Softcover ISBN:  9781470428396 
Product Code:  MEMO/253/1208 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 
Electronic ISBN:  9781470444112 
Product Code:  MEMO/253/1208.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 253; 2018; 107 ppMSC: Primary 03;
Let \(\mathcal A\) be a mathematical structure with an additional relation \(R\). The author is interested in the degree spectrum of \(R\), either among computable copies of \(\mathcal A\) when \((\mathcal A,R)\) is a “natural” structure, or (to make this rigorous) among copies of \((\mathcal A,R)\) computable in a large degree d. He introduces the partial order of degree spectra on a cone and begin the study of these objects. Using a result of Harizanov—that, assuming an effectiveness condition on \(\mathcal A\) and \(R\), if \(R\) is not intrinsically computable, then its degree spectrum contains all c.e. degrees—the author shows that there is a minimal nontrivial degree spectrum on a cone, consisting of the c.e. degrees.

Table of Contents

Chapters

1. Introduction

2. Preliminaries

3. Degree Spectra between the C.E. Degrees and the D.C.E. Degrees

4. Degree Spectra of Relations on the Naturals

5. A “Fullness” Theorem for 2CEA\xspace Degrees

6. Further Questions

A. Relativizing Harizanov’s Theorem on C.E. Degrees


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Let \(\mathcal A\) be a mathematical structure with an additional relation \(R\). The author is interested in the degree spectrum of \(R\), either among computable copies of \(\mathcal A\) when \((\mathcal A,R)\) is a “natural” structure, or (to make this rigorous) among copies of \((\mathcal A,R)\) computable in a large degree d. He introduces the partial order of degree spectra on a cone and begin the study of these objects. Using a result of Harizanov—that, assuming an effectiveness condition on \(\mathcal A\) and \(R\), if \(R\) is not intrinsically computable, then its degree spectrum contains all c.e. degrees—the author shows that there is a minimal nontrivial degree spectrum on a cone, consisting of the c.e. degrees.

Chapters

1. Introduction

2. Preliminaries

3. Degree Spectra between the C.E. Degrees and the D.C.E. Degrees

4. Degree Spectra of Relations on the Naturals

5. A “Fullness” Theorem for 2CEA\xspace Degrees

6. Further Questions

A. Relativizing Harizanov’s Theorem on C.E. Degrees