首页/论文/1908.11400

arXiv 1908.114002019-08math.LO

Distality Rank

Roland Walker

备注：: 32 pages

英文摘要：

Building on Pierre Simon's notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank $m$ such that $1\leq m \leq ω$. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called $m$-determinacy and show that theories of distality rank $m$ require certain products to be $m$-determined. Furthermore, for NIP theories, this behavior characterizes $m$-distality. If we narrow the scope to stable theories, we observe that $m$-distality can be characterized by the maximum cycle size found in the forking "geometry," so it coincides with $(m-1)$-triviality. On a broader scale, we see that $m$-distality is a strengthening of Saharon Shelah's notion of $m$-dependence.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Distality for the asymptotic couple of the field of logarithmic transseries

We show that the theory $T_{\log}$ of the asymptotic couple of the field of logarithmic transseries is distal. As distal theories are NIP (= the non-independence property), this provides a new proof that $T_{\log}$ is NIP. Finally, we show that $T_{\log}$ is not strongly dependent, and in particular, it is not $\operatorname{dp}$-minimal and it does not have finite $\operatorname{dp}$-rank.

2018-05-22math.LO

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 …

2012-10-27math.LO

Exact saturation in simple and NIP theories

A theory $T$ is said to have exact saturation at a singular cardinal $κ$ if it has a $κ$-saturated model which is not $κ^{+}$-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.

2015-10-09math.LO

NTP₂中的一些NIP-like现象

Some NIP-like phenomena in NTP$_{2}$

本文研究了NTP₂中的NIP-like现象，提出了NTP₂-光滑度量和distality概念，并探讨了有限交替定理及有限可满足性在NTP₂中的扩展问题。

2025-11-21math.LO

单形NIP理论中的不可区分元素

Indiscernibles in monadically NIP theories

本文研究了单形NIP理论中不可区分元素的性质，证明了平面图的distal扩展结果，并指出没有单形NIP理论能解释无限群。

2025-07-26math.LO

Non-forking and preservation of NIP and dp-rank

We investigate the question of whether the restriction of a NIP type $p\in S(B)$ which does not fork over $A\subseteq B$ to $A$ is also NIP, and the analogous question for dp-rank. We show that if $B$ contains a Morley sequence $I$ generated by $p$ over $A$, then $p\restriction AI$ is NIP and similarly preserves the dp-rank. This yields positive answers for generically stable NIP types and the analogous case of stable types. With similar techniques we also provide a new more direct proof for the latter. Moreover, we introduce a general construction of "trees whose open cones are models of some…

2019-12-14math.LO

更多相关论文收起更多论文

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.

2022-07-30math.COmath.LO

dp-有限与Noetherian NIP整环

Dp-finite and Noetherian NIP integral domains

该研究证明了NIP Noetherian整环的性质，并分类了dp-极小的Noetherian域。

2026-03-06math.ACmath.LO

依赖划分的后果

Consequences of Dependent Dividing on Burden

University of California, Berkeley（加州大学伯克利分校）

研究了依赖划分对负担的影响，证明其亚加性性质，并建立了负担与双 VC 密度的关系。

2026-02-23math.LO

Superrosy dependent groups having finitely satisfiable generics

We study a model theoretic context (finite thorn rank, NIP, with finitely satisfiable generics) which is a common generalization of groups of finite Morley rank and definably compact groups in o-minimal structures. We show that assuming thorn rank 1, the group is abelian-by-finite, and assuming thorn rank 2 the group is solvable by finite. Also a field is algebraically closed.

2007-06-04math.GRmath.LO

Ellis半群中的秩与模型论

Ranks in Ellis semigroups and model theory

本文推广Ellis半群中的秩概念至紧致豪斯多夫空间，并证明其取序数值等价于理论具有NIP性质，连接了动力系统与模型论稳定性。

2023-09-06math.DSmath.LO

Notes on trace equivalence

We introduce trace definability, a weak notion of interpretability, and trace equivalence, a weak notion of equivalence for first order structures and theories. In particular we get an interesting weak equivalence notion for $\mathrm{NIP}$ theories. We describe a close connection to indiscernible collapse. We also show that if $Q$ is a divisible subgroup of $(\mathbb{R};+)$ and $\mathcal{Q}$ is a dp-rank one expansion of $(Q;+,<)$ then exactly one of the following holds: $\mathrm{Th}(\mathcal{Q})$ trace defines $\mathrm{RCF}$ or $\mathcal{Q}$ is trace equivalent to a reduct of an ordered vecto…

2022-04-06math.LO