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Filters and Ultrafilters are useful mathematical constructs. (Do not bother to ask why these constructs have been given the name filter.)
Set S be a nonempty set. The collection of all subsets of S is called the Power Set of S. (Collection in this context means exactly the same as set. It is used to avoid excessive use of set.) For example, if S={a, b, c} then the power set of S, denoted P(S), is
Note that the empty set is considered a subset of any set and that the set itself is considered a subset.
A filter F on S is a collection of subsets of S in which two conditions hold:
It follows from these that the whole set S is a member of any filter.
The two conditions for defining a filter can be reduced to the one condition
For S={a, b, c} one filter is the collection
The power set of S is always a filter. The empty set ∅ is a special case concerning filters. Because ∅ is a subset of any subset of S and hence any subset of S is a superset of ∅ it follows that if ∅ belongs to a filter then all subsets belong and hence the filter has to be the power set of S. The power set of S might be called a trivial filter. A filter that does not contain the empty set ∅ is called a proper filter.
A filter of S such that for any subset A either A belongs to the filter or the complement of A in S belongs to the filter is called an ultrafilter. An ultrafilter U is then such that
Because an ultrafilter contains only A or its complement the trivial filter of the power set cannot be an ultrafilter. Thus an ultrafilter is a proper filter.
Consider the collection of subsets of S that contain an element a; i.e., all of the supersets of a and only the supersets of a. The example given above for a filter on the set S={a, b, c}
is such a collection. Note that the collection does not contain complement of {a}, {b, c}, or the complement of {a, b} which is {c}. This filter satisfies the conditions for being an ultrafilter. In general, the supersets of an element of S constitute an ultrafilter on S. It is called the principal ultrafilter generated by that element.
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