8 research outputs found
Strongly NIP almost real closed fields
The following conjecture is due to Shelah-Hasson: Any infinite strongly NIP
field is either real closed, algebraically closed, or admits a non-trivial
definable henselian valuation, in the language of rings. We specialise this
conjecture to ordered fields in the language of ordered rings, which leads
towards a systematic study of the class of strongly NIP almost real closed
fields. As a result, we obtain a complete characterisation of this class.Comment: To appear in MLQ Math. Log. Q. A previous version of this preprint
was part of arXiv:1810.10377. arXiv admin note: text overlap with
arXiv:2010.1183
DEFINABLE SETS IN DP-MINIMAL ORDERED ABELIAN GROUPS (Model theoretic aspects of the notion of independence and dimension)
This article surveys some recent results on ordered abelian groups (possibly with additional definable structure) from the subclass of NIP theories which are dp-minimal. To put these results in context, the first part of the article reviews and compares various other generalizations of a-minimality (such as local a-minimality and a-stability) and their consequences. It is useful to make the further assumption that there is a cardinal bound on the number of convex subgroups definable in elementary extensions of the structure. Under this hypothesis, some classic theorems on o-minimal structures, such as the monotonicity theorem for unary definable functions, can be suitably generalized
Vapnik-Chervonenkis density in some theories without the independence property, I
We recast the problem of calculating Vapnik-Chervonenkis (VC) density into
one of counting types, and thereby calculate bounds (often optimal) on the VC
density for some weakly o-minimal, weakly quasi-o-minimal, and -minimal
theories.Comment: 59