# Mekler’s construction and generalized stability

@article{Chernikov2017MeklersCA, title={Mekler’s construction and generalized stability}, author={Artem Chernikov and Nadja Hempel}, journal={Israel Journal of Mathematics}, year={2017}, volume={230}, pages={745-769} }

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…

## 8 Citations

### Mekler's construction and tree properties

- Mathematics, Computer Science
- 2019

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

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- 2021

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

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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…

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- 2021

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

- MathematicsForum of Mathematics, Sigma
- 2021

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

- Mathematics
- 2022

. 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…

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- MathematicsArXiv
- 2020

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.

### DISTALITY RANK

- MathematicsThe Journal of Symbolic Logic
- 2022

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…

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