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${\cal D}$-MAXIMAL SETS

Published online by Cambridge University Press:  22 December 2015

PETER A. CHOLAK ,
PETER GERDES  and
KAREN LANGE
PETER A. CHOLAK
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME NOTRE DAME, IN 46556-5683, USAE-mail: peter.cholak.1@nd.eduURL: http://www.nd.edu/∼cholak
PETER GERDES
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME NOTRE DAME, IN 46556-5683, USAE-mail: gerdes@invariant.org
KAREN LANGE
Affiliation:
DEPARTMENT OF MATHEMATICS WELLESLEY COLLEGE WELLESLEY, MA 02482, USAE-mail: karen.lange@wellesley.eduURL: http://palmer.wellesley.edu/∼klange/
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Abstract

Soare [20] proved that the maximal sets form an orbit in ${\cal E}$. We consider here ${\cal D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of ${\cal D}$-maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the ${\cal D}$-maximal sets. Although these invariants help us to better understand the ${\cal D}$-maximal sets, we use them to show that several classes of ${\cal D}$-maximal sets break into infinitely many orbits.

Keywords

computably enumerable sets under inclusionmaximal setsr-maximal setshhsimple sets
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 4 , December 2015 , pp. 1182 - 1210
Copyright
Copyright © The Association for Symbolic Logic 2015 

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