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Preprint Number 2003

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2003. JinHoo Ahn, Joonhee Kim, and Junguk Lee
On the Antichain Tree Property

Submission date: 16 June 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 by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and k-ATP, and provide a criterion for theories to have not ATP (being NATP).
Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure groups, pure fields, and valued fields. More precisely, we prove Mekler's construction for groups, Chatzidakis' style criterion for PAC fields, and the AKE-style principle for valued fields preserving NATP. And we give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras.

Mathematics Subject Classification: 2010: 03C45, 03C60, 12J10, 05C25

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