Non-minimalité de la clôture différentielle I : la preuve de S. Shelah
Non-minimalité de la clôture différentielle II : la preuve de M. Rosenlicht
Non-multidimensional theories without groups
On d-finiteness in continuous structures
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 expansions and extensions of powerful digraphs.
On many-sorted ω-categorical theories
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
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 does not fork over then the Lascar strong type of over coincides with the compact strong type of over and any global nonforking extension of is Borel definable over , (ii) analogous statements for Keisler measures and definable groups, including the fact that for definably amenable,...
On nonstructure of elementary submodels of a stable homogeneous structure
We assume that M is a stable homogeneous model of large cardinality. We prove a nonstructure theorem for (slightly saturated) elementary submodels of M, assuming M has dop. We do not assume that th(M) is stable.
On sets with rank one in simple homogeneous structures
We study definable sets D of SU-rank 1 in , 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 , 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
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
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 of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rank .
On the congruence lattice of stable algebras with definability of compact congruences
On the independence property.
On the number of countable models of stable theories
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
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.
P-NDOP and P-decompositions of -saturated models of superstable theories
Given a complete, superstable theory, we distinguish a class P of regular types, typically closed under automorphisms of ℭ and non-orthogonality. We define the notion of P-NDOP, which is a weakening of NDOP. For superstable theories with P-NDOP, we prove the existence of P-decompositions and derive an analog of the first author's result in Israel J. Math. 140 (2004). In this context, we also find a sufficient condition on P-decompositions that implies non-isomorphic models. For this, we investigate...
Powerful digraphs.
ℳ-rank and meager groups
Assume p* is a meager type in a superstable theory T. We investigate definability properties of p*-closure. We prove that if T has countable models then the multiplicity rank ℳ of every type p is finite. We improve Saffe’s conjecture.
ℳ-rank and meager types
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.
Relation entre le rang U et le poids