Filter (mathematics) - Filter On A Set

Filter On A Set

A special case of a filter is a filter defined on a set. Given a set S, a partial ordering ⊆ can be defined on the powerset P(S) by subset inclusion, turning (P(S),⊆) into a lattice. Define a filter F on S as a subset of P(S) with the following properties:

1. S is in F. (F is non-empty)
2. The empty set is not in F. (F is proper)
3. If A and B are in F, then so is their intersection. (F is closed under finite meets)
4. If A is in F and A is a subset of B, then B is in F, for all subsets B of S. (F is an upper set)

The first three properties imply that a filter on a set has the finite intersection property. Note that with this definition, a filter on a set is indeed a filter; in fact, it is a proper filter. Because of this, sometimes this is called a proper filter on a set; however, as long as the set context is clear, the shorter name is sufficient.

A filter base (or filter basis) is a subset B of P(S) with the following properties:

1. The intersection of any two sets of B contains a set of B
2. B is non-empty and the empty set is not in B

Given a filter base B, one may obtain a (proper) filter by including all sets of P(S) which contain a set of B. The resulting filter is said to be generated by or spanned by filter base B. Every filter is also a filter base, so the process of passing from filter base to filter may be viewed as a sort of completion.

If B and C are two filter bases on S, one says C is finer than B (or that C is a refinement of B) if for each B₀ ∈ B, there is a C₀ ∈ C such that C₀ ⊆ B₀.

• For filter bases B and C, if B is finer than C and C is finer than B, then B and C are said to be equivalent filter bases. Two filter bases are equivalent if and only if the filters they generate are equal.
• For filter bases A, B, and C, if A is finer than B and B is finer than C then A is finer than C. Thus the refinement relation is a preorder on the set of filter bases, and the passage from filter base to filter is an instance of passing from a preordering to the associated partial ordering.

Given a subset T of P(S) we can ask whether there exists a smallest filter F containing T. Such a filter exists if and only if the finite intersection of subsets of T is non-empty. We call T a subbase of F and say F is generated by T. F can be constructed by taking all finite intersections of T which is then filter base for F.