Ranked structures and arithmetic transfinite recursion

$\mathsf{ATR}_0$ is the natural subsystem of second-order arithmetic in which one can develop a decent theory of ordinals. We investigate classes of structures which are in a sense the ``well-founded part" of a larger, simpler class, for example, superatomic Boolean algebras (within the class of all Boolean algebras). The other classes we study are: well-founded trees, reduced Abelian $p$-groups, and countable, compact topological spaces. Using computable reductions between these classes, we show that Arithmetic Transfinite Recursion is the natural system for working with them: natural statements (such as comparability of structures in the class) are equivalent to $\mathsf{ATR}_0$. The reductions themselves are also objects of interest.