ON THE LOCAL MONOTONY OF ONE-PLACE FUNCTIONS DEFINABLE IN FINITELY QUILTED ORDERED STRUCTURES

Abstract

As it was proved by B. Kulpeshov, any cut in a weakly o-minimal
structures can have at most two extensions up to complete types, and the sets of
realizations of these types are convex in any elementary extensions. In this paper
we consider a generalization of weak o-minimality, namely notion of an n-quilted
structure: a totally ordered structure is said to be n-quilted if any cut has at most n
extensions up to complete types over the structure. Note that we omit here
condition that the set of all realizations of a type must be convex. In this article we
investigate the property of local monotonicity for unary functions definable in
finitely quilted ordered structures.