V. S. Harizanov | Searcharxiv

arXiv2008

Effective categoricity of equivalence Structures

An equivalence structure is a set with a single binary relation, satisfying sentences stating that the relation is an equivalence relation. A computable structure A is said to be $Δ^0_α$ categorical if for any computable structure B isomorphic to A there is a $Δ^0_α$ function witnessing that the two are isomorphic. The present paper gives an exact characterization of $Δ^0_α$ equivalence structures where $α= 1$ or $α\geq 3$. Extensive results for $α= 2$ are also given, and open cases are exhaustively described.

math.LO↗

arXiv2008

Effective categoricity of Abelian p-groups

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 $Δ^0_α$ categorical if for any computable structure B isomorphic to A there is a $Δ^0_α$ function witnessing that the two are isomorphic. The present paper seeks to characterize $Δ^0_α$ categoricity for Abelian p-groups, and results of this kind are given for broad classes of Abelian p-groups and values of $α$. The remaining open cases are exhaustively described.

math.LO↗