Divisible rigid groups. Morley rank

N. S. Romanovskii

Algebra i logika, Tome 61 (2022) no. 3, pp. 308-333

Résumé

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 $${\rm RM} (G)=\omega^{m-1}+\omega^{m-2}+\ldots+1.$$

Détail

Mots-clés : divisible $m$-rigid group
Keywords: Morley rank.