\begin{definition}[Definition:Restriction of Ordering]
Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$ be a subset of $S$.
Then the '''restriction of $\preceq$ to $T$''', denoted $\preceq \restriction_T$, is defined as:
:${\preceq \restriction_T} := {\preceq} \cap \paren {T \times T}$
viewing ${\preceq} \subseteq S \times S$ as a relation on $S$.
Here, $\times$ denotes Cartesian product.
Thence the '''restriction of $\preceq$ to $T$''' is an instance of a restriction of a relation.
\end{definition}