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ℳ-rank and meager types

Ludomir Newelski — 1995

Fundamenta Mathematicae

Assume T is superstable and small. Using the multiplicity rank ℳ we find locally modular types in the same manner as U-rank considerations yield regular types. We define local versions of ℳ-rank, which also yield meager types.

ℳ-rank and meager groups

Ludomir Newelski — 1996

Fundamenta Mathematicae

Assume p* is a meager type in a superstable theory T. We investigate definability properties of p*-closure. We prove that if T has < 2 0 countable models then the multiplicity rank ℳ of every type p is finite. We improve Saffe’s conjecture.

m-normal theories

Ludomir Newelski — 2001

Fundamenta Mathematicae

Originally, m-independence, ℳ -rank, m-stability and m-normality were defined only for small stable theories. Here we extend the definitions to an arbitrary small countable complete theory. Then we investigate these notions in the new, broader context. As a consequence we show that any superstable theory with < 2 countable models is m-normal. In particular, any *-algebraic group interpretable in such a theory is abelian-by-finite.

The diameter of a Lascar strong type

Ludomir Newelski — 2003

Fundamenta Mathematicae

We prove that a type-definable Lascar strong type has finite diameter. We also answer some other questions from [1] on Lascar strong types. We give some applications on subgroups of type-definable groups.

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