Fundamental confusion on set theory and permutations

Question

I am confused on the following: A set does not have any order. Now I read that a permutation is a bijection of a set. But doesn't this imply an order?
I mean a bijection is a one-to-one function from a set A to a set B. So if a permutation is a essentially a shuffling of a set and it is equivalent to a bijection doesn't that mean that this shuffling "explicitely" assumes an ordering in the first place? Otherwise there is no permutation of a set as a set has no order.
Can someone help me clear this out?

Answer 1

No, the term "permutation", in the lack of further context, means just a bijection. Sometimes when a set is presupposed to have a particular structure a permutation might mean a structure-preserving bijection, or the proper term an automorphism.

If you really want to think about permutation only in context of partial orders, then you can always consider the partial order $\{(x,x)\mid x\in X\}$, where no two distinct elements are comparable.

Answer 2

If the set is finite with $n$ elements, then a order is a bijection between $n$ and the set, this order is defined by $f(1)$ is the first element, $f(2)$ second element... etc. This is sense of a permutation.