On Variants of CM-triviality

We introduce a generalization of CM-triviality relative to a fixed invariant collection of partial types, in analogy to the Canonical Base Property defined by Pillay, Ziegler and Chatzidakis which generalizes one-basedness. We show that, under this condition, a stable field is internal to the family, and a group of finite Lascar rank has a normal nilpotent subgroup such that the quotient is almost internal to the family.

[1]  F. Campana,et al.  Algébricité et compacité dans l'espace des cycles d'un espace analytique complexe , 1980 .

[2]  Frank O. Wagner,et al.  Ample thoughts , 2013, J. Symb. Log..

[3]  Andreas Baudisch.  On Superstable Groups , 1989, Easter Conference on Model Theory.

[4]  Andreas Baudisch,et al.  A new uncountably categorical group , 1996 .

[5]  Zo'e Chatzidakis,et al.  A note on canonical bases and one-based types in supersimple theories , 2011, 1201.0134.

[6]  Ehud Hrushovski,et al.  A New Strongly Minimal Set , 1993, Ann. Pure Appl. Log..

[7]  Martin Ziegler,et al.  Jet spaces of varieties over differential and difference fields , 2003 .

[8]  田中 克己,et al.  ω-stable groups , 1988 .

[9]  Anand Pillay,et al.  Weakly normal groups , 1985, Logic Colloquium.

[10]  Anand Pillay,et al.  The geometry of forking and groups of finite Morley rank , 1995, Journal of Symbolic Logic.

[11]  Frank O. Wagner.  CM-Triviality and Stable Groups , 1998, J. Symb. Log..

[12]  Akira FUJIKI,et al.  On the douady space of a compact complex space in the category , 1982, Nagoya Mathematical Journal.

[13]  P. Dangerfield.  Logic , 1996, Aristotle and the Stoics.

[14]  Rahim Moosa,et al.  On canonical bases and internality criteria , 2007, 0705.4078.

[15]  Anand Pillay.  A Note on CM-Triviality and The Geometry of Forking , 2000, J. Symb. Log..

[16]  Anand Pillay,et al.  Quantifier elimination for algebraic $D$-groups , 2005 .

[17]  Ehud Hrushovski,et al.  On the canonical base property , 2012, 1205.5981.

[18]  Andreas Baudisch,et al.  A Free Pseudospace , 2000, J. Symb. Log..

[19]  Akira Fujiki.  On the Douady space of a compact complex space in the category ${\cal C}$ , 1982 .

[20]  M. Ziegler,et al.  Ample Hierarchy , 2012, 1210.2552.