Simon said in [#41](https://forum.azimuthproject.org/discussion/comment/20242/#Comment_20242)

> It seems that Matthew is saying that it is not necessary that a monoidal preorder has the monoidal product distributing over joins for it to be closed.

No, I didn't mean that.

You may be confused with my reply to your comment in [#27](https://forum.azimuthproject.org/discussion/comment/20221/#Comment_20221). In #27 you wrote:

> In the lattice theoretic world, you impose that (A) the monoidal product distributes over joins, whereas in the category theoretic world you impose that (B) the monoidal product preserves the order. Are these equivalent?

(I have added labels (A) and (B) to your question.)

I argued in [#29](https://forum.azimuthproject.org/discussion/comment/20214/#Comment_20214) that (A) implies (B), and tried to provide a counter example to show (B) does not imply (A).

As far as I can tell, I didn't mention closedness in comment #29.