Automorphisms of the Lattice of Recursively Enumerable Sets and Hyperhypersimple Sets

Abstract

This chapter discusses the automorphisms of the lattice of recursively enumerable (r.e.) sets and hyperhypersimple (hh-simple) sets. While Maass proved a sufficient criterion for hh-simple sets to be automorphic, in Herrman, those properties of these sets has been analyzed; and it can be concluded when such sets are not automorphic even if their r.e. superset structures are isomorphic. The Lachlan's construction of hh-simple sets is universal from the point of view of their lattice position. The automorphism analysis of the hh-simple sets is an extensive topic for itself. There are still many open problems and questions. The main problem is to find a necessary and sufficient condition for two hh-simple sets to be automorphic. The isomorphism type of the family P∗I(A) for the hh-simple set A could be such a condition. © 1989, Elsevier Inc. All rights reserved.