Bektur Baizhanov : O-minimality and stability

Posted by , part of the Logic Seminar.

At

Oct. 21, 2008, 4 p.m.

In

SEO 612

Abstract

In my talk I present our (with Viktor Verbovskiy) generalization of the notion o-minimality, which allows us to apply technics of stability theory to investigation of ordered structures. Let $T$ be a complete theory, having $\emptyset$-definable relation of linear order. Let $M$ be a model of $T$, $M$ be a model of $T$, $A\subseteq M$. For any 1-formula $\phi(x)$ we define a convex hull $C_{\phi} (x)$ as $ C_{\phi}(x) \stackrel{\triangle}{=} \exists y,z (\phi(y) \land \phi (z) \land y \le x \le z) $; and for an one-type $p\in S^1(A)$ we define a convex hull $c(p)$ as $ c(p) \stackrel{\triangle}{=} \{ C_{\phi} \mid \phi \in p\}. $ Denote $S^1_{c(p)}(A)=\{q\in S^1(A)| c(p)\subseteq q\}$. The model $M$ is o-stable in $\lambda$ if for all $A\subseteq M$, $|A| \le \lambda$, for any 1-type $p$ $|S^1_{c(p)}|\le \lambda.$ Theory $T$ is o-stable, if every model of $T$ is. As in stability theory we can introduce the notions of o-$\omega$-stability and o-superstability. Notice that Morley's Theorem for o-$\omega$-stability holds. It follows from definition that o-minimal theory and weakly o-minimal theory are o-$\omega$-stable and quasi o-minimal theory is o-superstable. Theorem 1 Let $T$ be o-stable theory, then $T$ does not have the independence property. Theorem 2 Any infinite model $(M;=, <)$ has o-superstable theory. B. Baizhanov and V. Verbovskiy O-stable theories, preprint, 2008.

seminar begins with tea.