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The isomorphism problem for computable Abelian p-groups of bounded length

Published online by Cambridge University Press:  12 March 2014

Wesley Calvert
Wesley Calvert*
Affiliation:
Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame, Indiana 46556, USA, E-mail: wcalvert@nd.edu
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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.

Keywords

ClassificationComputableUlmBack-and-forth
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 1 , March 2005 , pp. 331 - 345
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1]Ash, C. J., Labelling systems and r.e. structures, Annals of Pure and Applied Logic, vol. 47 (1990), pp. 99–119.CrossRefGoogle Scholar
[2]Ash, C. J. and Knight, J. F., Computable structures and the hyperarithmetical hierarchy, Elsevier, 2000.Google Scholar
[3]Barker, E., Back and forth relations for reduced Abelian p-groups. Annals of Pure and Applied Logic, vol, 75 (1995), pp. 223–249.CrossRefGoogle Scholar
[4]Calvert, W., The isomorphism problem for classes of computable fields. Archive for Mathematical Logic, vol. 43 (2004), pp. 327–336.CrossRefGoogle Scholar
[5]Friedman, H., Simpson, S., and Smith, R., Countable algebra and set existence axioms. Annals of Pure and Applied Logic, vol. 25 (1983). pp. 141–181.CrossRefGoogle Scholar
[6]Friedman, H. and Stanley, L., A Borel reducibility theory for classes of countable structures, this Journal, vol. 54 (1989), pp. 894–914.Google Scholar
[7]Goncharov, S. S. and Knight, J. F., Computable structure and non-structure theorems, Algebra and Logic, vol. 41 (2002), pp. 351–373.CrossRefGoogle Scholar
[8]Hirschfeldt, D., Khoussainov, B., Shore, R., and Slinko, A. M., Degree spectra and computable dimensions in algebraic structures, Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71–113.CrossRefGoogle Scholar
[9]Hjorth, G., Classification and orbit equivalence relations, American Mathematical Society, 1999.CrossRefGoogle Scholar
[10]Kaplansky, I., Infinite Abelian groups, University of Michigan Press, 1969.Google Scholar
[11]Khisamiev, N. G., Constructive Abelian p-groups, Siberian Advances in Mathematics, vol. 2 (1992), pp. 68–113.Google Scholar
[12]Lin, C., The effective content of Ulm's theorem, Aspects of effective algebra (Crossley, J. N., editor), Upside Down A Book Company, 1979, pp. 147–160.Google Scholar
[13]Lin, C., Recursively presented Abelian groups: Effective p-group theory I, this Journal, vol. 46 (1981), pp. 617–624.Google Scholar
[14]Morozov, A. S., Functional trees and automorphisms of models, Algebra and Logic, vol. 32 (1993), pp. 28–38.CrossRefGoogle Scholar
[15]Nies, A., Undecidable fragments of elementary theories, Algebra Universalis, vol. 35 (1996), pp. 8–33.CrossRefGoogle Scholar
[16]Rabin, M. O. and Scott, D., The undecidability of some simple theories, preprint.Google Scholar
[17]Richman, F., The constructive theory of countable Abelian p-groups, Pacific Journal of Mathematics, vol. 45 (1973), pp. 621–624.CrossRefGoogle Scholar
[18]Rogers, L., The structure of p-trees: Algebraic systems related to Abelian groups, Abelian group theory: 2nd New Mexico state conference. Springer-Verlag, 1976, pp. 57–72.Google Scholar
[19]Simpson, S., Subsystems of second order arithmetic. Springer-Verlag, 1999.CrossRefGoogle Scholar