## Journal of Symbolic Logic

### Expansion of a Model of a Weakly O-Minimal Theory by a Family of Unary Predicates

Bektur Sembiuly Baizhanov

#### Abstract

A subset A $\subseteq$ M of a totally ordered structure M is said to be convex, if for any a, b $\in A : [a < b \rightarrow \forall t(a < t < b \rightarrow t \in A)]$. A complete theory of first order is weakly o-minimal (M. Dickmann [D]) if any model M is totally ordered by some $\emptyset$-definable formula and any subset of M which is definable with parameters from M is a finite union of convex sets. We prove here that for any model M of a weakly o-minimal theory T, any expansion M$^+$ of M by a family of unary predicates has a weakly o-minimal theory iff the set of all realizations of each predicate is a union of a finite number of convex sets (Theorem 63), that solves the Problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories.

#### Article information

Source
J. Symbolic Logic, Volume 66, Issue 3 (2001), 1382-1414.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183746567

Mathematical Reviews number (MathSciNet)
MR1856749

Zentralblatt MATH identifier
0992.03047

JSTOR