@John – point taken re "let's not fool anyone into thinking every monoidal preorder needs to have \\(\otimes\\) be the join or meet" – one of the difficulties I have with monoidal categories is in grasping how we only care about the formal combinatory properties of the monoidal product, and not what it "actually is". So I tend to "reach" for familiar products, eg meet or cartesian or tensor, which can be misleading.

As for some counterexamples...

• Take any monoid (with at least 2 elements) – put the discrete partial order on the underlying set – we now have a (trivial) monoidal preorder where \\(\otimes\\) is not join or meet (since these operations don't exist).

• Take any non-commutative monoidal preorder, eg words on an alphabet ordered by length, or monotone endomaps on a poset ordered pointwise – the \\(\otimes\\) cannot be join or meet since it isn't commutative.

• Take the natural numbers and addition with the usual ordering – if \\(n, m\\) are non-zero then \\(n + m\\) is neither \\(\mathrm{min}(n, m)\\) nor \\(\mathrm{max}(n, m)\\), so here we have a commutative monoidal product where \\(\otimes\\) is neither join nor meet.

I'm working on a couple more where the \\(\otimes\\) operation is commutative and idempotent, but isn't join or meet. How "lattice-like" can \\(\otimes\\) get before it is forced to be a join or meet wrt the underlying preorder?

As for some counterexamples...

• Take any monoid (with at least 2 elements) – put the discrete partial order on the underlying set – we now have a (trivial) monoidal preorder where \\(\otimes\\) is not join or meet (since these operations don't exist).

• Take any non-commutative monoidal preorder, eg words on an alphabet ordered by length, or monotone endomaps on a poset ordered pointwise – the \\(\otimes\\) cannot be join or meet since it isn't commutative.

• Take the natural numbers and addition with the usual ordering – if \\(n, m\\) are non-zero then \\(n + m\\) is neither \\(\mathrm{min}(n, m)\\) nor \\(\mathrm{max}(n, m)\\), so here we have a commutative monoidal product where \\(\otimes\\) is neither join nor meet.

I'm working on a couple more where the \\(\otimes\\) operation is commutative and idempotent, but isn't join or meet. How "lattice-like" can \\(\otimes\\) get before it is forced to be a join or meet wrt the underlying preorder?