Definition:Order Category/Definition 1

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Definition

Let $\struct {S, \preceq}$ be an ordered set.


One can interpret $\struct {S, \preceq}$ 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 $\tuple {a, b}$.


The category that so arises is called an order category.


Also see

  • Results about order categories can be found here.


Sources