Mekler’s construction and generalized stability

  title={Mekler’s construction and generalized stability},
  author={Artem Chernikov and Nadja Hempel},
  journal={Israel Journal of Mathematics},
Mekler’s construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class 2 and exponent p > 2, but not finitely generated in general). Even though this construction is not a bi-interpretation, it is known to preserve some model-theoretic tameness properties of the original structure including stability and simplicity. We demonstrate that k-dependence of the theory is preserved, for all k ∈ N, and that NTP2 is preserved. We apply this result… 

Mekler's construction and tree properties

It is shown that the construction of a pure group from any given structure preserves NTP$_1$(NSOP$_2$) and NSOP$-stability for any cardinal $\kappa$, and it is obtained that if there is a theory of finite language which is non-simple NSOP $_1$, or which is NSOP_2 but has SOP#_1, then there is an pure group theory with the same properties.

On the Antichain Tree Property

In this note, we investigate a new model theoretical tree property, called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed

Title On n-dependent groups and fields II Permalink

  • Mathematics
  • 2019
We continue the study of n-dependent groups, fields and related structures. We demonstrate that n-dependence is witnessed by formulas with all but one variable singletons, provide a type-counting

Artin-Schreier extensions and combinatorial complexity in henselian valued fields

We give explicit formulas witnessing IP, IP n or TP2 in fields with Artin-Schreier extensions. We use them to control p -extensions of mixed characteristic henselian valued fields, allowing us most

On n-dependent groups and fields II

Abstract We continue the study of n-dependent groups, fields and related structures, largely motivated by the conjecture that every n-dependent field is dependent. We provide evidence toward this

Preservation of NATP

. We prove several preservation theorems for NATP and furnish several examples of NATP. First, we prove preservation of NATP for the parametrization and sum of the theories of Fra¨ıss´e limits of

Hypergraph regularity and higher arity VC-dimension

It is shown that when H is a k'-uniform hypergraph with small VC_k-dimension for some k<k', the decomposition of H given by hypergraph regularity only needs the first k levels, and that on most of the resulting k-ary cylinder sets, the density of H is either close to 0 or close to 1.


We develop distality rank as a property of first-order theories and give examples for each rank $m$ such that $1\leq m \leq \omega$. For NIP theories, we show that distality rank is invariant under



Mekler's construction preserves CM-triviality

Groups and fields with NTP2

NTP2 is a large class of first-order theories defined by Shelah and generalizing simple and NIP theories. Algebraic examples of NTP2 structures are given by ultra-products of p-adics and certain

Classification theory - and the number of non-isomorphic models, Second Edition

  • S. Shelah
  • Mathematics
    Studies in logic and the foundations of mathematics
  • 1990

Stability of nilpotent groups of class 2 and prime exponent

  • A. Mekler
  • Mathematics
    Journal of Symbolic Logic
  • 1981
It is suggested that the problem of characterizing the wo-stable groups is intractable.

Strongly dependent theories

We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel

Definable groups for dependent and 2-dependent theories

Let T be a (first order complete) dependent theory, C a kappa-saturated model of T and G a definable subgroup which is abelian. Among subgroups of bounded index which are the union of < kappa type

On Kim-independence

We study NSOP$_{1}$ theories. We define Kim-independence, which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that


The dividing order of a theory is defined—a generalization of Poizat’s fundamental order from stable theories—and some equivalent characterizations under the assumption of NTP2 are given.