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Algebra Logika, 2022 Volume 61, Number 3, Pages 308–333 (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

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.

UDC: 512.5:510.6

Received: 20.05.2022
Revised: 28.10.2022

DOI: 10.33048/alglog.2022.61.303



© Steklov Math. Inst. of RAS, 2024