# Mekler’s construction and generalized stability

@article{Chernikov2017MeklersCA,
title={Mekler’s construction and generalized stability},
journal={Israel Journal of Mathematics},
year={2017},
volume={230},
pages={745-769}
}
• Published 11 August 2017
• Mathematics
• 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…
8 Citations
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$, or which is NSOP_2 but has SOP#_1, then there is an pure group theory with the same properties. • Mathematics • 2021 In this note, we investigate a new model theoretical tree property, called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed • Mathematics • 2019 We continue the study of n-dependent groups, fields and related structures. We demonstrate that n-dependence is witnessed by formulas with all but one variable singletons, provide a type-counting We give explicit formulas witnessing IP, IP n or TP2 in ﬁelds with Artin-Schreier extensions. We use them to control p -extensions of mixed characteristic henselian valued ﬁelds, allowing us most • Mathematics Forum of Mathematics, Sigma • 2021 Abstract We continue the study of n-dependent groups, fields and related structures, largely motivated by the conjecture that every n-dependent field is dependent. We provide evidence toward this • Mathematics • 2022 . We prove several preservation theorems for NATP and furnish several examples of NATP. First, we prove preservation of NATP for the parametrization and sum of the theories of Fra¨ıss´e limits of • Mathematics ArXiv • 2020 It is shown that when H is a k'-uniform hypergraph with small VC_k-dimension for some k<k', the decomposition of H given by hypergraph regularity only needs the first k levels, and that on most of the resulting k-ary cylinder sets, the density of H is either close to 0 or close to 1. We develop distality rank as a property of first-order theories and give examples for each rank$m$such that$1\leq m \leq \omega$. For NIP theories, we show that distality rank is invariant under ## References SHOWING 1-10 OF 22 REFERENCES • Mathematics • 2012 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 • A. Mekler • Mathematics Journal of Symbolic Logic • 1981 It is suggested that the problem of characterizing the wo-stable groups is intractable. 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 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 • Mathematics Journal of the European Mathematical Society • 2020 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
• Mathematics
The Journal of Symbolic Logic
• 2014
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.