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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.
Received: 20.05.2022 Revised: 28.10.2022
Citation:
N. S. Romanovskii, “Divisible rigid groups. Morley rank”, Algebra Logika, 61:3 (2022), 308–333
Linking options:
https://www.mathnet.ru/eng/al2712 https://www.mathnet.ru/eng/al/v61/i3/p308
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Abstract page: | 115 | Full-text PDF : | 41 | References: | 24 |
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