51 research outputs found
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
On properties of (weakly) small groups
A group is small if it has countably many complete -types over the empty
set for each natural number n. More generally, a group is weakly small if
it has countably many complete 1-types over every finite subset of G. We show
here that in a weakly small group, subgroups which are definable with
parameters lying in a finitely generated algebraic closure satisfy the
descending chain conditions for their traces in any finitely generated
algebraic closure. An infinite weakly small group has an infinite abelian
subgroup, which may not be definable. A small nilpotent group is the central
product of a definable divisible group with a definable one of bounded
exponent. In a group with simple theory, any set of pairwise commuting elements
is contained in a definable finite-by-abelian subgroup. First corollary : a
weakly small group with simple theory has an infinite definable
finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal
solvable group A of derived length n is contained in an A-definable almost
solvable group of class n
Interpretations of Presburger Arithmetic in Itself
Presburger arithmetic PrA is the true theory of natural numbers with
addition. We study interpretations of PrA in itself. We prove that all
one-dimensional self-interpretations are definably isomorphic to the identity
self-interpretation. In order to prove the results we show that all linear
orders that are interpretable in (N,+) are scattered orders with the finite
Hausdorff rank and that the ranks are bounded in terms of the dimension of the
respective interpretations. From our result about self-interpretations of PrA
it follows that PrA isn't one-dimensionally interpretable in any of its finite
subtheories. We note that the latter was conjectured by A. Visser.Comment: Published in proceedings of LFCS 201
Computability Theory (hybrid meeting)
Over the last decade computability theory has seen many new and
fascinating developments that have linked the subject much closer
to other mathematical disciplines inside and outside of logic.
This includes, for instance, work on enumeration degrees that
has revealed deep and surprising relations to general topology,
the work on algorithmic randomness that is closely tied to
symbolic dynamics and geometric measure theory.
Inside logic there are connections to model theory, set theory, effective descriptive
set theory, computable analysis and reverse mathematics.
In some of these cases the bridges to seemingly distant mathematical fields
have yielded completely new proofs or even solutions of open problems
in the respective fields. Thus, over the last decade, computability theory
has formed vibrant and beneficial interactions with other mathematical
fields.
The goal of this workshop was to bring together researchers representing
different aspects of computability theory to discuss recent advances, and to
stimulate future work
On theories of random variables
We study theories of spaces of random variables: first, we consider random
variables with values in the interval , then with values in an arbitrary
metric structure, generalising Keisler's randomisation of classical structures.
We prove preservation and non-preservation results for model theoretic
properties under this construction: i) The randomisation of a stable structure
is stable. ii) The randomisation of a simple unstable structure is not simple.
We also prove that in the randomised structure, every type is a Lascar type
- …