Filter (mathematics)

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. Special cases of filters include ultrafilters, which are filters that cannot be enlarged, and describe nonconstructive techniques in mathematical logic. Filters on sets were introduced by Henri Cartan in 1937. Nicolas Bourbaki, in their book Topologie Générale, popularized filters as an alternative to E. H. Moore and Herman L. Smith's 1922 notion of a net; order filters generalize this notion from the specific case of a power set under inclusion to arbitrary partially ordered sets. Nevertheless, the theory of power-set filters retains interest in its own right, in part for substantial applications in topology. Fix a partially ordered set (poset) P. Intuitively, a filter F is a subset of P whose members are elements large enough to satisfy some criterion. For instance, if x ∈ P, then the set of elements above x is a filter, called the principal filter at x. (If x and y are incomparable elements of P, then neither the principal filter at x nor y is contained in the other.) Similarly, a filter on a set S contains those subsets that are sufficiently large to contain some given . For example, if S is the real line and x ∈ S, then the family of sets including x in their interior is a filter, called the neighborhood filter at x. The in this case is slightly larger than x, but it still does not contain any other specific point of the line. The above considerations motivate the upward closure requirement in the definition below: "large enough" objects can always be made larger. To understand the other two conditions, reverse the roles and instead consider F as a "locating scheme" to find x. In this interpretation, one searches in some space X, and expects F to describe those subsets of X that contain the goal. The goal must be located somewhere; thus the empty set ∅ can never be in F.