I am currently reading Bourbaki Algebra and in section 2 of chapter one they define when two ordered sequences are similar as this:
Two ordered sequence $(x_i)_{i \in I}$ and $(y_k)_{k \in K}$ are similar if there exists an ordered set isomorphism $f$ of $I$ onto $K$ such that $y_{f(i)} = x_i$ for all $i \in I$. What do they mean by ordered set isomorphism? Do they mean that the function is an ordered set? If so why do they go on and endow the function with an ordered relation?
*Note: Definition of ordered sequence $(x_i)_{i \in I}$ is a finite family of elements of $E$ whose index $I$ is totally ordered set.
Asserting that $f$ is an ordered set isomorphism means that $f$ is a bijection such that both $f$ and $f^{-1}$ preserve the order (that is $x\leqslant y\implies f(x)\leqslant f(y)$).