ISRAEL JOURNAL OF MATHEMATICS,Vol. 31. No. 1, 1978 r-MAXIMAL MAJOR SUBSETS BY MANUEL LERMAN, RICHARD A. SHORE AND ROBERT I. SOARE* ABSTRACT The question of which r.e. sets A possess major subsets B which are also r-maximal in A (A C,~B) arose in attempts to extend Lachlan's decision procedure for the V3-theory of ~g*, the lattice of r.e. sets modulo finite sets, and Soare's theorem that A and B are automorphic if their lattice of supersets -~*(A) and &e*(B) are isomorphic finite Boolean algebras. We characterize the r.e. sets A with some B C,m A as those with a A3 function that for each recursive R~ specifies R, or/~ as infinite on ~, and to be preferred in the construction of B. There are r.e. A and B with .LP*(A) and ~*(B) isomorphic to the atomless Boolean algebra such that A has an rm subset and B does not. Thus (~f*,A) and (~f*, B) are not even elementarily equivalent. In every non-zero r.e. degree there are r.e. sets with and without rm subsets. However the class F of degrees of simple sets with no rm subsets satisfies H1 ___ F C/~z. Introduction This paper grew out of a meeting of two different lines of investigation into the structure of ~f*, the lattice of recursively enumerable sets modulo finite sets. The first was an attempt to extend Lachlan's decision procedure for the V3-theory of ~* [2] by adding on a predicate to distinguish maximal sets. (This should be viewed as a first step towards a decision procedure for higher quantifier levels since they can be reduced to the V3-1evel by adding on the appropriate predicates. This approach is being attempted by Lerman and Soare [8].) As in Lachlan's procedure one begins by trying to rule out as many sentences as possible by considering certain "canonical" configurations of r.e. sets. It turns out that an important new ingredient involves deciding whether or not certain simple sets, in particular whether hyperhypersimple (hhs) sets, have major subsets which are also r-maximal in them. (Warning: All sets and degrees named in this paper will be r.e.) With this convention in mind we define the following notions: B is a major subset of A, written B CmA, iff A - B is infinite and T The authors were partially supported by NSF Grants MCS 76-07258, MCS 77-04013 and MCS 77-01965 respectively. Received September 4, 1977