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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:
[1] S. S. Goncharov and V. P. Dobritsa, ”An example of a constructive Abelian group with nonconstructivizable reduced subgroup,” 4th All-Union Conf. on Math. Logic, Kishinev (1976), p. 33. · Zbl 0603.20049
[2] N. G. Khisamiev, ”Strongly constructive Abelian P - groups,” Algebra Logika,22, No. 2, 198–217 (1983). · Zbl 0568.20052
[3] A. I. Mal’tsev, ”Positive and negative enumerations,” Dokl. Akad. Nauk SSSR,160, No. 2, 278–280 (1965).
[4] Yu. L. Ershov, Problems of Solubility and Constructive Models [in Russian], Nauka, Moscow (1980).
[5] Yu. L. Ershov, Enumeration Theory [in Russian], Nauka, Moscow (1977).
[6] M. I. Kargapolov and Yu. I. Merzlyakov, Foundations of Group Theory [in Russian], 3rd ed., Nauka, Moscow (1982). · Zbl 0508.20001
[7] A. I. Mal’tsev, Algorithms and Recursive Functions [in Russian], Nauka, Moscow (1965).
[8] A. G. Kurosch, ”Primitive torsionfreie Abelsche Gruppen vom endliche Range,” Ann. Math.,38, No. 1, 175–203 (1937). · Zbl 0016.01501 · doi:10.2307/1968518
[9] A. I. Mal’tsev, ”Torsion-free Abelian groups of finite rank,” Mat. Sb.,4, No. 1, 45–67 (1938).
[10] V. P. Dobritsa, ”On constructivizable Abelian groups,” Sib. Mat. Zh.,22, No. 3, 208–213 (1981). · Zbl 0473.03037
[11] V. P. Dobritsa, ”Some constructivizations of Abelian groups,” Sib. Mat. Zh.,24, No. 2, 18–25 (1983). · Zbl 0528.20038
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