Logic Seminar
Beibut Kulpeshov
Institute of Informatics and Control Problems, Almaty
Binarity and convexity rank in $\aleph_0$--categorical weakly o-minimal structures
Abstract: This talk concerns the notion of weak o-minimality originally and deeply studied by D. Macpherson, D.
Marker and C. Steinhorn [TAMS, 2000]. Real closed fields with a proper convex valuation ring provide
an important example of weakly o-minimal structures. A. Pillay and C. Steinhorn have described all $
\aleph_0$--categorical o-minimal theories [TAMS, 1986]. Their description implies binarity for these
theories. Here we present some results on $\aleph_0$--categorical weakly o-minimal theories, and
discuss some connections between two notions: binarity and convexity rank. Recall that convexity rank
for a formula with one free variable was introduced by the speaker in [JSL, 1998]. In particular, a theory
has convexity rank 1 if there is no definable (with parameters) equivalence relation with infinitely many
infinite convex classes. It is obvious an o-minimal theory has convexity rank 1. Firstly, we give a
description of $\aleph_0$--categorical binary weakly o-minimal theories of convexity rank 1 [A & L,
2005]. Further, we present some technique on 2--formulas which was originally introduced by B.S.
Baizhanov. At last, by using this technique we obtain a criterion for binarity of $\aleph_0$--categorical
weakly o-minimal theories in terms of convexity rank (the main result of the talk).
Tuesday November 29, 2005 at 4:00 PM in SEO 427