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Algebra Logika, 2011, Volume 50, Number 6, Pages 802–821 (Mi al517)  

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

Universal theories for rigid soluble groups

A. G. Myasnikova, N. S. Romanovskiibc

a Schaefer School of Engineering and Science, Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ, USA
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
c Novosibirsk State University, Novosibirsk, Russia

Abstract: A group is said to be $p$-rigid, where $p$ is a natural number, if it has a normal series of the form
$$ G=G_1>G_2>…>G_p>G_{p+1}=1, $$
whose quotients $G_i/G_{i+1}$ are Abelian and are torsion free when treated as $\mathbb Z[G/G_i]$-modules. Examples of rigid groups are free soluble groups. We point out a recursive system of universal axioms distinguishing $p$-rigid groups in the class of $p$-soluble groups. It is proved that if $F$ is a free $p$-soluble group, $G$ is an arbitrary $p$-rigid group, and $W$ is an iterated wreath product of $p$ infinite cyclic groups, then $\forall$-theories for these groups satisfy the inclusions
$$ \mathcal A(F)\supseteq\mathcal A(G)\supseteq\mathcal A(W). $$
We construct an $\exists$-axiom distinguishing among $p$-rigid groups those that are universally equivalent to $W$. An arbitrary $p$-rigid group embeds in a divisible decomposed $p$-rigid group $M=M(\alpha_ 1,…,\alpha_ p)$. The latter group factors into a semidirect product of Abelian groups $A_1A_2…A_p$, in which case every quotient $M_i/M_{i+1}$ of its rigid series is isomorphic to $A_i$ and is a divisible module of rank $\alpha_i$ over a ring $\mathbb Z[M/M_i]$. We specify a recursive system of axioms distinguishing among $M$-groups those that are Muniversally equivalent to $M$. As a consequence, it is stated that the universal theory of $M$ with constants in $M$ is decidable. By contrast, the universal theory of $W$ with constants is undecidable.

Keywords: $p$-rigid group, universal theory of group, decidable theory.

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English version:
Algebra and Logic, 2012, 50:6, 539–552

Bibliographic databases:

UDC: 512.54.05
Received: 01.03.2011

Citation: A. G. Myasnikov, N. S. Romanovskii, “Universal theories for rigid soluble groups”, Algebra Logika, 50:6 (2011), 802–821; Algebra and Logic, 50:6 (2012), 539–552

Citation in format AMSBIB
\by A.~G.~Myasnikov, N.~S.~Romanovskii
\paper Universal theories for rigid soluble groups
\jour Algebra Logika
\yr 2011
\vol 50
\issue 6
\pages 802--821
\jour Algebra and Logic
\yr 2012
\vol 50
\issue 6
\pages 539--552

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    This publication is cited in the following articles:
    1. N. S. Romanovskii, “Universal theories for free solvable groups”, Algebra and Logic, 51:3 (2012), 259–263  mathnet  crossref  mathscinet  zmath  isi
    2. Romanovskiy N.S., “Presentations for Rigid Solvable Groups”, J. Group Theory, 15:6 (2012), 793–810  crossref  mathscinet  zmath  isi  elib  scopus
    3. S. G. Afanas'eva, N. S. Romanovskii, “Rigid metabelian pro-$p$-groups”, Algebra and Logic, 53:2 (2014), 102–113  mathnet  crossref  mathscinet  isi
    4. D. V. Ovchinnikov, “Automorphisms of divisible rigid groups”, Algebra and Logic, 53:2 (2014), 133–139  mathnet  crossref  mathscinet  isi
    5. Myasnikov A.G. Romanovskii N.S., “Logical Aspects of the Theory of Divisible Rigid Groups”, Dokl. Math., 90:3 (2014), 697–698  crossref  mathscinet  zmath  isi  elib  scopus
    6. 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
    7. N. S. Romanovskii, “Partially divisible completions of rigid metabelian pro-$p$-groups”, Algebra and Logic, 55:5 (2016), 376–386  mathnet  crossref  crossref  isi
    8. E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Algebraic geometry over algebraic structures. VI. Geometric equivalence”, Algebra and Logic, 56:4 (2017), 281–294  mathnet  crossref  crossref  isi
    9. N. S. Romanovskii, “Divisible rigid groups. Algebraic closedness and elementary theory”, Algebra and Logic, 56:5 (2017), 395–408  mathnet  crossref  crossref  isi
    10. A. G. Myasnikov, N. S. Romanovskii, “Divisible rigid groups. II. Stability, saturation, and elementary submodels”, Algebra and Logic, 57:1 (2018), 29–38  mathnet  crossref  crossref  isi
    11. N. S. Romanovskii, “Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination”, Algebra and Logic, 57:6 (2019), 478–489  mathnet  crossref  crossref  isi
    12. Myasnikov A.G., Romanovskii N.S., “Characterization of Finitely Generated Groups By Types”, Int. J. Algebr. Comput., 28:8, SI (2018), 1613–1632  crossref  mathscinet  zmath  isi  scopus
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