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Degree spectra of relations on computable structures in the presence of Δ20 isomorphisms

Published online by Cambridge University Press:  12 March 2014

Denis R. Hirschfeldt
Denis R. Hirschfeldt*
Affiliation:
School of Mathematical and Computing Sciences, Victoria University of Wellington, New Zealand
*
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637, USA, E-mail: drh@math.uchicago.edu
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Abstract

We give some new examples of possible degree spectra of invariant relations on Δ20-categorical computable structures, which demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a Δ20-categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [23] to establish the same result for computable relations on computable linear orderings.

We also place our results in the context of the study of what types of degree-theoretic constructions can be carried out within the degree spectrum of a relation on a computable structure, given some restrictions on the relation or the structure. From this point of view we consider the cases of Δ20-categorical structures, linear orderings, and 1-decidable structures, in the last case using the proof of a result of Ash and Nerode [3] to extend results of Harizanov [14].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 2 , June 2002 , pp. 697 - 720
Copyright
Copyright © Association for Symbolic Logic 2002

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