On the antichain tree property

In this paper, we investigate a new model theoretical tree property (TP), 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 pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras.

Automated summary

In this paper, the authors investigate a new type of tree property in model theory called the antichain tree property (ATP). They develop combinatorial techniques for ATP and show that ATP can always be witnessed by a formula with a single free variable. They also prove the equivalence of ATP and k-ATP and provide a criterion for theories to not have ATP (being NATP). The authors present algebraic examples of ATP and NATP, including pure groups, pure fields, and valued fields. They conclude by constructing an antichain tree in Skolem arithmetic and atomless Boolean algebras.