\begin{definition}[Definition:Ideal (Order Theory)]
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 $S$ {{iff}} it is both a lower set and a directed set.
That is, $I$ is an '''ideal''' {{iff}}:
:$\forall x \in I, y \in P: y \preceq x \implies y \in I$
:$\forall x, y \in I: \exists z \in I: x \preceq z$ and $y \preceq z$
Category:Definitions/Order Theory
\end{definition}