_Preorders._

A preorder is a category such that, for every two objects \$$a,b \$$, there is at most one morphism \$$a \to b \$$. That is, there either is or is not a morphism from \$$a \$$ to \$$b \$$, but there are never two morphisms \$$a \$$ to \$$b \$$. If there is a morphism \$$a \to b \$$, we write \$$a \leq b \$$; if there is not a morphism \$$a \to b \$$, we don’t. For example, there is a preorder \$$P \$$ whose objects are the positive integers \$$Ob(P) = \mathbb{N}_{≥1} \$$ and whose hom-sets are given by \$$P(a,b) := \\\{x \in N | x∗a = b \\\} \$$ This is a preorder because either \$$P(a,b) \$$ is empty (if \$$b \$$ is not divisible by \$$a \$$) or contains exactly one element.

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(a) What is the identity on 12?

(b) Show that if \$$x : a \to b \$$ and \$$y : b \to c \$$ are morphisms, then there is a morphism \$$y◦x \$$ to serve as their composite.

(c) Would it have worked just as well to take \$$P \$$ to have all of \$$\mathbb{N} \$$ as objects, rather than just the positive integers?