Isomorphism types of Rogers semilattices in the analytical hierarchy

Abstract

A numbering of a countable family S is a surjective map from the set of natural numbers ω onto S. A numbering ν is reducible to a numbering µ if there is an effective procedure which given a ν-index of an object from S, computes a µ-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 ⊂ P (ω) belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers m = n, any non-trivial Rogers semilattice of a Π 1 m -computable family cannot be isomorphic to a Rogers semilattice of a Π 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.

Rogers semilattices of punctual numberings

“…Goncharov and Sorbi (1997) started developing the theory of generalized computable numberings. Their approach initiated a fruitful line of research, which is focused on numberings for families of sets which belong to various levels of recursion-theoretic hierarchies: the arithmetical hierarchy (Badaev et al 2006;Bazhenov et al 2019c;Podzorov 2008), the Ershov hierarchy (Badaev and Lempp 2009;Goncharov et al 2002;Herbert et al 2019;Ospichev 2015), the analytical hierarchy (Bazhenov et al 2019d(Bazhenov et al , 2020bDorzhieva 2019), etc. We refer the reader to Goncharov (2008, 2000) for further background on numberings in these hierarchies.…”