Definition:Ideal (Order Theory)

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

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

Then $I$ is an ideal of $S$ it is both a lower set and a directed set.

That is, $I$ is an ideal :


 * $\forall x \in I, y \in P: y \preceq x \implies y \in I$
 * $\forall x, y \in I: \exists z \in I: x \preceq z$ and $y \preceq z$