Matthew - whoops, a bunch of my easy-to-find monoidal poset structures on \\( (\mathbb{C}, +, 0 ) \\) were actually monoidal preorders. It sounds like maybe you know these.

By the way, any monoidal preorder on a group (like the complex numbers with \\(+\\) as its monoid operation) obeys

\[ a \le b \iff a + k \le b + k .\]

So I don't think your "translated lexicographic orders" are different from the ordinary lexicographic order. But I seem to be making mistakes...