Mekler's construction and tree properties

2019/03/17 by JinHoo Ahn, Ahn, JinHoo
Computer Science · Mathematics · #03C45 #05C25 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.1903.07087

openalex publication_date 2019/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Mekler constructed a way to produce a pure group from any given structure where the construction preserves κ-stability for any cardinal κ. Not only the stability, it is known that his construction preserves various model-theoretic properties such as simplicity, NIP, and NTP₂. Inspired by the last result, we show that the construction also preserves NTP₁(NSOP₂) and NSOP₁. As a corollary, we obtain that if there is a theory of finite language which is non-simple NSOP₁, or which is NSOP₂ but has SOP₁, then there is a pure group theory with the same properties, respectively.

Related

• Mekler's construction and generalized stability
• On model-theoretic tree properties
• More on tree properties
• An improved proof of the Almost Stability Theorem
• Theories without the tree property of the second kind