> **Puzzle 71.** Can you make the complex numbers, ℂ, into a commutative monoidal poset with the usual + and 0 and some concept of ≤? If so, how many ways can you do this?

We can always just do \$$x \preceq_1 y \iff \mathfrak{Re}(x) \leq \mathfrak{Re}(y)\$$ where \$$\mathfrak{Re}(x)\$$ is the real component of \$$x\$$.

But this is just a special case of looking at the magnitudes of the [*orthogonal projection*](https://en.wikibooks.org/wiki/Linear_Algebra/Orthogonal_Projection_Onto_a_Line) of two points in \$$\mathcal{C}\$$ onto a common point. Specifically, \$$\mathfrak{Re}(x)\$$ is the magnitude of \$$x\$$ projected onto \$$1 + i0\$$. We could pick any point in \$$\mathbb{C}\$$ and do this. The magnitude of the orthogonal projection of \$$p + iq\$$ onto \$$x + i y\$$ is \$$p x + y q\$$, or the usual [dot-product \$$(\cdot)\$$](https://en.wikipedia.org/wiki/Dot_product) from linear algebra.

So we have \$$2^{\aleph_0}\$$ other examples, each one corresponding to

$$x \preceq_p y \iff p \cdot x \leq p \cdot y$$

for each point \$$p \in \mathbb{C}\$$.

[wikipedia_norm]: https://en.wikipedia.org/wiki/Norm_(mathematics)