A note on Lascar strong types in simple theories

Let T be a countable, small simple theory. In this paper, we prove for such T, the notion of Lascar Strong type coincides with the notion of a strong type,over an arbitrary set

An independence theorem for NTP2 theories

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 from stable 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

The Lascar groups and the 1st homology groups in model theory

Let $p$ be a strong type of an algebraically closed tuple over B=\acl^{\eq}(B) in any theory $T$. Depending on a ternary relation \indo^* satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$), the first homology group $H^*_1(p)$ can be introduced, similarly to \cite{GKK1}. We show that there is a canonical surjective homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that the map factors naturally via a surjection from the `relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of $p$ is independent from the choice of \indo^*, and can be written simply as $H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$ unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.Comment: 30 pages, no figures, this merged with the article arXiv:1504.0772