1,177 research outputs found
External definability and groups in NIP theories
We prove that many properties and invariants of definable groups in NIP
theories, such as definable amenability, G/G^{00}, etc., are preserved when
passing to the theory of the Shelah expansion by externally definable sets,
M^{ext}, of a model M. In the light of these results we continue the study of
the "definable topological dynamics" of groups in NIP theories. In particular
we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in
some new cases, including definably amenable groups in o-minimal structures.Comment: 28 pages. Introduction was expanded and some minor mistakes were
corrected. Journal of the London Mathematical Society, accepte
Definability of groups in -stable metric structures
We prove that in a continuous -stable theory every type-definable
group is definable. The two main ingredients in the proof are:
\begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson
ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to
prove the theorem in case the metric is invariant under the group action; and
\item Results concerning the existence of translation-invariant definable
metrics on type-definable groups and the extension of partial definable metrics
to total ones. \end{enumerate
Dimensional groups and fields
We shall define a general notion of dimension, and study groups and rings
whose interpretable sets carry such a dimensio. In particular, we deduce chain
conditions for groups, definability results for fields and domains, and show
that pseudofinite groups contain big finite-by-abelian subgroups, and
pseudofinite groups of dimension 2 contain big soluble subgroups
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Generic Automorphisms and Green Fields
We show that the generic automorphism is axiomatisable in the green field of
Poizat (once Morleyised) as well as in the bad fields which are obtained by
collapsing this green field to finite Morley rank. As a corollary, we obtain
"bad pseudofinite fields" in characteristic 0. In both cases, we give geometric
axioms. In fact, a general framework is presented allowing this kind of
axiomatisation. We deduce from various constructibility results for algebraic
varieties in characteristic 0 that the green and bad fields fall into this
framework. Finally, we give similar results for other theories obtained by
Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories
having the definable multiplicity property. We also close a gap in the
construction of the bad field, showing that the codes may be chosen to be
families of strongly minimal sets.Comment: Some minor changes; new: a result of the paper (Cor 4.8) closes a gap
in the construction of the bad fiel
G-Compactness and Groups
Lascar described E_KP as a composition of E_L and the topological closure of
EL. We generalize this result to some other pairs of equivalence relations.
Motivated by an attempt to construct a new example of a non-G-compact theory,
we consider the following example. Assume G is a group definable in a structure
M. We define a structure M_0 consisting of M and X as two sorts, where X is an
affine copy of G and in M_0 we have the structure of M and the action of G on
X. We prove that the Lascar group of M_0 is a semi-direct product of the Lascar
group of M and G/G_L. We discuss the relationship between G-compactness of M
and M_0. This example may yield new examples of non-G-compact theories.Comment: 18 page
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