Israel Journal of Mathematics

, Volume 31, Issue 1, pp 1–18

r-Maximal major subsets

  • Manuel Lerman
  • Richard A. Shore
  • Robert I. Soare
Article

DOI: 10.1007/BF02761377

Cite this article as:
Lerman, M., Shore, R.A. & Soare, R.I. Israel J. Math. (1978) 31: 1. doi:10.1007/BF02761377

Abstract

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

Copyright information

© Hebrew University 1978

Authors and Affiliations

  • Manuel Lerman
    • 1
  • Richard A. Shore
    • 1
  • Robert I. Soare
    • 1
  1. 1.Department of MathematicsCornell UniversityIthacaUSA

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