I wrote:

> Another important thing to do is this: whenever someone defines some sort of gadget, like a "preorder", and then defines a nice special case with extra features, like a "poset", you need to look for examples of posets that aren't preorders.

As Jonathan pointed out, that was wrong: there aren't any posets that aren't preorders. I meant to say this:

> Another important thing to do is this: whenever someone defines some sort of gadget, like a "preorder", and then defines a nice special case with extra features, like a "poset", you need to look for examples of preorders that aren't posets.