Category:Order Categories

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This category contains results about Order Categories.
Definitions specific to this category can be found in Definitions/Order Categories.

Let $\left({S, \preceq}\right)$ be an ordered set.

One can interpret $\left({S, \preceq}\right)$ as being a category, with:

Objects:         The elements of $S$
Morphisms: 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 $\left({a, b}\right)$.

The category that so arises is called an order category.