\begin{definition}[Definition:Order Generating]
Let $\left({S, \preceq}\right)$ be a preordered set.
Let $X$ be a subset of $S$.
Then $X$ is '''order generating''' {{iff}}
:$\forall x \in S: x^\succeq \cap X$ admits an infimum and $x = \inf \left({x^\succeq \cap X}\right)$
\end{definition}