Definition:Ideal (Order Theory)

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

Let $I \subseteq S$ be a non-empty subset of $S$.

Then $I$ is an ideal of $S$ $I$ satisifies the ideal axioms:

Also see

 * Definition:Join Semilattice Ideal
 * Join Semilattice Ideal iff Ordered Set Ideal
 * Definition:Lattice Ideal
 * Equivalence of Definitions of Lattice Ideal
 * Definition:Filter