Definition:Filter

Definition
Let $\struct {S, \preccurlyeq}$ be an ordered set.

A subset $\FF \subseteq S$ is called a filter of $\struct {S, \preccurlyeq}$ (or a filter on $\struct {S, \preccurlyeq}$)  $\FF$ satisfies the filter axioms:

Also see

 * Definition:Filter on Set


 * Definition:Filter (Meet Semilattice), where a filter is defined in the context of a meet semilattice


 * Meet Semilattice Filter iff Ordered Set Filter, where it is shown that in a meet semilattice the definition of (meet semilattice) filter is equivalent to the definition of (ordered set) filter.


 * Definition:Filter (Lattice), where a filter is defined in the context of a lattice


 * Equivalence of Definitions of Lattice Filter, where it is shown that in a lattice the definition of (lattice) filter is equivalent to the definition of (meet semilattice) filter.


 * Definition:Ideal (Order Theory), where the concept of ideal, the dual concept of a filter, is defined.