Jonathan wrote:

> As John notes, however, these are perfectly serviceable commutative monoidal preorders.

I didn't say that. I said:

> given two points in the complex let us say that \\(z \le z'\\) if \\(z\\) is closer to the origin than \\(z'\\), i.e.

> \[ |z| \le |z'| .\]

> This defines a preorder, not a partial order. So my next question was: does this make \\((\mathbb{C}, +, 0) \\) into a monoidal preorder?

There's something that needs to be checked here. If it works, we've got a monoidal preorder. If it doesn't, we don't.

> As John notes, however, these are perfectly serviceable commutative monoidal preorders.

I didn't say that. I said:

> given two points in the complex let us say that \\(z \le z'\\) if \\(z\\) is closer to the origin than \\(z'\\), i.e.

> \[ |z| \le |z'| .\]

> This defines a preorder, not a partial order. So my next question was: does this make \\((\mathbb{C}, +, 0) \\) into a monoidal preorder?

There's something that needs to be checked here. If it works, we've got a monoidal preorder. If it doesn't, we don't.