Summary: UIC Model Theory Seminar, November 29, 2005
Binarity and convexity rank in 0­categorical weakly o-minimal
theories
Beibut Kulpeshov
Institute of Informatics and Control Problems, Almaty
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 impor-
tant example of weakly o-minimal structures. A. Pillay and C. Steinhorn
have described all 0­categorical o-minimal theories [TAMS, 1986]. Their
description implies binarity for these theories. Here we present some results
on 0­categorical 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 infi-
nite convex classes. It is obvious an o-minimal theory has convexity rank 1.
Firstly, we give a description of 0­categorical binary weakly o-minimal the-
ories of convexity rank 1 [A & L, 2005]. Further, we present some technique
on 2­formulas which was originally introduced by B.S. Baizhanov. At last,

Source: Aschenbrenner, Matthias - Department of Mathematics, University of California at Los Angeles

Collections: Mathematics