rMaximal major subsets
 Manuel Lerman,
 Richard A. Shore,
 Robert I. Soare
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The question of which r.e. setsA possess major subsetsB which are alsormaximal inA (A⊂_{rm} B) arose in attempts to extend Lachlan’s decision procedure for the αεtheory of ℰ^{*}, the lattice of r.e. sets modulo finite sets, and Soare’s theorem thatA andB are automorphic if their lattice of supersets ℒ^{*}(A) and ℒ^{*}(B) are isomorphic finite Boolean algebras. We characterize the r.e. setsA with someB⊂_{rm} A as those with a Δ_{3} function that for each recursiveR _{ i } specifiesR _{ i } or \(\bar R_i \) as infinite on \(\bar A\) and to be preferred in the construction ofB. There are r.e.A andB with ℒ^{*}(A) and ℒ^{*}(B) isomorphic to the atomless Boolean algebra such thatA has anrm subset andB does not. Thus 〈ℰ^{*},A〉 and 〈ℰ^{*},B〉 are not even elementarily equivalent. In every nonzero r.e. degree there are r.e. sets with and withoutrm subsets. However the classF of degrees of simple sets with norm subsets satisfies \(H_1 \subseteq F \subseteq \bar L_2 \) .
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 Title
 rMaximal major subsets
 Journal

Israel Journal of Mathematics
Volume 31, Issue 1 , pp 118
 Cover Date
 19780301
 DOI
 10.1007/BF02761377
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
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 Authors

 Manuel Lerman ^{(1)}
 Richard A. Shore ^{(1)}
 Robert I. Soare ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Cornell University, 14853, Ithaca, NY, USA