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Question 1.6 - Preorders

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?

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