Isomorphism types of Rogers semilattices in the analytical hierarchy

TL;DR

This paper investigates the structure of Rogers semilattices for families within the analytical hierarchy, establishing non-isomorphism results based on the hierarchy levels using computability and degree spectrum techniques.

Contribution

It proves that Rogers semilattices of different levels in the analytical hierarchy are non-isomorphic, extending understanding of their structural complexity.

Findings

Rogers semilattices at different hierarchy levels are non-isomorphic.

The proof utilizes degree spectra of linear orders.

Results connect computability theory with the structure of semilattices.

Abstract

A numbering of a countable family $S$ is a surjective map from the set of natural numbers $\omega$ onto $S$. A numbering $\nu$ is reducible to a numbering $\mu$ if there is an effective procedure which given a $\nu$-index of an object from $S$, computes a $\mu$-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. The paper studies Rogers semilattices for families $S \subset P(\omega)$ belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers $m\neq n$, any non-trivial Rogers semilattice of a $\Pi^1_m$-computable family cannot be isomorphic to a Rogers semilattice of a $\Pi^1_n$-computable family. One of the key ingredients of the proof is an application of the result by Downey and Knight on degree spectra of linear orders.