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Jump degrees of torsion-free abelian groups

Published online by Cambridge University Press:  12 March 2014

Brooke M. Andersen ,
Asher M. Kach ,
Alexander G. Melnikov  and
Reed Solomon
Brooke M. Andersen
Affiliation:
Department of Mathematics and Computer Science, Assumption College, Worcester, MA 01609-1296, USA, E-mail: brandersen@assumption.edu
Asher M. Kach
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA, E-mail: kach@math.uchicago.edu, E-mail: asher.kach@gmail.com
Alexander G. Melnikov
Affiliation:
Department of Computer Science, University of Auckland, Auckland 1142, New Zealand, E-mail: a.melnikov@cs.auckland.ac.nz, E-mail: alexander.g.melnikov@gmail.com
Reed Solomon
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA, E-mail: david.solomon@uconn.edu
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Abstract

We show, for each computable ordinal α and degree a > 0(α), the existence of a torsion-free abelian group with proper αth jump degree a.

Keywords

torsion-free abelian groupjump degree
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 4 , December 2012 , pp. 1067 - 1100
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1]Ash, C. J., Jockusch, C. G. Jr., and Knight, J. F., Jumps of orderings, Transactions of the American Mathematical Society, vol. 319 (1990), no. 2, pp. 573–599.CrossRefGoogle Scholar
[2]Ash, C. J. and Knight, J., Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland, Amsterdam, 2000.Google Scholar
[3]Coles, Richard J., Downey, Rod G., and Slaman, Theodore A., Every set has a least jump enumeration, Journal of the London Mathematical Society. Second Series, vol. 62 (2000), no. 3, pp. 641–649.CrossRefGoogle Scholar
[4]Downey, Rod and Montalbán, Antonio, The isomorphism problem for torsion-free abelian groups is analytic complete, Journal of Algebra, vol. 320 (2008), no. 6, pp. 2291–2300.CrossRefGoogle Scholar
[5]Downey, Rodney and Knight, Julia F., Orderings with ath jump degree 0(α), Proceedings of the American Mathematical Society, vol. 114 (1992), no. 2, pp. 545–552.Google Scholar
[6]Downey, Rodney G., On presentations of algebraic structures, Complexity, logic, and recursion theory, Lecture Notes in Pure and Applied Mathematics, vol. 187, Dekker, New York, 1997, pp. 157–205.Google Scholar
[7]Fokina, Ekaterina, Knight, Julia F., Maher, C., Melnikov, Alexander G., and Quinn, Sara, Classes of Vim type, and relations between the class of rank-homogeneous trees and other classes, accepted.Google Scholar
[8]Friedman, Harvey M., Simpson, Stephen G., and Smith, Rick L., Countable algebra and set existence axioms, Annals of Pure and Applied Logic, vol. 25 (1983), no. 2, pp. 141–181.CrossRefGoogle Scholar
[9]Fuchs, László, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, vol. 36, Academic Press, New York, 1970.Google Scholar
[10]Fuchs, László, Infinite abelian groups. Vol. II, Pure and Applied Mathematics, vol. 36-II, Academic Press, New York, 1973.Google Scholar
[11]Hjorth, Greg, The isomorphism relation on countable torsion free abelian groups, Fundamenta Mathematicae, vol. 175 (2002), no. 3, pp. 241–257.CrossRefGoogle Scholar
[12]Jockusch, Carl G. Jr. and Soare, Robert I., Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), no. 1-2, pp. 39–64.CrossRefGoogle Scholar
[13]Jockusch, Carl G. Jr. and Soare, Robert I., Boolean algebras, Stone spaces, and the iterated Turing jump, this Journal, vol. 59 (1994), no. 4, pp. 1121–1138.Google Scholar
[14]Knight, Julia F., Degrees coded in jumps of orderings, this Journal, vol. 51 (1986), no. 4, pp. 1034–1042.Google Scholar
[15]Melnikov, Alexander G., Enumerations and completely decomposable torsion-free abelian groups, Theory of Computing Systems, vol. 45 (2009), no. 4, pp. 897–916.CrossRefGoogle Scholar
[16]Richter, Linda Jean, Degrees of structures, this Journal, vol. 46 (1981), no. 4, pp. 723–731.Google Scholar
[17]Smith, Rick L., Two theorems on autostability in p-groups, Logic Year 1979–80, Lecture Notes in Mathematics, vol. 859, Springer, Heidelberg, 1981, pp. 302–311.CrossRefGoogle Scholar
[18]Solomon, Reed, Reverse mathematics and fully ordered groups, Notre Dame Journal of Formal Logic, vol. 39 (1998), no. 2, pp. 157–189.CrossRefGoogle Scholar