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Partially-Ordered Sets ( Posets ) and Hom-sets

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Show that a poset (partially-ordered set) is the same thing as a category in which all Hom-sets have at most one element.

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This is only true if we consider isomorphic objects in a category to be the same.
If we have such a category C then define a partial order on the set of objects
O : a <= b iff there is a morphism from a to b.
If a <= b and b <= c, then we can compose the morphisms to ...

Solution Summary

Partially-Ordered Sets ( Posets ) and Hom-sets are investigated. The solution is detailed and well presented.