Recursion-Theoretic Ranking and Compression

For which sets A does there exist a mapping, computed by a total or partial recursive function, such that the mapping, when its domain is restricted to A, is a 1-to-1, onto mapping to Σ * ? And for which sets A does there exist such a mapping that respects the lexicographical ordering within A? Both cases are types of perfect, minimal hash functions. The complexity-theoretic versions of these notions are known as compression functions and ranking functions. The present paper defines and studies the recursion-theoretic versions of compression and ranking functions, and in particular studies the question of which sets have, or lack, such functions.