Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-16T02:21:57.827Z Has data issue: false hasContentIssue false

Jump degrees of torsion-free abelian groups

Published online by Cambridge University Press:  12 March 2014

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

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 573599.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. 641649.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. 22912300.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. 545552.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. 157205.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. 141181.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. 241257.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. 3964.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. 11211138.Google Scholar
[14]Knight, Julia F., Degrees coded in jumps of orderings, this Journal, vol. 51 (1986), no. 4, pp. 10341042.Google Scholar
[15]Melnikov, Alexander G., Enumerations and completely decomposable torsion-free abelian groups, Theory of Computing Systems, vol. 45 (2009), no. 4, pp. 897916.CrossRefGoogle Scholar
[16]Richter, Linda Jean, Degrees of structures, this Journal, vol. 46 (1981), no. 4, pp. 723731.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. 302311.CrossRefGoogle Scholar
[18]Solomon, Reed, Reverse mathematics and fully ordered groups, Notre Dame Journal of Formal Logic, vol. 39 (1998), no. 2, pp. 157189.CrossRefGoogle Scholar