Logic Seminar
Bektur Baizhanov
Institute of Informatics and Control Problems, Almaty, Kazakhstan
O-minimality and stability
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.
Tuesday October 21, 2008 at 4:00 PM in SEO 612