Michael Hong:

> So pretty much the preorder needs to have a top or bottom element in order to define a well behaved identity for the monoid.

That's not true. The real numbers form a perfectly fine monoid with addition as the monoid operation. The identity element is 0. The real numbers has no top or bottom element!

Same for the integers, or the rational numbers, or many other examples we'll be interested in.

This is just the confusion I was warning about in [Comment 34](https://forum.azimuthproject.org/discussion/comment/17987/#Comment_17987). If the monoid operation \$$\otimes\$$ in our monoidal preorder is the join, the identity has to be the bottom element... so our preorder needs to have a bottom element. If the monoid operation is the meet, the identity has to be be the top element... so our preorder needs to have a top element. But there's no reason on God's green earth why the monoid operation needs to be the join or meet! And there's no reason a monoidal preorder needs to have a top or bottom element.