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**On Lascar rank and Morley rank of definable groups in differentially closed fields.**
*(English)*
Zbl 1018.03034

J. Symb. Log. 67, No. 3, 1189-1196 (2002); corrigendum ibid. 74, No. 4, 1436-1437 (2009).

The authors prove that in an \(\omega\)-stable theory, if Morley and Lascar ranks coincide for any type of monomial Morley rank, then Morley and Lascar ranks coincide for definable (imaginary) groups. The hypothesis is in particular satisfied for differentially closed fields with several commuting derivations.

(Note that in Fact 2, there is a typo: The last inequality should be strict.)

In a first appendix the authors give an example of an \(\omega\)-stable group of different Morley and Lascar ranks. Note that the Morley rank of \(R^*+\cdots+R^*\) is \(\omega\cdot n\), and not \(\omega^n\) as stated. Similarly \(\text{RM}(G^*)=\omega^2\), and not \(\omega^\omega\).

In the second appendix they prove that in a non-multidimensional theory where all dimensions are associated to a strongly minimal set, Morley rank equals Lascar rank, and the rank is definable. In particular, the Morley degree is bounded. This was previously shown by B. Poizat [Groupes stables, Nur Al-Mantiq Wal-Ma’rifah, Villeurbanne (1987; Zbl 0633.03019), p. 50], and restated by the reviewer [Stable groups, London Mathematical Society Lecture Note Series 240, Cambridge University Press, Cambridge (1997; Zbl 0897.03037), Proposition 4.7.10].

(Note that in Fact 2, there is a typo: The last inequality should be strict.)

In a first appendix the authors give an example of an \(\omega\)-stable group of different Morley and Lascar ranks. Note that the Morley rank of \(R^*+\cdots+R^*\) is \(\omega\cdot n\), and not \(\omega^n\) as stated. Similarly \(\text{RM}(G^*)=\omega^2\), and not \(\omega^\omega\).

In the second appendix they prove that in a non-multidimensional theory where all dimensions are associated to a strongly minimal set, Morley rank equals Lascar rank, and the rank is definable. In particular, the Morley degree is bounded. This was previously shown by B. Poizat [Groupes stables, Nur Al-Mantiq Wal-Ma’rifah, Villeurbanne (1987; Zbl 0633.03019), p. 50], and restated by the reviewer [Stable groups, London Mathematical Society Lecture Note Series 240, Cambridge University Press, Cambridge (1997; Zbl 0897.03037), Proposition 4.7.10].

Reviewer: Frank Wagner (Villeurbanne)

### MSC:

03C60 | Model-theoretic algebra |

03C45 | Classification theory, stability, and related concepts in model theory |

### Keywords:

Lascar rank; Morley rank; definable group; differentially closed field; generic; \(\omega\)-stable theory
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\textit{A. Pillay} and \textit{W. Y. Pong}, J. Symb. Log. 67, No. 3, 1189--1196 (2002; Zbl 1018.03034)

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### References:

[1] | Some applications of ordinal dimensions to the theory of differentially closedfields 65 pp 347– (2000) |

[2] | Geometric stability theory 32 (1996) |

[3] | DOI: 10.1016/0168-0072(86)90035-7 · Zbl 0599.03034 · doi:10.1016/0168-0072(86)90035-7 |

[4] | Transactions of the American Mathematical Society 292 pp 451– (1985) |

[5] | Lascar and Morley ranks differ in differentially closed fields 64 pp 1280– (1999) |

[6] | The Model theory of differential fields with finitely many commuting derivations 65 pp 885– (2000) |

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