Divisible rigid groups. Morley rank. (English. Russian original) Zbl 1515.03170
Algebra Logic 61, No. 3, 207-224 (2022); translation from Algebra Logika 61, No. 3, 308-333 (2022).
Summary: 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 \((n_1, \dots, n_m)\) 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 \({G}^{n_1+\dots +{n}_m} \). Then the Morley rank of \(G^{( \alpha )}\) with respect to \(G\) is equal to \(\alpha \). This implies that \(\mathrm{RM} (G) = \omega^{m-1} + \omega^{m -2} + \ldots + 1\).
References:
| [1] | Romanovskii, NS, Divisible rigid groups. Algebraic closedness and elementary theory, Algebra and Logic, 56, 5, 395-408 (2017) · Zbl 1403.20050 · doi:10.1007/s10469-017-9461-6 |
| [2] | Myasnikov, AG; Romanovskii, NS, Divisible rigid groups. II. Stability, saturation, and elementary submodels, Algebra and Logic, 57, 1, 29-38 (2018) · Zbl 1485.03126 · doi:10.1007/s10469-018-9476-7 |
| [3] | Romanovskii, NS, Divisible rigid groups. III. Homogeneity and quantifier elimination, Algebra and Logic, 57, 6, 478-489 (2018) · Zbl 1468.03048 · doi:10.1007/s10469-019-09518-2 |
| [4] | Romanovskii, NS, Divisible rigid groups. IV. Definable subgroups, Algebra and Logic, 59, 3, 237-252 (2020) · Zbl 1484.03066 · doi:10.1007/s10469-020-09596-7 |
| [5] | N. Romanovskii, “Rigid solvable groups. Algebraic geometry and model theory,” in Groups and Model Theory. GAGTA Book 2, O. Kharlampovich et al. (Eds.), De Gruyter, Berlin (2021), pp. 193-229. · Zbl 1528.20050 |
| [6] | W. Hodges, Model Theory, Enc. Math. Appl., 42, Cambridge Univ. Press, Cambridge (1993). · Zbl 0789.03031 |
| [7] | Romanovskii, NS, Equational Noetherianness of rigid soluble groups, Algebra and Logic, 48, 2, 147-160 (2009) · Zbl 1245.20036 · doi:10.1007/s10469-009-9045-1 |
| [8] | Romanovskii, NS, Irreducible algebraic sets over divisible decomposed rigid groups, Algebra and Logic, 48, 6, 449-464 (2009) · Zbl 1245.20054 · doi:10.1007/s10469-009-9071-z |
| [9] | Romanovskii, NS, Irreducibility of an affine space in algebraic geometry over a group, Algebra and Logic, 52, 3, 262-265 (2013) · Zbl 1284.20048 · doi:10.1007/s10469-013-9239-4 |
| [10] | Yu. L. Ershov and E. A. Palyutin, Mathematical Logic [in Russian], 6th edn., Fizmatlit, Moscow (2011). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
