Automorphisms of the lattice of recursively enumerable sets. Part II: Low sets

Let S denote the lattice of recursively enumerable (r.e.) sets under inclusion, and let #* denote the quotient lattice of S modulo the ideal 3F of finite sets. For A e ê let A* denote the equivalence class in <ƒ* which contains A. An r.e. set A is maximal if A* is a coatom (maximal element) of <f *. Let Aut ê (Aut <f *) denote the group of automorphisms of ê (<ƒ*). We prove that, for any two maximal sets A and B, there exists O e Aut ê such that 0 ( ^ ) = J 5 . It follows that for each k^l the group Aut ê* is &-ply transitive on its coatoms. This demonstrates much more uniformity of structure of ê than was supposed, and answers a question of Martin and Lachlan [1, p. 36]. We also use automorphisms to relate the structure of an r.e. set to its degree, particularly for degrees d which are high (d'=0") or low (i/ '=0'), and as corollaries we answer questions and extend results of Lachlan, Martin, Sacks, Yates, and others. The proofs involve infinite-injury priority arguments like those of Sacks [11], [12], and [13], but here an altogether different method is needed to resolve conflicts between opposing requirements. The numbering of results in §1 and §2 corresponds to that of [15] where full proofs will appear. The results in §3 Will appear in [16] and [17].