Binary convexity rank in almost omega-categorical weakly o-minimal theories

Abstract

The present paper concerns the notion of weak o-minimality that was initially deeply studied by D. Macpherson, D. Marker and C. Steinhorn. A subset A of a linearly ordered structure M is convex if for all a, b Î A and c Î M whenever a < c < b we have c Î A. A weakly o-minimal structure is a linearly ordered structure M = áM, =, <, …ñ such that any definable (with parameters) subset of M is a union of finitely many convex sets in M. A criterion for equality of the binary convexity ranks for non-weakly orthogonal non-algebraic 1-types in almost omega-categorical weakly o-minimal theories in case of existing an element of the set of realizations of one of these types the definable closure of which has a non-empty intersection with the set of realizations of another type is found.