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Nonconstructivizability of the reduced part of a strongly constructive torsion-free Abelian group. (English. Russian original) Zbl 0601.20051
Algebra Logic 24, 69-76 (1985); translation from Algebra Logika 24, No. 1, 108-118 (1985).
S. S. Goncharov presented, at the 5th All-Union (USSR) Conference on Mathematical Logic, the following open problem: is the reduced part of a constructive, torsion-free abelian group constructivizable? In the present paper, the authors give a negative answer to this problem.
An abelian group A is called strongly constructive (constructive), if the fundamental set of the group A is the set $$\omega$$ of all natural numbers, and there exists an algorithm, which for any (atomic) formula $$\phi (x_ 0,...,x_{s-1})$$ of the restricted predicate calculus of the group signature, and any numbers $$n_ 0,...,n_{s-1}\in \omega$$, defines whether or not $$\phi (n_ 0,...,n_{s-1})$$ is true in A. The group A is called (strongly) constructivizable, if it is isomorphic to some (strongly) constructive group. The authors prove that there exists a strongly constructivizable torsion-free abelian group whose reduced part is nonconstructivizable (Corollary 2).
Reviewer: K.Honda

##### MSC:
 20K20 Torsion-free groups, infinite rank 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 03D20 Recursive functions and relations, subrecursive hierarchies
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##### References:
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