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A numbering of a countable family $S$ is a surjective map from the set ofnatural numbers $\omega$ onto $S$. A numbering $\nu$ is reducible to anumbering $\mu$ if there is an effective procedure which given a $\nu$-index ofan object from $S$, computes a $\mu$-index for the same object. Thereducibility between numberings gives rise to a class of upper semilattices,which are usually called Rogers semilattices. The paper studies Rogerssemilattices for families $S \subset P(\omega)$ belonging to various levels ofthe analytical hierarchy. We prove that for any non-zero natural numbers $m\neqn$, any non-trivial Rogers semilattice of a $\Pi^1_m$-computable family cannotbe isomorphic to a Rogers semilattice of a $\Pi^1_n$-computable family. One ofthe key ingredients of the proof is an application of the result by Downey andKnight on degree spectra of linear orders.

Isomorphism types of Rogers semilattices in the analytical hierarchy