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First-order definability and algebraicity of the sets of annihilating and generating collections of elements for some relatively free solvable groups

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Abstract

We study the first-order definable, Diophantine, and algebraic subsets in the set of all ordered sets generating a group or generating a group as a normal subgroup for some relatively free solvable groups.

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References

  1. Mal’tsev A. I., “On the equation zxyx −1 y −1 z −1 = aba −1 b −1 in a free group,” Algebra i Logika, 1, No. 5, 45–50 (1962).

    MathSciNet  MATH  Google Scholar 

  2. Baumslag G., Myasnikov A. G., and Remeslennikov V. N., “Algebraic geometry over groups. I. Algebraic sets and ideal theory,” J. Algebra, 219, No. 1, 16–79 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  3. Roman’kov V. A., “Test elements for free solvable groups of rank 2,” Algebra and Logic, 40, No. 2, 106–111 (2001).

    Article  MathSciNet  Google Scholar 

  4. Remeslennikov V. N. and Sokolov V. G., “Some properties of Magnus embeddings,” Algebra i Logika, 9, No. 5, 566–578 (1970).

    MATH  MathSciNet  Google Scholar 

  5. Neumann H., Varieties of Groups [Russian translation], Mir, Moscow (1969).

    Google Scholar 

  6. Rhemtulla A. and Akhavan-Malayeri M., “Commutator length of abelian-by-nilpotent groups,” Glasgow Math. Z., 40, No. 1, 117–121 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  7. Timoshenko E. I., “On universal theories of metabelian groups and the Shmel’kin embedding,” Siberian Math. J., 42, No. 5, 981–986 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  8. Mal’tsev A. I., “On free solvable groups,” Dokl. Akad. Nauk SSSR, 130, No. 3, 495–498 (1960).

    Google Scholar 

  9. Ershov Yu. L., Decidability Problems and Constructive Models [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  10. Penzin Yu. G., “Undecidability of the integer theory with addition and the coprimality predicate, ” in: Abstracts: 3 All-Union Conference on Mathematical Logic [in Russian], Novosibirsk, 1974, pp. 149–153.

  11. Bachmuth S., “Automorphisms of free metabelian groups,” Trans. Amer. Math. Soc., 118, 93–104 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  12. Gupta C. K. and Timoshenko E. I., “Automorphic and endomorphic reducibility and primitive endomorphisms of free metabelian groups,” Comm. Algebra, 25, 3057–3070 (1997).

    MATH  MathSciNet  Google Scholar 

  13. Gupta Ch. K. and Timoshenko E. I., “Test rank for some free polynilpotent groups,” Algebra and Logic, 42, No. 1, 20–28 (2003).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Manitoba University, Winnipeg, Canada

    Ch. K. Gupta

  2. Novosibirsk State University of Architecture and Building, Novosibirsk, Russia

    E. I. Timoshenko

Authors
  1. Ch. K. Gupta
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  2. E. I. Timoshenko
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Additional information

Original Russian Text Copyright © 2006 Gupta Ch. K. and Timoshenko E. I

The authors were supported by the Russian Foundation for Basic Research (Grant 05.01.00292) and the Scientific Program “Basic Research in Higher School: Universities of Russia” of the Ministry for Education of the Russian Federation (Grant UR.04.01.031).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 4, pp. 769–779, July–August, 2006.

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Gupta, C.K., Timoshenko, E.I. First-order definability and algebraicity of the sets of annihilating and generating collections of elements for some relatively free solvable groups. Sib Math J 47, 634–642 (2006). https://doi.org/10.1007/s11202-006-0074-x

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  • DOI: https://doi.org/10.1007/s11202-006-0074-x

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