Expansion of a model of a weakly o-minimal theory by a family of unary predicates

Journal of Symbolic Logic 66 (3):1382-1414 (2001)
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 . 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
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Roman Wencel (2008). Weakly o-Minimal Nonvaluational Structures. Annals of Pure and Applied Logic 154 (3):139-162.
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