Can someone explain the phrase "order-preserving permutation of a totally ordered set"?

Question

If a set is totally ordered, then any nontrivial permutation would destroy that order, so I'm clearly missing something.

I have seen this exact phrase in many papers and books, mostly those dealing with ordered groups.

Answer 1

It is not true that any nontrivial permutation destroys a total ordering. This is true for finite sets, but "most" infinite sets are counter examples. For example, $x\to x+1$ preserves the total order on the integers, rationals, and reals.

Another example is to take $\mathbb{Q}$ and double every number bigger than $0$, and leave the rest fixed.

Just these two will let you produce many, as the composition of order preserving permutations is an order preserving permutation.