Characterization of Recursively Enumerable Sets with Supersets Effectively Isomorphic to all Recursively Enumerable Sets

Abstract

We show that the lattice of supersets of a recursively enumerable (r.e.) set $A$ is effectively isomorphic to the lattice of all r.e. sets if and only if the complement $\bar A$ of $A$ is infinite and $\{ e|{W_e} \cap \bar A\;{\text {finite}}\}\;{\leqslant _{1}}\emptyset ''$ (i.e. $\bar A$ is $\text {semilow}_{1.5}$). It is obvious that the condition “$\bar {A}\; \text {semilow}_{1.5}$” is necessary. For the other direction a certain uniform splitting property (the "outer splitting property") is derived from $\text {semilow}_{1.5}$ and this property is used in an extension of Soare’s automorphism machinery for the construction of the effective isomorphism. Since this automorphism machinery is quite complicated we give a simplified proof of Soare’s Extension Theorem before we add new features to this argument.