Filter (mathematics) - General Definition

General Definition

A non-empty subset F of a partially ordered set (P,≤) is a filter if the following conditions hold:

1. For every x, y in F, there is some element z in F such that z ≤ x and z ≤ y. (F is a filter base)
2. For every x in F and y in P, x ≤ y implies that y is in F. (F is an upper set)
3. A filter is proper if it is not equal to the whole set P. This is sometimes omitted from the definition of a filter.

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.

The smallest filter that contains a given element p is a principal filter and p is a principal element in this situation. The principal filter for p is just given by the set {x in P | p ≤ x} and is denoted by prefixing p with an upward arrow: .

The dual notion of a filter, i.e. the concept obtained by reversing all ≤ and exchanging ∧ with ∨, is ideal. Because of this duality, the discussion of filters usually boils down to the discussion of ideals. Hence, most additional information on this topic (including the definition of maximal filters and prime filters) is to be found in the article on ideals. There is a separate article on ultrafilters.