### Filter

A structure of a collection of subsets. It ensures that
if a set is in the collection, then all sets which contain
it are also in the collection.
Let $S$ be a nonempty set. A filter on $S$ is a nonempty
collection $F$ of subsets of $S$ having the following
properties:
- $\emptyset \not\in F$
- If $A,B \in F$, then $A \union B \in F$
- If $A \in F$ and $A \subseteq B \subseteq S$ then $B \in F$ ("closed going up")

Some definitions allow $\emptyset \in F$. In these contexts,
if a filter does not contain $\emptyset$, then it is a
proper filter.
### Ideal

The "opposite" of a filter.
- $\emptyset \not\in F$
- If $A,B \in F$, then $A \union B \in F$
- If $A \in F$ and $B \subseteq A \subseteq S$ then $B \in F$ ("closed going down")

I believe that Matroids are Ideals.
### UltraFilter

A variation of a filter in which the collection ensures that
either a subset or its complement is in the collection ---
thus it is the maximal proper filter --- it cannot be extended
to a larger proper filter.
- $\emptyset \not\in F$
- If $A,B \in F$, then $A \union B \in F$
- If $A \in F$ and $A \subseteq B \subseteq S$ then $B \in F$ ("closed going up")
- For any $A \subset S$, either $A \in F$ or $S \setminus A \in F$