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.