On the orbits of hyperhypersimple sets

This paper contributes to the question of under which conditions recursively enumerable sets with isomorphic lattices of recursively enumerable-supersets are automorphic in the lattice of all recursively enumerable sets. We show that hyperhypersimple sets (i.e. sets where the recursively enumerable supersets form a Boolean algebra) are automorphic if there is a Z-definable isomorphism between their lattices of supersets. Lerman, Shore and Soare have shown that this is not true if one replaces Z3 by Z2. ?