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${\cal D}$-MAXIMAL SETS
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- Journal:
- The Journal of Symbolic Logic / Volume 80 / Issue 4 / December 2015
- Published online by Cambridge University Press:
- 22 December 2015, pp. 1182-1210
- Print publication:
- December 2015
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- Article
- Export citation
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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.
