an:05563899
Zbl 1177.03046
Calvert, Wesley; Cenzer, Douglas; Harizanov, Valentina S.; Morozov, Andrei
Effective categoricity of abelian \(p\)-groups
EN
Ann. Pure Appl. Logic 159, No. 1-2, 187-197 (2009).
0168-0072
2009
j
03D45 03C57
computable structures; abelian \(p\)-groups; computable categoricity; \(\Delta_2^0\)-categoricity; relative categoricity; Scott family
In this paper the authors investigate algorithmic properties of \(p\)-groups and their characters, providing a connection between equivalence structures and abelian \(p\)-groups. The main focus of the paper is the categoricity of abelian \(p\)-groups. It is shown that every computably categorical abelian \(p\)-group is relatively computably categorical. (A structure \({\mathcal A}\) is relatively categorical if for every structure \({\mathcal B}\) isomorphic to \({\mathcal A}\), there is an isomorphism that is computable relative to the atomic diagram of \({\mathcal B}\).) Studying \(\Delta_2^0\)-isomorphisms of abelian \(p\)-groups, the authors characterize those groups that are relatively \(\Delta_2^0\)-categorical. (A structure \({\mathcal A}\) is relatively \(\Delta_2^0\)-categorical if for every structure \({\mathcal B}\) isomorphic to \({\mathcal A}\), there is an isomorphism that is \(\Delta_2^0\)-relative to the atomic diagram of \({\mathcal B}\).) The paper contains also a list of open problems.
Marat M. Arslanov (Kazan)