\begin{definition}[Definition:Principal Ideal of Ordered Set/Definition 2]
Let $\left({S, \preceq}\right)$ be a preordered set.
Let $I$ be an ideal in $S$.
Then $I$ is '''principal (ideal)''' {{iff}}
:$\exists x \in S: I = x^\preceq$
where $x^\preceq$ denotes the lower closure of $x$.
\end{definition}