In answer to John's question posed in Comment 23: To prove that \\((\mathbb{C}, + , 0)\\) is a monoidal preorder, we would need to check that if \\(y \leq y' \\) and \\(z \leq z'\\) then \\(y + z \leq y' + z'\\). Unfortunately, this doesn't hold! For example using the "distance to the origin" definition of \\(\leq \\) $$1 \leq 1 \text{ and }1 \leq i \text{ but } 2 \nleq 1 + i.$$