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Degree Spectra of Relations on a Cone


About this Title

Matthew Harrison-Trainor

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 253, Number 1208
ISBNs: 978-1-4704-2839-6 (print); 978-1-4704-4411-2 (online)
DOI: https://doi.org/10.1090/memo/1208
Published electronically: March 29, 2018
Keywords:computable structure, degree spectrum of a relation, cone of Turing degrees

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Preliminaries
  • Chapter 3. Degree Spectra between the C.E. Degrees and the D.C.E. Degrees
  • Chapter 4. Degree Spectra of Relations on the Naturals
  • Chapter 5. A “Fullness” Theorem for 2-\cea Degrees
  • Chapter 6. Further Questions
  • Appendix A. Relativizing Harizanov’s Theorem on C.E. Degrees

Abstract


Let be a mathematical structure with an additional relation . We are interested in the degree spectrum of , either among computable copies of when is a âĂIJnaturalâĂİ structure, or (to make this rigorous) among copies of computable in a large degree d. We introduce the partial order of degree spectra on a cone and begin the study of these objects. Using a result of HarizanovâĂŤthat, assuming an effectiveness condition on and , if is not intrinsically computable, then its degree spectrum contains all c.e. degreesâĂŤwe see that there is a minimal non-trivial degree spectrum on a cone, consisting of the c.e. degrees. We show that this does not generalize to d.c.e. degrees by giving an example of two incomparable degree spectra on a cone. We also give a partial answer to a question of Ash and Knight: they asked whether (subject to some effectiveness conditions) a relation which is not intrinsically must have a degree spectrum which contains all of the -CEA degrees. We give a positive answer to this question for by showing that any degree spectrum on a cone which strictly contains the degrees must contain all of the 2-CEA degrees. We also investigate the particular case of degree spectra on the structure . This work represents the beginning of an investigation of the degree spectra of âĂIJnaturalâĂİ structures, and we leave many open questions to be answered.

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