Divisible Rigid Groups. Morley Rank

Abstract

Let G be a countable saturated model of the theory 𝔗_m of divisible m-rigid groups. Fix the splitting G₁G₂ . . .G_m of a group G into a semidirect product of Abelian groups. With each tuple (n₁, . . . , n_m) of nonnegative integers we associate an ordinal α = ω^m−1n_m+ . . . + ωn₂ + n₁ and denote by G^(α) the set \( {G}_1^{n_1}\times {G}_2^{n_2}\times \dots \times {G}_m^{n_m} \), which is definable over G in \( {G}^{n_1+\dots +{n}_m} \). Then the Morley rank of G^(α) with respect to G is equal to α. This implies that RM (G) = ω^m−1 + ω^m−2 + . . . + 1.