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Zbl 0555.20034
Schmitt, Peter H.
Undecidable theories of valuated Abelian groups. (English)
[J] Mém. Soc. Math. Fr., Nouv. Sér. 16, 67-76 (1984). ISSN 0249-633X

Although written in the context of model theory, this paper should be of interest to abelian group theorists, since the main results state that the theory of finitely valued p-valuated groups whose underlying group is the countable direct sum of copies of ${\bbfZ}(p\sp 9)$ is hereditarily undecidable, and so is the theory of countable valued p-valuated countable p-local torsion-free abelian groups. One consequence of these results is that it is probably hopeless to seek a complete classification of these classes. \par The proof is based on the fact that the theory of countable $p\sp 9$- bounded groups with a distinguished subgroup is reducible to the theory of valuated groups; but the former theory was proved hereditarily undecidable by {\it W. Baur} [Proc. Am. Math. Soc. 55, 125-128 (1976; Zbl 0328.02032)].
[Ph.Schultz]

MSC 2000:: *20K25 Direct sums, etc. (abelian groups)
20A15 Appl. of logic to group theory
03C65 Models of other mathematical theories
20K20 Torsion free abelian groups, infinite rank

Keywords: finitely valued p-valuated groups; hereditarily undecidable; countable valued p-valuated countable p-local torsion-free abelian groups

Citations: Zbl 0328.02032

Cited in: Zbl 0604.03012

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