, Volume 31, Issue 1, pp 1-18

r-Maximal major subsets

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Abstract

The question of which r.e. setsA possess major subsetsB which are alsor-maximal inA (Arm 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 someBrm 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 \(H_1 \subseteq F \subseteq \bar L_2 \) .

The authors were partially supported by NSF Grants MCS 76-07258, MCS 77-04013 and MCS 77-01965 respectively.