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

Oops, thanks for the correction.

I was thinking about translation because I was noticing that if \$$\leq\$$ is a monoidal poset then for every [affine transformation](https://en.wikipedia.org/wiki/Affine_transformation) \$$A\$$ there corresponds a monoidal preorder \$$\preceq_A\$$, where

$$x \preceq_A y \iff A(x) \leq A(y)$$

I was thinking of this because affine transforms preserve addition and little else. Since affine transformations can all be expressed as \$$A(x) = \mathbf{a} x + \mathbf{b}\$$, I was erroneously thinking that translation would lead to distinct posets/preorders.

An affine transformation without a translation term is just a [linear map](https://en.wikipedia.org/wiki/Linear_map).

So now I am wondering: if I have two isomorphic monoidal posets \$$\langle \mathbb{C}, +, 0, \leq_1\\rangle\$$ and \$$\langle \mathbb{C}, +, 0, \leq_2\\rangle\$$, then is their isomorphism function \$$\phi : \mathbb{C} \to \mathbb{C}\$$ a linear map?