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 \$$\backslash$bar\A\\$ of \$A\$ is infinite and \$$\backslash$\e$\backslash$mid W\_e $\backslash$cap $\backslash$bar\A\ finite$\backslash$\ $\backslash$leqslant\_1 $\backslash$varnothing''\$ (i.e. \$$\backslash$bar\A\\$ is \$$\backslash$operatorname\semilow\\_\1.5\\$). It is obvious that the condition "\$$\backslash$bar\A\ $\backslash$operatorname\semilow\_\1.5\\$" is necessary. For the other direction a certain uniform splitting property (the "outer splitting property") is derived from \$$\backslash$operatorname\semilow\\_\1.5\\$ and this property is used in an extension of Soare's automorphism machinery for the construction of the effective isomorphism. Since this automorphis, machinery is quite complicated we give a simplified proof of Soare's Extension Theorem before we add new features to this argument.