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 \$$ $$1 \leq 1 \text{ and }1 \leq i \text{ but } 2 \nleq 1 + i.$$