### 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:
1. $\emptyset \not\in F$
2. If $A,B \in F$, then $A \union B \in F$
3. 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.
1. $\emptyset \not\in F$
2. If $A,B \in F$, then $A \union B \in F$
3. 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.
1. $\emptyset \not\in F$
2. If $A,B \in F$, then $A \union B \in F$
3. If $A \in F$ and $A \subseteq B \subseteq S$ then $B \in F$ ("closed going up")
4. For any $A \subset S$, either $A \in F$ or $S \setminus A \in F$