Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T18:07:07.403Z Has data issue: false hasContentIssue false
  • English
  • Français

Isomorphisms of splits of computably enumerable sets

Published online by Cambridge University Press:  12 March 2014

Peter A. Cholak  and
Leo A. Harrington
Peter A. Cholak
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683, E-mail: [email protected]
Leo A. Harrington
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that if A and are automorphic via Φ then the structures (A) and () are Δ30-isomorphic via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 3 , September 2003 , pp. 1044 - 1064
Copyright
Copyright © Association for Symbolic Logic 2003

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] Cholak, Peter, Automorphisms of the lattice of recursively enumerable sets, Memoirs of the American Mathematical Society, vol. 113 (1995), no. 541.Google Scholar
[2] Cholak, Peter and Harrington, Leo A., Extension theorems, orbits and automorphisms of the computably enumerable sets. Submitted to Annals of Mathematics, 2002.Google Scholar
[3] Cholak, Peter and Harrington, Leo A., On the definability of the double jump in the computably enumerable sets, Journal of Mathematical Logic, vol. 2 (2002), no. 2, pp. 261–296, A copy can be found at: http://www.nd.edu/~cholak.CrossRefGoogle Scholar
[4] Cholak, Peter A. and Harrington, Leo A., Definable encodings in the computably enumerable sets. The Bulletin of Symbolic Logic, vol. 6 (2000), no. 2, pp. 185–196, A copy can be found at: http://www.nd.edu/~cholak.CrossRefGoogle Scholar
[5] Downey, Rod and Stob, Michael, Friedberg splittings of recursively enumerable sets, Annals of Pure and Applied Logic, vol. 59 (1993), no. 3, pp. 175–199, Fourth Asian Logic Conference (Tokyo, 1990).Google Scholar
[6] Downey, Rod and Stob, Michael, Splitting theorems in recursion theory. Annals of Pure and Applied Logic, vol. 65 (1993), no. 1, p. 106 pp.Google Scholar
[7] Harrington, Leo A. and Nies, André, Coding in the partial order of enumerable sets. Advances in Mathematics, vol. 133 (1998), no. 1, pp. 133–162.Google Scholar
[8] Harrington, Leo A. and Soare, Robert I., The ΄3 0-automorphism method and noninvariant classes of degrees, Journal of the American Mathematical Society, vol. 9 (1996), no. 3, pp. 617–666.Google Scholar
[9] Maass, W., On the orbit of hyperhypersimple sets, this Journal, vol. 49 (1984), pp. 51–62.Google Scholar
[10] Montagna, Franco and Sorbi, Andrea, Universal recursion-theoretic properties of r. e. preordered structures, this Journal, vol. 50 (1985), no. 2, pp. 397–406.Google Scholar
[11] Nies, A., Intervals of the lattice of computably enumerable sets and effective Boolean algebras. The Bulletin of the London Mathematical Society, vol. 29 (1997), pp. 683–92.Google Scholar
[12] Nies, André, Effectively dense Boolean algebras and their applications, Transactions of the American Mathematical Society, vol. 352 (2000), no. 11, pp. 4989–5012 (electronic).Google Scholar
[13] Soare, Robert I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer–Verlag, Heidelberg, 1987.Google Scholar
[14] Stob, M., The structure and elementary theory of the recursive enumerable sets, Ph.D. thesis , University of Chicago, 1979.Google Scholar