Automorphisms of the totally ordered group Z^n with lexicographical order

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

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Here's a counterexample: on Z^2, f(x,y)=(x,y+x). More generally, the order-preserving automorphisms of 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).

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Thanks, awesome! After Charles' answer I first thought that I was too dumb for the induction step. –  user717 Nov 2 '09 at 14:45
I also want to point out that the lexicographic order on $\mathbb{Z}^n$ has no minimal positive element, that is precisely the reason that the argument given by Eric holds. –  Stines Oct 21 '10 at 21:02