Limitwise monotonic sequences and degree spectra of structures

We study effective monotonic approximations of sets and sequences of sets. We show that there is a sequence of sets which has no uniform computable monotonic approximation, but has an xcomputable monotonic approximation, for every hyperimmune degree x. We also construct a Σ2 set which is not limitwise monotonic, but is x-limitwise monotonic relative to every nonzero ∆2 degree x. We show that if a sequence of sets is uniformly limitwise monotonic in x for all, except countably many, degrees x, then it has to be uniformly limitwise monotonic. Finally, we apply these results to investigate degree spectra of abelian groups, equivalence relations, and א1-categorical structures.

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