> **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)

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)