On the Antichain Tree Property

TL;DR

This paper introduces the antichain tree property (ATP) in model theory, develops combinatorial techniques to analyze it, and provides algebraic examples and criteria for theories to have or lack ATP, including applications to groups, fields, and Boolean algebras.

Contribution

It establishes fundamental properties of ATP, shows its equivalence to k-ATP, and extends the analysis to various algebraic structures, offering new tools for model-theoretic classification.

Findings

ATP is witnessed by formulas with a single free variable.

The class of formulas without ATP is closed under disjunction.

Criteria for theories to be NATP are provided, with algebraic examples included.

Abstract

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 group, 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.