On VC-minimal theories and variants
نویسندگان
چکیده
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablity and show that this lies strictly between VCminimality and dp-minimality. To do this we prove a general result about set systems with independence dimension ≤ 1. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.
منابع مشابه
Forking in VC-minimal theories
We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M , generalizing a result of Dolich on o-minimal theories in [4].
متن کاملVapnik-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 P -minimal theories.
متن کاملConvexly orderable Groups and Valued Fields
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also co...
متن کاملOn VC-minimal fields and dp-smallness
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.
متن کاملAdditivity of the Dp-rank
The main result is the prove of the linearity of the dp-rank. We also prove that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types and discuss the possible relations between dp-rank and VC-density.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Arch. Math. Log.
دوره 52 شماره
صفحات -
تاریخ انتشار 2013