Order (and small disorder)

In most textbooks, one finds that a category represents a preorder exactly when there is at most one arrow between any pair of objects. But, surprisingly, the same is told about partial orders!

In fact, both statements are correct. The key is in the word pair.

If pair means ordered pair, i.e., an element of the Cartesian product over the set of elements with itself, then the statement identifies exactly the categories representing preorders. But, when the word pair means a set composed by two elements, not necessarily distinct, then the statement identifies exactly the categories representing partial orders.

Ironically, to represent partial orders (which are “more ordered” than preorders), we must use a disordered notion of pair!