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:Ideal (Join Semilattice), where an ideal is defined in the context of a join semilattice


 * Join Semilattice Ideal iff Ordered Set Ideal, where it is shown that in a join semilattice the definition of (join semilattice) ideal is equivalent to the definition of (ordered set) ideal.


 * Definition:Ideal (Lattice), where an ideal is defined in the context of a lattice


 * Equivalence of Definitions of Lattice Ideal, where it is shown that in a lattice the definition of (lattice) ideal is equivalent to the definition of (join semilattice) ideal.


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