\begin{definition}[Definition:Order Category/Definition 1]
Let $\struct {S, \preceq}$ be an ordered set.
One can interpret $\struct {S, \preceq}$ as being a category, with:
{{DefineCategory
|ob = The elements of $S$
|mor = Precisely one morphism $a \to b$ for every $a, b \in S$ with $a \preceq b$
}}
More formally, we let the morphisms be the elements of the relation ${\preceq} \subseteq S \times S$.
Thus, $a \to b$ in fact denotes the ordered pair $\tuple {a, b}$.
The category that so arises is called an '''order category'''.
\end{definition}