>take a set X and decree that everything in this set is less than or equal to everything else.

I think I will always remember this example John. Thanks. Your example made me realize why I have trouble sometimes with the most basic things. I keep thinking in terms of specific examples while trying to keep tabs on the abstraction which causes problems if the example in my head isn't abstract enough to capture the greater picture.

>So in a typical preorder, you should never expect to reason from inequalities to equations. For that, you want a poset.

This opened up a lot for me. Thanks.