Keith wrote:

> I still don't understand why \\( \\{ \mathbb{C},+,0, \le, ||z|| \\} \\) doesn't count as a commutative monoidal poset.

The main reason is that a commutative monoidal poset consists of 4 things:

* a set
* a binary operation on that set
* an element of that set
* and a binary relation on that set

obeying some rules. You've listed 5. There's no role for that fifth item in a commutative monoidal poset.