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Algebra Logika, 2010, Volume 49, Number 5, Pages 654–669 (Mi al459)  

This article is cited in 5 scientific papers (total in 5 papers)

Equations and algebraic geometry over profinite groups

S. G. Melesheva

Novosibirsk State Univ., Novosibirsk, Russia

Abstract: The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved that if the set $\pi(G)$ of prime divisors of the profinite period of a group $G$ is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups, the finiteness of a set $\pi(G)$ gives rise to equational Noetherianness. The concept of a standard linear pro-$p$-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent pro-$p$-groups and free metabelian pro-$p$-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-$p$-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined. For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language of universal theories and the notion of discriminability.

Keywords: profinite group, equation, equationally Noetherian group, universal theory over profinite group.

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English version:
Algebra and Logic, 2010, 49:5, 444–455

Bibliographic databases:

UDC: 512.542
Received: 15.11.2009
Revised: 30.05.2010

Citation: S. G. Melesheva, “Equations and algebraic geometry over profinite groups”, Algebra Logika, 49:5 (2010), 654–669; Algebra and Logic, 49:5 (2010), 444–455

Citation in format AMSBIB
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\paper Equations and algebraic geometry over profinite groups
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\vol 49
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\pages 654--669
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\jour Algebra and Logic
\yr 2010
\vol 49
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\pages 444--455
\crossref{https://doi.org/10.1007/s10469-010-9108-3}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. G. Afanas'eva, N. S. Romanovskii, “Rigid metabelian pro-$p$-groups”, Algebra and Logic, 53:2 (2014), 102–113  mathnet  crossref  mathscinet  isi
    2. 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
    3. 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
    4. N. S. Romanovskii, “Partially divisible completions of rigid metabelian pro-$p$-groups”, Algebra and Logic, 55:5 (2016), 376–386  mathnet  crossref  crossref  isi
    5. Casals-Ruiz M., Kazachkov I., Remeslennikov V., “Pro-Hall R-Groups and Groups Discriminated By the Free Pro-P Group”, J. Group Theory, 19:3 (2016), 391–403  crossref  mathscinet  zmath  isi  scopus
  • Алгебра и логика Algebra and Logic
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