Abelian p-groups and the Halting problem

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