It is easy to see that the totally ordered group $\mathbb{Z}$ (the integers) with the natural order has no non-trivial automorphisms. Is this also true for $\mathbb{Z}^n$ with the lexicographical order?



Automorphisms of the totally ordered group $\mathbb{Z}{^n}$ with lexicographical order
Here's a counterexample: on $\mathbb{Z}^2$, $f(x,y)=(x,y+x)$. 


More generally, the order-preserving automorphisms of $\mathbb{Z}^n$ are exactly the upper triangular matrices with 1s on the diagonal (this should be easy to see by combining Charles's argument with my example in the case $n=2$, and then the generalization to arbitrary $n$ isn't too hard).


