Effective categoricity of Abelian p-groups

TL;DR

This paper characterizes the levels of computable categoricity for Abelian p-groups, extending known classifications and identifying open cases across broad classes and various levels of the arithmetical hierarchy.

Contribution

It provides a comprehensive characterization of $ ext{Delta}^0_eta$ categoricity for Abelian p-groups, including broad classes and remaining open cases.

Findings

Characterized $ ext{Delta}^0_eta$ categoricity for broad classes of Abelian p-groups.

Identified and exhaustively described remaining open cases.

Extended the classification of computable structures beyond known results.

Abstract

Let p be a fixed prime. An Abelian p-group is an Abelian group (not necessarily finitely generated) in which every element has for its order some power of p. The countable Abelian p-groups are classified by Ulm's theorem, and Khisamiev characterized the Abelian p-groups with computable copies. A computable structure A is said to be $\Delta^0_\alpha$ categorical if for any computable structure B isomorphic to A there is a $\Delta^0_\alpha$ function witnessing that the two are isomorphic. The present paper seeks to characterize $\Delta^0_\alpha$ categoricity for Abelian p-groups, and results of this kind are given for broad classes of Abelian p-groups and values of $\alpha$. The remaining open cases are exhaustively described.