\begin{definition}[Definition:Greatest/Ordered Set/Subset]
Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$ be a subset of $S$.
An element $x \in T$ is '''the greatest element of $T$''' {{iff}}:
:$\forall y \in T: y \preceq \restriction_T x$
where $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.
\end{definition}