If and are two complete types over a set , and are set to be almost orthogonal if there is a unique complete type in the variables extending . That is, if and , then .
If and are stationary types in a stable theory, then one can easily check that and are almost orthogonal if and only if for all realizing and realizing .
In a stable theory , two stationary types and are orthogonal if and are almost orthogonal for every set containing the bases of and of . Here and denote the unique non-forking extensions of and to . It turns out that if and fail to be almost orthogonal for some , then and also fail to be almost orthogonal for all . Therefore, it suffices to check the orthogonality at sufficiently large sets , and orthogonality depends only on the parallelism class of and .
Roughly speaking, and are orthogonal if there are no interesting relations between realizations of and realizations of . For example, if and are the generic types of two strongly minimal sets and , then and are orthogonal if and only if there are no finite-to-finite correspondences between and , i.e., no definable sets with projecting onto and onto with finite fibers in both directions.
The relation of non-orthogonality is an equivalence relation on strongly minimal sets, or more generally, on stationary types of U-rank 1. If and are two non-orthogonal types of rank 1, then and have the same underlying geometry. A theory is uncountably categorical if and only if it is $\omega$-stable and unidimensional (e.g. every pair of stationary non-algebraic types is non-orthogonal). ::: ::: :::