RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra Logika: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (238 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 512.5

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
\Bibitem{RemRom05} \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 \mathnet{http://mi.mathnet.ru/al133} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2195022} \zmath{https://zbmath.org/?q=an:1104.20028} \transl \jour Algebra and Logic \yr 2005 \vol 44 \issue 5 \pages 336--347 \crossref{https://doi.org/10.1007/s10469-005-0032} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-27544473308} 

• http://mi.mathnet.ru/eng/al133
• http://mi.mathnet.ru/eng/al/v44/i5/p601

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
3. Daniyarova E., Myasnikov A., Remeslennikov V., “Unification theorems in algebraic geometry”, Aspects of Infinite Groups, Algebra and Discrete Mathematics, 1, 2008, 80–111
4. N. S. Romanovskii, “Irreducible algebraic sets over divisible decomposed rigid groups”, Algebra and Logic, 48:6 (2009), 449–464
5. Myasnikov A., Romanovskiy N., “Krull dimension of solvable groups”, J. Algebra, 324:10 (2010), 2814–2831
6. N. S. Romanovskii, “Coproducts of rigid groups”, Algebra and Logic, 49:6 (2010), 539–550
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
8. N. S. Romanovskii, “Irreducibility of an affine space in algebraic geometry over a group”, Algebra and Logic, 52:3 (2013), 262–265
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
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
11. Lysenok I., Ushakov A., “Spherical Quadratic Equations in Free Metabelian Groups”, Proc. Amer. Math. Soc., 144:4 (2016), 1383–1390
12. S. G. Afanas'eva, “Algebraic sets in a divisible $2$-rigid group”, Siberian Math. J., 59:2 (2018), 202–206
•  Number of views: This page: 268 Full text: 72 References: 43 First page: 1