In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in topology where they originate. The dual notion of a filter is an ideal.
Filters were introduced by Henri Cartan in 1937 and subsequently used by Bourbaki in their book Topologie Générale as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. L. Smith.
A non-empty subset F of a partially ordered set (P,≤) is a filter if the following conditions hold:
While the above definition is the most general way to define a filter for arbitrary posets, it was originally defined for lattices only. In this case, the above definition can be characterized by the following equivalent statement: A non-empty subset F of a lattice (P,≤) is a filter, if and only if it is an upper set that is closed under finite meets (infima), i.e., for all x, y in F, we find that x ∧ y is also in F.
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