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. (shrink)

The main result of this paper is Theorem 3.1 which is a criterion for weak o-minimality of a linearly ordered structure in terms of realizations of 1-types. Here we also prove some other properties of weakly o-minimal structures. In particular, we characterize all weakly o-minimal linear orderings in the signature $\{ . Moreover, we present a criterion for density of isolated types of a weakly o-minimal theory. Lastly, at the end of the paper we present some remarks on the Exchange (...) Principle for algebraic closure in a weakly o-minimal structure. (shrink)

Orthogonality of all families of pairwise weakly orthogonal 1-types for ℵ0-categorical weakly o-minimal theories of finite convexity rank has been proved in 6. Here we prove orthogonality of all such families for binary 1-types in an arbitrary ℵ0-categorical weakly o-minimal theory and give an extended criterion for binarity of ℵ0-categorical weakly o-minimal theories . © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.