The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length

2004/06/24 by Calvert, Wesley · 1 citation
#03C57 #20K10 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.math/0406505

Abstract

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian p-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated.

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• Computability of Separation Axioms in Countable Second Countable Spaces

Related

• The isomorphism problem for classes of computable fields
• A first-order theory of Ulm type
• Effective categoricity of Abelian p-groups
• Procountable groups are not classifiable by countable structures
• The classifications of countably based profinite abelian groups