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Filters and Ultrafilters

Filter#

Definition. A filter on a nonempty set \(S\) is a collection \(F\) of subsets of \(S\) s.t.

  1. \(S\in F\) and \(\emptyset \not\in F\) (no empty set, with the whole set),
  2. if \(X\in F\) and \(Y\in F\) then \(X\cap Y \in F\) (closed under intersection),
  3. if \(X,Y\subset S\), \(X\in F\) and \(X\subset Y\) then \(Y\in F\) (closed upwardly).

Definition. An ideal on a nonempty set \(S\)is a collection \(I\) of subsets of \(S\) s.t.

  1. \(S\not\in I\) and \(\emptyset \in I\) (with empty set, without the whole set).
  2. if \(X\in I\) and \(Y\in I\) then \(X\cup Y \in I\), (closed under union).
  3. if \(X,Y\subset S\), \(X\in I\) and \(Y\subset X\), then \(Y\in I\) (closed downwardly).

We see that if \(F\) is a filter on \(S\) then \(I = \{S-X:X\in F\} = \mathcal{P}(S) - F\) is an ideal on \(S\) and vice versa. \(F\) and \(I\) are dual to each other when they are in this relation.