Search

We initiate a systematic study of the class of theories without the tree property of the second kind - NTP2. Most importantly, we show: the burden is "sub-multiplicative" in arbitrary theories (in particular, if a theory has TP2 then there is a formula with a single variable witnessing this); NTP2 is equivalent to the generalized Kim's lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters - so the dp-rank of a 1-type in any theory is always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic (0, 0) is NTP2 (strong, of finite burden) if and only if the residue field is NTP2 (the residue field and the value group are strong, of finite burden respectively), so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2. © 2013 Elsevier B.V.

We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski’s terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP2 theory. We also define the dividing order of a theory—a generalization of Poizat’s fundamental order fromstable theories—and give some equivalent characterizations under the assumption of NTP2. The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy.

We prove that many properties and invariants of definable groups in NIP theories (i.e. theories without the independence property), such as definable amenability, G/G^{00}, etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, M^{\operatorname {ext} }, of a model M. In the light of these results, we continue the study of the 'definable topological dynamics' of groups in \operatorname {NIP} theories. In particular, we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in some new cases, including definably amenable groups in o-minimal structures. © 2014 London Mathematical Society.

We present a complete finite axiomatization of the unrestricted implication problem for inclusion and conditional independence atoms in the context of dependence logic. For databases, our result implies a finite axiomatization of the unrestricted implication problem for inclusion, functional, and embedded multivalued dependencies in the unirelational case.