A preservation theorem for theories without the tree property of the first kind

Abstract

We prove that the NTP 1 property of a geometric theory T is inherited by theories of lovely pairs and H-structures associated to T . We also provide a class of examples of nonsimple geometric NTP 1 theories. IntroductionOne theme of research in model theory is to inquire whether some well-known properties are preserved under a certain unary predicate expansions of a given structure. One of the motivations for this is that positive theorems of this kind often allows us to obtain interesting and complicatedlooking theories which still satisfy some strong tameness conditions. The study of expansions by unary predicates reaches back to the paper of Poizat on beautiful pairs [10], and has been developed in various directions ever since. Remarkable papers on this subject include for example [5], where the stability condition is examined, and [1], which is dedicated to studying expansions in simple theories (which generalize the stable ones).The well-known equivalence TP ⇔ TP 1 ∨ TP 2 due to Shelah [11] (where TP denotes the tree property while TP 1 and TP 2 denote the tree properties of the first and second kind, respectively), suggests two natural generalizations of simple theories, namely NTP 1 theories and NTP 2 theories (i.e., theories without TP 1 and TP 2 , respectively). So far, NTP 1 and NTP 2 theories have been studied much less extensively than the simple ones (i.e. theories without TP). However, recently some interesting results on these theories began to appear, notably [6] and [7]. In particular, natural examples of non-simple NTP 1 thoeries were provided in [6], namely: ω-free PAC fields, linear spaces with a generic bilinear form and a class of theories obtained by the "pfc" construction.The study of expansions in the NTP 2 context was undertaken in [2], where it was shown that the NTP 2 property is preserved under "dense and codense" unary predicate expansions of geometric structures where the unary predicate is assumed to define either an algebraically independent subset or an elementary substructure. In the present paper, we prove that the NTP 1 property is also preserved under such expansions. One of the main ingredients in our proof is the recently proved fact (due to Chernikov and Ramsey [6]) that the TP 1 property can, in any TP 1 theory, always be witnessed by some formula in a single free variable. We also prove (in Section 4) that an NTP 1 nonsimple geometric theory can be obtained from any Fraïssé limit which has a simple *

Generic variations and NTP $$_1$$ 1

“…The following observation comes from [5]. For any language L, a new language L + is obtained by replacing each n-ary…”

Vector spaces with a dense-codense generic submodule

“…The approach we follow is to check the property by doing a formula-by-formula analysis, separating the cases when the corresponding definable set is small (algebraic over the predicate) or large. To show the preservation of NSOP 1 we build on ideas presented in [23], the proofs for the preservation of NTP 1 , NTP 2 generalize ideas presented in [2,18]. In Section 6, we study independence notions in the expansion, assuming the original theory has a good notion of independence.…”

“…In this section, we prove the expansion preserves other nice properties such as N T P 2 , NTP 1 and NSOP 1 . To prove the preservation of N T P 2 and NTP 1 , we will follow the approach from [2], [18], but will need to modify several parts of the arguments. The main difference is that in our setting the definable subsets in G involve not only the induced structure from V (as is the case in [2,18]) but also the structure that it carries as an R-module.…”

“…To prove the preservation of N T P 2 and NTP 1 , we will follow the approach from [2], [18], but will need to modify several parts of the arguments. The main difference is that in our setting the definable subsets in G involve not only the induced structure from V (as is the case in [2,18]) but also the structure that it carries as an R-module. We will follow a similar approach to show the preservation of NSOP 1 using as a guide the ideas from [23].…”

“…Now we deal with NTP 1 (also called NSOP 2 ). We will now follow the strategy from [18], emphasizing the main differences that are needed to adapt the arguments to the new setting. We start with an appropriate notion of tree-indiscernability (see [18,Def.…”