1,262 research outputs found
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
Sheaves on T-topologies
The aim of this paper is to give a unifying description of various
constructions (subanalytic, semialgebraic, o-minimal site) using the notion of
T-topology. We then study the category of T-sheaves.Comment: 31 pages, uses xy-pic, revised versio
NIP omega-categorical structures: the rank 1 case
We classify primitive, rank 1, omega-categorical structures having
polynomially many types over finite sets. For a fixed number of 4-types, we
show that there are only finitely many such structures and that all are built
out of finitely many linear orders interacting in a restricted number of ways.
As an example of application, we deduce the classification of primitive
structures homogeneous in a language consisting of n linear orders as well as
all reducts of such structures.Comment: Substantial changes made to the presentation, especially in sections
3 and
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