Abelian p-groups and the Halting problem

Abstract We investigate which effectively presented abelian p-groups are isomorphic relative to the halting problem. The standard approach to this and similar questions uses the notion of Δ 2 0 -categoricity (to be defined). We partially reduce the description of Δ 2 0 -categorical p-groups of Ulm type 1 to the analogous problem for equivalence structures. Using this reduction, we solve a problem left open in [5] . For the sake of the reduction mentioned above, we introduce a new notion of effective Δ 2 0 -categoricity that lies strictly in-between plain Δ 2 0 -categoricity and relative Δ 2 0 -categoricity (to be defined). We then reduce the problem of classifying effective Δ 2 0 -categoricity to a question stated in terms of Σ 2 0 -sets. Among other results, we show that for c.e. Turing degrees bounding such sets is equal to being complete.

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