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Title: The Local Structure of the Enumeration Degrees
Author: Soskova, Mariya Ivanova
ISNI:       0000 0001 3472 1103
Awarding Body: University of Leeds
Current Institution: University of Leeds
Date of Award: 2008
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This thesis discusses properties of the local structure of the enumeration degrees. We begin with some historical background of the subject. vye give motivation for investigating the properties of the local structure ofthe enumeration degrees and discuss the basic concepts and methods used throughout the thesis. Chapter 2 presents evidence that the study of the structure of enumeration degrees can provide a richer understanding of the structure of the Thring degrees. We prove that -- -' - . there exists a II~ enumeration degree which is the bottom of a cone within which the II~ enumeration degrees cannot be cupped to O~. As a corollary we obtain a generalization of Harrington's non-splitting theorem for the ~g Thring degrees. Chapters 3 and 4 are dedicated to the study of properties, specific to the properly ~g enumeration degrees. In Chapter 3 we construct a properly ~g enumeration degree above which there is no splitting of O~. Degrees with this property can be used to define a filter in the local structure of the enumeration degrees that consists entirely of properly ~g enumeration degrees and O~. In Chapter 4 we strengthen the result obtained by Cooper, Li, Sorbi and Yang of the existence of a non-bounding enumeration degree by constructing a I-generic enumeration degree that does not bound a minimal pair. Degrees with this property· can be used to define an ideal consisting of properly ~g enumeration degrees and Oe. Chapters 5 and 6 concern the cupping properties of ~g enumeration degrees and the sub-classes of the ~g enumeration degrees related to the finite and w- levels of the Ershov hierarchy. In Chapter 5 we complement a result by Cooper, Sorbi and Yi by showing that every non-zero .6.g enumeration degree can be cupped by a partial low .6.g enumeration degree. On the other hand we show that one cannot computably list a sequence of degrees which contains a cupping partner for every .6.g enumeration degree. In Chapter 6 we concentrate on the smaller subclasses, where the situation improves. We prove that every non-zero w-c.e. enumeration degree can be cupped by a 3-c.e. enumeration degree and as the 3-c.e. enumeration degrees are computably enumerable this property constitutes a difference between the .6.g enumeration degrees and the w-c.e. enumeration degrees. Furthermore we establish a structural difference between the class of II~ enumeration degrees and the 3-c.e. enumeration degrees by proving that one cannot find a single Eg enumeration degree that cups every non-zero 3-c.e. enumeration degree to O~. Finally in Chapter 7 we show that the structure of the 3-c.e. enumeration degrees is far from trivial as there exists a Lachlan Il()n-spJ~tting pair with top a II~enuIlleration degree and bottom a 3-c.e. enumeration degree.
Supervisor: Not available Sponsor: Not available
Qualification Name: University of Leeds, 2008 Qualification Level: Doctoral
EThOS ID:  DOI: Not available