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

Wesley Calvert
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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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Wesley Calvert. 2004-06-24. The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length. https://arxiv.org/abs/math/0406505

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