# Definition:Ideal (Order Theory)

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 $S$ if and only if $I$ satisifies the ideal axioms:
 $(1)$ $:$ $I$ is a lower section of $S$: $\ds \forall x \in I: \forall y \in S:$ $\ds y \preceq x \implies y \in I$ $(2)$ $:$ $I$ is a directed subset of $S$: $\ds \forall x, y \in I: \exists z \in I:$ $\ds x \preceq z \text{ and } y \preceq z$