Anindya wrote:

> Intuitively I think the best way of looking at this is that the partial order is too irregular to admit a monoidal structure: \\(a\\) and \\(b\\) don't have a join, \\(c\\) and \\(d\\) don't have a meet, there is no top element, nor is there a bottom one... basically \\(\leq\\) is too badly behaved for any \\(\otimes\\) operation to respect it.

There's probably something to this, but let's not fool anyone into thinking every monoidal preorder needs to have \\(\otimes\\) be the join or meet! Think of six counterexamples.