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Degrees of unsolvability of first order decision problems for finitely presented groups

Author(s): Oleg V. Belegradek
Journal: Proc. Amer. Math. Soc. 124 (1996), 623-625.
MSC (1991): Primary 03D40, 03D30, 20F10, 20F18
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Abstract: We show that for any arithmetical $m$-degree $\mathbf{d}$ there is a first order decision problem $\mathbf{P}$ such that $\mathbf{P}$ has $m$-degree $\mathbf{d}$ for the free 2-step nilpotent group of rank 2. This implies a conjecture of Sacerdote.

References:

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O. V. Belegradek, The Mal$^\prime $cev correspondence revisited, Proc. Int. Conf. on Algebra Dedicated to the Memory of A. I. Mal$^\prime $cev (L. A. Bokut$^\prime $, Yu. L. Ershov, and A. I. Kostrikin, eds.), Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. (37--59), MR 93m:03059.

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A. I. Mal$^\prime $tsev, A correspondence between rings and groups, Mat. Sb. (N.S.) 50 (1960), 257--
266; English transl., The metamathematics of algebraic systems. Collected papers: 1936--1967, North-Holland, 1971, pp. 124--137.

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G. S. Sacerdote, Some unsolvable decision problems in group theory, Proc. Amer. Math. Soc. 36 (1972), 231--238, MR 47:8660.

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Additional Information:

Oleg V. Belegradek
Affiliation: Kemerovo State University, Kemerovo 650043, Russia
Email: beleg@kaskad.kemerovo.su

DOI: 10.1090/S0002-9939-96-03209-1
PII: S 0002-9939(96)03209-1
Keywords: First order decision problem, $m$-degree
Received by editor(s): August 19, 1994
Additional Notes: The author was partially supported by the AMS fSU Aid Fund.
Communicated by: Andreas R. Blass
Copyright of article: Copyright 1996, American Mathematical Society

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