Non-Orthogonality of 1-types in Theories with a Linear Order

Non-orthogonality of complete types is an important concept for such classes of first-order theories as o-minimal, weakly-o-minimal and quite o-minimal theories. This concept is used in studying countable spectrum of such theories, since orthogonality affects omission and realization of types. Further study of the Vaught’s conjecture for small ordered theories requires the use of the relation between incomplete types, in particular, convex closures of 1-types. In this paper, two notions of non-orthogonality of convex incomplete types are introduced. Connections between different kinds of non-orthogonality are shown. Theorems on preservation of properties of types under non-orthogonality are proven.