thumb|The power set lattice of the set , with upper set colored dark green. This upper set is a , and even a . It is not an , because including also the light green elements extends it to the larger nontrivial filter . Since the latter cannot be extended further, is an ultrafilter. In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal.
thumb|The power set lattice of the set , with upper set colored dark green. This upper set is a , and even a . It is not an , because including also the light green elements extends it to the larger nontrivial filter . Since the latter cannot be extended further, is an ultrafilter. In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal.