Alex wrote:

> Anyways, using the bullet-proof method of imagining parallelograms sliding around on the surface of a 3D plot, I think a superset of Matthew's set of solutions is

> \[ x \preceq_{p,F} y \iff F(p \cdot x) \leq F(p \cdot y) \]

> for each point \\(p \in \mathbb{C}\\) and monotone function \\(F : \mathbb{R} \to \mathbb{R} \\). I think this is \\(2^{2^{\aleph_0}}\\) solutions.

How does the function \\(F\\) affect \\(\preceq_{p,F}\\)? For example, \\(F(x) = 2x\\) gives the same partial order as \\(F(x) = e^x \\).

> Anyways, using the bullet-proof method of imagining parallelograms sliding around on the surface of a 3D plot, I think a superset of Matthew's set of solutions is

> \[ x \preceq_{p,F} y \iff F(p \cdot x) \leq F(p \cdot y) \]

> for each point \\(p \in \mathbb{C}\\) and monotone function \\(F : \mathbb{R} \to \mathbb{R} \\). I think this is \\(2^{2^{\aleph_0}}\\) solutions.

How does the function \\(F\\) affect \\(\preceq_{p,F}\\)? For example, \\(F(x) = 2x\\) gives the same partial order as \\(F(x) = e^x \\).