Corrigendum: Correction to "Undecidability of L(Finfty) and Other Lattices of r.e. Substructures"

In [l], the author gave a proof that a wide class of lattices of r.e. substructures had undecidable first-order theories. The proof consisted of two parts. First an r.e. set A was constructed so that A had the following two properties: (i) A had the lifting property (see below). (ii) A had an r.e. superset B such that the lattice L*(A, B) of r.e. sets containing A and contained in B (modulo finite sets) was effectively isomorphic to $*, the lattice of r.e. sets modulo finite sets. The second part of the proof was a purely algebraic argument that showed such sets A and B could be used to code enough of 8* into the relevant lattices to establish the undecidability of these lattices. Unfortunately the proof (in [l]) of the first part contained an error. Specifically the set A constructed in [l] fails to have property (ii) for all r.e. supersets B of A. The error occurs in Lemma 3.5 where the proof only shows that there is an embedding of 8* into L*(A, B). We repair this flaw here. That is, we construct an r.e. set A with properties (i) and (ii) above (with B = CO). We recall the following definition from [l].