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Algebra Logika, 2005, Volume 44, Number 5, Pages 601–621 (Mi al133)  

This article is cited in 12 scientific papers (total in 12 papers)

Irreducible Algebraic Sets in Metabelian Groups

V. N. Remeslennikov, N. S. Romanovskiia

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We present the construction for a $u$-product $G_1\circ G_2$ of two $u$-groups $G_1$ and $G_2$, and prove that $G_1\circ G_2$ is also a $u$-group and that every $u$-group, which contains $G_1$ and $G_2$ as subgroups and is generated by these, is a homomorphic image of $G_1\circ G_2$. It is stated that if $G$ is a $u$-group then the coordinate group of an affine space $G^n$ is equal to $G \circ F_n$, where $F_n$ is a free metabelian group of rank $n$. Irreducible algebraic sets in $G$ are treated for the case where $G$ is a free metabelian group or wreath product of two free Abelian groups of finite ranks.

Keywords: $u$-group, $u$-product, coordinate group of an affine space, free metabelian group, free Abelian group

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English version:
Algebra and Logic, 2005, 44:5, 336–347

Bibliographic databases:

UDC: 512.5
Received: 23.02.2005

Citation: V. N. Remeslennikov, N. S. Romanovskii, “Irreducible Algebraic Sets in Metabelian Groups”, Algebra Logika, 44:5 (2005), 601–621; Algebra and Logic, 44:5 (2005), 336–347

Citation in format AMSBIB
\by V.~N.~Remeslennikov, N.~S.~Romanovskii
\paper Irreducible Algebraic Sets in Metabelian Groups
\jour Algebra Logika
\yr 2005
\vol 44
\issue 5
\pages 601--621
\jour Algebra and Logic
\yr 2005
\vol 44
\issue 5
\pages 336--347

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    This publication is cited in the following articles:
    1. N. S. Romanovskii, “Algebraic sets in metabelian groups”, Algebra and Logic, 46:4 (2007), 274–280  mathnet  crossref  mathscinet  zmath  isi
    2. M. G. Amaglobeli, “Algebraic sets and coordinate groups for a free nilpotent group of nilpotency class 2”, Siberian Math. J., 48:1 (2007), 3–7  mathnet  crossref  mathscinet  zmath  isi
    3. Daniyarova E., Myasnikov A., Remeslennikov V., “Unification theorems in algebraic geometry”, Aspects of Infinite Groups, Algebra and Discrete Mathematics, 1, 2008, 80–111  mathscinet  zmath  isi
    4. N. S. Romanovskii, “Irreducible algebraic sets over divisible decomposed rigid groups”, Algebra and Logic, 48:6 (2009), 449–464  mathnet  crossref  mathscinet  zmath  isi
    5. Myasnikov A., Romanovskiy N., “Krull dimension of solvable groups”, J. Algebra, 324:10 (2010), 2814–2831  crossref  mathscinet  zmath  isi  elib  scopus
    6. N. S. Romanovskii, “Coproducts of rigid groups”, Algebra and Logic, 49:6 (2010), 539–550  mathnet  crossref  mathscinet  isi
    7. E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Algebraic geometry over algebraic structures. II. Foundations”, J. Math. Sci., 185:3 (2012), 389–416  mathnet  crossref
    8. N. S. Romanovskii, “Irreducibility of an affine space in algebraic geometry over a group”, Algebra and Logic, 52:3 (2013), 262–265  mathnet  crossref  mathscinet  isi
    9. S. G. Afanas'eva, “The coordinate group of an affine space over a rigid metabelian pro-$p$-group”, Algebra and Logic, 53:3 (2014), 187–190  mathnet  crossref  mathscinet  isi
    10. N. S. Romanovskii, “Algebraic sets in a finitely generated rigid $2$-step solvable pro-$p$-group”, Algebra and Logic, 54:6 (2016), 478–488  mathnet  crossref  crossref  mathscinet  isi
    11. Lysenok I., Ushakov A., “Spherical Quadratic Equations in Free Metabelian Groups”, Proc. Amer. Math. Soc., 144:4 (2016), 1383–1390  crossref  mathscinet  zmath  isi  elib  scopus
    12. S. G. Afanas'eva, “Algebraic sets in a divisible $2$-rigid group”, Siberian Math. J., 59:2 (2018), 202–206  mathnet  crossref  crossref  isi  elib
  • Алгебра и логика Algebra and Logic
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