As a monoid, \\(\langle \mathbb{C}, +, 0 \rangle\\) is isomorphic to \\(\langle \mathbb{R}^2, +, (0, 0) \rangle\\), which has a natural partial order given by the product on \\(\langle \mathbb{R}, \le \rangle\\) -- in other words, we have \\((x, y) \le (x', y')\\) iff \\(x \le x' \land y \le y'\\). The monoid structure respects this order, since if \\(x \le x'\\) and \\(r\\) is any real number, we definitely have \\(x + r \le x' + r\\) and \\(r + x \le r + x'\\); this lifts to the product straightforwardly. We can carry this order across the isomorphism back to the complex numbers, giving a monoidal poset structure on \\(\mathbb{C}\\).

This is much like Maria's construction in #6, but -- if I understand correctly -- we avoid identifying numbers whose components have the same magnitude, e.g. \\(1 + i\\) and \\(-1 - i\\).

We can incorporate Matthew's ideas as true posets here: since rotations about the origin are automorphisms of the complex plane, we can similarly rotate our partial order. For instance, a 90-degree rotation would give us a poset where \\(-1 + i \le -2 + 2i\\). For every line of the complex plane passing through the origin, we can obtain a rotation that maps this line to the \\(x\\)-axis; every such line gives another induced partial order.

This is much like Maria's construction in #6, but -- if I understand correctly -- we avoid identifying numbers whose components have the same magnitude, e.g. \\(1 + i\\) and \\(-1 - i\\).

We can incorporate Matthew's ideas as true posets here: since rotations about the origin are automorphisms of the complex plane, we can similarly rotate our partial order. For instance, a 90-degree rotation would give us a poset where \\(-1 + i \le -2 + 2i\\). For every line of the complex plane passing through the origin, we can obtain a rotation that maps this line to the \\(x\\)-axis; every such line gives another induced partial order.