I'm in the opposite camp there, @Jonathan – I'd say joins/meets in preorders are just coproducts/products, and we'd only ever expect them to be defined "up to isomorphism". So I'd draw the opposite conclusion – any monoidal operation on a codiscrete preorder is automatically a join and a meet!

Re your other point, I'm thinking about a monoidal preorder where we can "throw things away", ie \\(x \otimes y \leq x\\), or equivalently \\(y \leq I\\) for all \\(y\\). In this case \\(x \otimes y\\) is a lower bound of \\(x\\) and \\(y\\). And if we can "duplicate things", ie \\(z \leq z \otimes z\\), then we can prove \\(\otimes\\) is the meet operation (since any lower bound \\(z \leq z \otimes z \leq x \otimes y\\)).

But what if we can "dispose" but not "duplicate"? Then \\(\otimes\\) cannot be meet since it isn't idempotent. My hunch is that the "manufacturing" monoidal preorders would be an example of this, because we can chuck stuff away but not turn one girder into two.

Re your other point, I'm thinking about a monoidal preorder where we can "throw things away", ie \\(x \otimes y \leq x\\), or equivalently \\(y \leq I\\) for all \\(y\\). In this case \\(x \otimes y\\) is a lower bound of \\(x\\) and \\(y\\). And if we can "duplicate things", ie \\(z \leq z \otimes z\\), then we can prove \\(\otimes\\) is the meet operation (since any lower bound \\(z \leq z \otimes z \leq x \otimes y\\)).

But what if we can "dispose" but not "duplicate"? Then \\(\otimes\\) cannot be meet since it isn't idempotent. My hunch is that the "manufacturing" monoidal preorders would be an example of this, because we can chuck stuff away but not turn one girder into two.