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


 * Meet Semilattice Filter iff Ordered Set Filter
 * Equivalence of Definitions of Lattice Filter


 * Definition:Ideal (Order Theory)