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Back and forth relations for reduced abelian \(p\)-groups. (English) Zbl 0842.03029
The known general results on \(\Delta^0_\alpha\)-categoricity, \(\Delta^0_\alpha\)-stability, as well as those on interplay between definability of a relation by some effective formulas and their properties in recursive structures [C. J. Ash, Trans. Am. Math. Soc. 298, 497-514 (1986; Zbl 0631.03017); C. J. Ash, Ann. Pure Appl. Logic 34, 1-14 (1987; Zbl 0617.03016); C. J. Ash and J. F. Knight, Ann. Pure Appl. Logic 46, 211-234 (1990; Zbl 0712.03020); E. Barker, Ann. Pure Appl. Logic 39, 105-130 (1988; Zbl 0651.03034)] are applied in the concrete case of reduced Abelian \(p\)-groups. In particular, the author proves that if \(G\) is a countable reduced Abelian \(p\)-group with a recursive sequence of Ulm invariants, then there is a recursive copy of \(G\) for which the height function is recursive; the set of elements of height \(\geq \omega \alpha\) is proved to be formally \(\Pi^0_{2\alpha}\) and the set of elements of height \(\geq \omega\alpha + n\) is proved to be formally \(\Sigma^0_{2\alpha + 1}\); the same is also proved for elements of power \(p\). The author proves these bounds to be the best ones in natural cases. A series of results on \(\Delta^0_\alpha\)-categoricity, \(\Delta^0_\alpha\)-stability, and pairs of structures are also obtained.

MSC:
03C57 Computable structure theory, computable model theory
03D45 Theory of numerations, effectively presented structures
20K99 Abelian groups
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