 
Summary: UIC Model Theory Seminar, November 29, 2005
Binarity and convexity rank in 0categorical weakly ominimal
theories
Beibut Kulpeshov
Institute of Informatics and Control Problems, Almaty
This talk concerns the notion of weak ominimality 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 ominimal structures. A. Pillay and C. Steinhorn
have described all 0categorical ominimal theories [TAMS, 1986]. Their
description implies binarity for these theories. Here we present some results
on 0categorical weakly ominimal 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 ominimal theory has convexity rank 1.
Firstly, we give a description of 0categorical binary weakly ominimal 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,
