r-Maximal major subsets

Abstract

The question of which r.e. setsA possess major subsetsB which are alsor-maximal inA (A⊂_rm B) arose in attempts to extend Lachlan’s decision procedure for the αε-theory 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 Δ₃ 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 non-zero r.e. degree there are r.e. sets with and withoutrm subsets. However the classF of degrees of simple sets with norm subsets satisfies$H_1 \subseteq F \subseteq \bar L_2$.