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Divisible rigid groups. II. Stability, saturation, and elementary submodels
A. G. Myasnikova,
N. S. Romanovskiibc a Schaefer School of Engineering and Science, Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken NJ 07030-5991, USA
b Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
c Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090 Russia
Abstract:
A group
$G$ is said to be rigid if it contains a normal series
$$
G=G_1>G_2>\dots>G_m>G_{m+1}=1,
$$
whose quotients
$G_i/G_{i+1}$ are Abelian and, treated as right
$\mathbb Z[G/G_i]$-modules, are torsion-free. A rigid group
$G$ is divisible if elements of the quotient
$G_i/G_{i+1}$ are divisible by nonzero elements of the ring
$\mathbb Z[G/G_i]$. Every rigid group is embedded in a divisible one.
Previously, it was stated that the theory
$\mathfrak T_m$ of divisible
$m$-rigid groups is complete. Here, it is proved that this theory is
$\omega$-stable. Furthermore, we describe saturated models, study elementary submodels of an arbitrary model, and find a representation for a countable saturated model in the form of a limit group in the Fraïssé system of all finitely generated
$m$-rigid groups. Also, it is proved that the theory
$\mathfrak T_m$ admits quantifier elimination down to a Boolean combination of
$\forall\exists$-formulas.
Keywords:
divisible rigid group, theory, model, stability, saturation, $\forall\exists$-formula.
UDC:
512.5+
510.6 Received: 10.08.2017
Revised: 19.12.2017
DOI:
10.17377/alglog.2018.57.103
Fulltext:
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