Embedding Computable Infinitary Equivalence nto P-Groups

We examine the relation between the uniformity of a collection of operators witnessingTuring computable embeddings, and the existence of an operator witnessingthe universality of a class. The primary equivalence relation studied here is computableinfinitary Σα equivalence. This project of exploiting uniformity of Turingcomputable embeddings to construct a limit embedding is carried out entirely in thecontext of countable reduced abelian p-groups. One may look at this program as eithera project in the computable structure theory of abelian p-groups, or as a projectin the construction of limits of sequences of uniform Turing computable operators. In an attempt to explore the boundary between computable infinitary Σα equivalenceand isomorphism, we show that for any computable , certain classes of countablereduced abelian p-groups are universal for ∼cα under Turing computable embedding.Further, the operators witnessing these embeddings are extremely uniform. Exploiting the uniformity of the embeddings, we produce operators which are,in some sense, limits of the embeddings witnessing the universality of the classesof countable reduced abelian p-groups. This is approached in three dierent ways:transnite recursion on ordinal notation, Barwise-Kreisel Compactness, and hyperarithemeticalsaturation. Finally, we work in admissible set theory, and use BarwiseCompactness and ΣA-saturation to generalize selected results.