r-Maximal major subsets
- Cite this article as:
- Lerman, M., Shore, R.A. & Soare, R.I. Israel J. Math. (1978) 31: 1. doi:10.1007/BF02761377
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The question of which r.e. setsA possess major subsetsB which are alsor-maximal inA (A⊂rmB) 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⊂rmA as those with a Δ3 function that for each recursiveRi specifiesRi 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\(H_1 \subseteq F \subseteq \bar L_2 \).