Matthew, I think crossing it out (but not deleting) with a note directing people to your comment 47 might be sensible. Can you prove the following? That is to say, is the following a theorem?

> If \\(\mathcal{V}\\) is a monoidal partial order which has all joins and for which the monoidal product distributes over joins, then \\(\mathcal{V}\\) is closed.

Or maybe even the following is true.

> Suppose \\(\mathcal{V}\\) is a monoidal partial order which has all joins, then \\(\mathcal{V}\\) is closed if and only if the monoidal product distributes over joins.

> If \\(\mathcal{V}\\) is a monoidal partial order which has all joins and for which the monoidal product distributes over joins, then \\(\mathcal{V}\\) is closed.

Or maybe even the following is true.

> Suppose \\(\mathcal{V}\\) is a monoidal partial order which has all joins, then \\(\mathcal{V}\\) is closed if and only if the monoidal product distributes over joins.