Theories of Rogers Semilattices of Analytical Numberings

The paper studies Rogers semilattices, i.e. upper semilattices induced by the reducibility between numberings. Under the assumption of Projective Determinacy, we prove that for every non-zero natural number $$n$$ , there are infinitely many pairwise elementarily non-equivalent Rogers semilattices for $$\Sigma^{1}_{n}$$ -computable families.