@Jonathan – let's take a look at your minimal-poset-that-isn't-a-semilattice above.

suppose it has a monoidal structure that respects the order, with \\(d\\) as the unit \\(I\\).

then \\(a\otimes b \ge a\otimes d = a\otimes I = a\\)

and \\(a\otimes b \ge d\otimes b = I\otimes b = b\\)

this is a contradiction, since \\(a\\) and \\(b\\) do not have a common upper bound

similar arguments show the unit can't be \\(a\\), \\(b\\) or \\(c\\) either.

so there is no monoidal structure that respects the ordering – QED

suppose it has a monoidal structure that respects the order, with \\(d\\) as the unit \\(I\\).

then \\(a\otimes b \ge a\otimes d = a\otimes I = a\\)

and \\(a\otimes b \ge d\otimes b = I\otimes b = b\\)

this is a contradiction, since \\(a\\) and \\(b\\) do not have a common upper bound

similar arguments show the unit can't be \\(a\\), \\(b\\) or \\(c\\) either.

so there is no monoidal structure that respects the ordering – QED