${\cal D}$-MAXIMAL SETS. (22nd December 2015)

• Record Type: Journal Article
• Authors: CHOLAK, PETER A.; GERDES, PETER; LANGE, KAREN
• Is Part Of: Journal of symbolic logic. Volume 80:Number 4(2015)
• Page Start: 1182
• Page End: 1210
• Publication Date: 2015-12-22
• Subjects: computably enumerable sets under inclusion, -- maximal sets, -- r-maximal sets, -- hhsimple sets; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2015.3
• Languages: English
• ISSNs: 0022-4812

Abstract

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.