.

Definition 4.3 (Ultrafilter). Let $J$ be an infinite set and $F$ a filter on $J$ . We say $F$ is an ultrafilter if every filter $G$ on $J$ satisfying $F \subseteq G$ also satisfies $G = F$ .

Proposition 4.4. Assuming that:

$J$ a set
$F$ a filter on $J$

Then

F

is an ultrafilter if and only if for every

A \subseteq J

either

A \in F

J ∖ A \in F

Proof. Example Sheet 1. □

Proposition 4.5. Assuming that:

$J$ is a set
$F$ a filter on $J$

Then there is an ultrafilter

U

such that

F \subseteq U

Proof (sketch). Let $X$ be the set of filters extending $F$ , and partially order it by inclusion. Note every chain has an upper bound (take the union), so by Zorn’s lemma we have a maximal element, which is a ultrafilter (by definition). □