1,931 research outputs found
An independence theorem for NTP2 theories
We establish several results regarding dividing and forking in NTP2 theories.
We show that dividing is the same as array-dividing. Combining it with
existence of strictly invariant sequences we deduce that forking satisfies the
chain condition over extension bases (namely, the forking ideal is S1, in
Hrushovski's terminology). Using it we prove an independence theorem over
extension bases (which, in the case of simple theories, specializes to the
ordinary independence theorem). As an application we show that Lascar strong
type and compact strong type coincide over extension bases in an NTP2 theory.
We also define the dividing order of a theory -- a generalization of Poizat's
fundamental order from stable theories -- and give some equivalent
characterizations under the assumption of NTP2. The last section is devoted to
a refinement of the class of strong theories and its place in the
classification hierarchy
Henselian valued fields and inp-minimality
We prove that every ultraproduct of -adics is inp-minimal (i.e., of burden
). More generally, we prove an Ax-Kochen type result on preservation of
inp-minimality for Henselian valued fields of equicharacteristic in the RV
language.Comment: v.2: 15 pages, minor corrections and presentation improvements;
accepted to the Journal of Symbolic Logi
Distal and non-distal NIP theories
We study one way in which stable phenomena can exist in an NIP theory. We
start by defining a notion of 'pure instability' that we call 'distality' in
which no such phenomenon occurs. O-minimal theories and the p-adics for example
are distal. Next, we try to understand what happens when distality fails. Given
a type p over a sufficiently saturated model, we extract, in some sense, the
stable part of p and define a notion of stable-independence which is implied by
non-forking and has bounded weight. As an application, we show that the
expansion of a model by traces of externally definable sets from some adequate
indiscernible sequence eliminates quantifiers
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