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On d-finiteness in continuous structures

Itaï Ben Yaacov, Alexander Usvyatsov (2007)

Fundamenta Mathematicae

We observe that certain classical results of first order model theory fail in the context of continuous first order logic. We argue that this happens since finite tuples in a continuous structure may behave as infinite tuples in classical model theory. The notion of a d-finite tuple attempts to capture some aspects of the classical finite tuple behaviour. We show that many classical results involving finite tuples are valid in continuous logic upon replacing "finite" with "d-finite". Other results,...

On many-sorted ω-categorical theories

Enrique Casanovas, Rodrigo Peláez, Martin Ziegler (2011)

Fundamenta Mathematicae

We prove that every many-sorted ω-categorical theory is completely interpretable in a one-sorted ω-categorical theory. As an application, we give a short proof of the existence of non-G-compact ω-categorical theories.

On NIP and invariant measures

Ehud Hrushovski, Anand Pillay (2011)

Journal of the European Mathematical Society

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = tp ( b / A ) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd ( A ) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G 000 = G 00 for G definably amenable,...

On sets with rank one in simple homogeneous structures

Ove Ahlman, Vera Koponen (2015)

Fundamenta Mathematicae

We study definable sets D of SU-rank 1 in e q , where ℳ is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a ’canonically embedded structure’, which inherits all relations on D which are definable in e q , and has no other definable relations. Our results imply that if no relation symbol of the language of ℳ has arity higher than 2, then there is a close relationship between triviality of dependence and being a reduct of a binary...

On special partial types and weak canonical bases in simple theories

Ziv Shami (2013)

Fundamenta Mathematicae

We define the notion of a weak canonical base for a partial type in a simple theory. We prove that members of a certain family of partial types, which we call special partial types, admit a weak canonical base; this family properly contains the family of amalgamation bases.

On the Cantor-Bendixson rank of metabelian groups

Yves Cornulier (2011)

Annales de l’institut Fourier

We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence ( G n ) of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rank  ω n .

On the number of countable models of stable theories

Predrag Tanović (2001)

Fundamenta Mathematicae

We prove: Theorem. If T is a countable, complete, stable, first-order theory having an infinite set of constants with different interpretations, then I(T,ℵ₀) ≥ ℵ₀.

On variants of CM-triviality

Thomas Blossier, Amador Martin-Pizarro, Frank O. Wagner (2012)

Fundamenta Mathematicae

We introduce a generalisation 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 generalises 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.

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