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Algebra i logika, 2022, Volume 61, Number 3, Pages 308–333
DOI: https://doi.org/10.33048/alglog.2022.61.303
(Mi al2712)
 

Divisible rigid groups. Morley rank

N. S. Romanovskiiab

a Novosibirsk State University
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
References:
Abstract: Let $G$ be a countable saturated model of the theory $\mathfrak{T}_m$ of divisible $m$-rigid groups. Fix the splitting $G_1G_2\ldots G_m$ of a group $G$ into a semidirect product of Abelian groups. With each tuple $(n_1,\ldots,n_m)$ of nonnegative integers we associate an ordinal
$$\alpha=\omega^{m-1}n_m+\ldots+\omega n_2+n_1$$
and denote by $G^{(\alpha)}$ the set $G_1^{n_1}\times G_2^{n_2}\times\ldots\times G_m^{n_m}$, which is definable over $G$ in $G^{n_1+\ldots+n_m}$. Then the Morley rank of $G^{(\alpha)}$ with respect to $G$ is equal to $\alpha$. This implies that
$${\rm RM} (G)=\omega^{m-1}+\omega^{m-2}+\ldots+1.$$
Keywords: divisible $m$-rigid group, Morley rank.
Funding agency Grant number
Russian Science Foundation 19-11-00039
Received: 20.05.2022
Revised: 28.10.2022
Document Type: Article
UDC: 512.5:510.6
Language: Russian
Citation: N. S. Romanovskii, “Divisible rigid groups. Morley rank”, Algebra Logika, 61:3 (2022), 308–333
Citation in format AMSBIB
\Bibitem{Rom22}
\by N.~S.~Romanovskii
\paper Divisible rigid groups. Morley rank
\jour Algebra Logika
\yr 2022
\vol 61
\issue 3
\pages 308--333
\mathnet{http://mi.mathnet.ru/al2712}
\crossref{https://doi.org/10.33048/alglog.2022.61.303}
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