> I'm still not making myself clear. Your lemma says that having having arbitrary joins suffices, you do not need distributivity in order to be closed. But didn't you give an example in comment #29 with joins which is not closed? Does your example not contradict the lemma?

Okay, I understand.

This thing I tried to show in [comment #21](https://forum.azimuthproject.org/discussion/comment/20204/#Comment_20204)

> **Lemma.** If \$$\mathcal{V}\$$ is monoidal partial order which also an [upper semilattice](https://en.wikipedia.org/wiki/Semilattice#Complete_semilattices) with arbitrary joins, then it is closed.

This contradicts a counter example I invented in [comment #13](https://forum.azimuthproject.org/discussion/comment/20139/#Comment_20139) in Lecture 61, and later mentioned in comment #29.

The lemma is wrong. Can I cross it out? If I define \$$x \Rightarrow y := \bigvee\lbrace z\mid x\otimes z \leq y\rbrace\$$, then the most I can show is \$$x \otimes z \leq y \text{ implies } z \leq x \Rightarrow y\$$.

As an aside, I feel a little honored you are scrutinizing me like this. This feels a bit like practice for actual journal review! It's just a shame I'm too dumb to come up with anything original so it's sort of wasted on me.