If {\displaystyle p(x)} and {\displaystyle q(y)} are two complete types over a set {\displaystyle C}, {\displaystyle p} and {\displaystyle q} are set to be almost orthogonal if there is a unique complete type in the variables {\displaystyle (x,y)} extending {\displaystyle p(x)\cup q(y)}. That is, if {\displaystyle a,a'\models p} and {\displaystyle b,b'\models q}, then {\displaystyle ab\equiv _{C}a'b'}.

If {\displaystyle p(x)} and {\displaystyle q(y)} are stationary types in a stable theory, then one can easily check that {\displaystyle p(x)} and {\displaystyle q(y)} are almost orthogonal if and only if {\displaystyle a\downarrow _{C}b} for all {\displaystyle a} realizing {\displaystyle p(x)} and {\displaystyle b} realizing {\displaystyle q(y)}.

In a stable theory {\displaystyle T}, two stationary types {\displaystyle p} and {\displaystyle q} are orthogonal if {\displaystyle p|C} and {\displaystyle q|C} are almost orthogonal for every set {\displaystyle C} containing the bases of {\displaystyle p} and of {\displaystyle q}. Here {\displaystyle p|C} and {\displaystyle q|C} denote the unique non-forking extensions of {\displaystyle p} and {\displaystyle q} to {\displaystyle C}. It turns out that if {\displaystyle p|C} and {\displaystyle q|C} fail to be almost orthogonal for some {\displaystyle C}, then {\displaystyle p|C'} and {\displaystyle q|C'} also fail to be almost orthogonal for all {\displaystyle C'\supseteq C}. Therefore, it suffices to check the orthogonality at sufficiently large sets {\displaystyle C}, and orthogonality depends only on the parallelism class of {\displaystyle p} and {\displaystyle q}.

Roughly speaking, {\displaystyle p} and {\displaystyle q} are orthogonal if there are no interesting relations between realizations of {\displaystyle p} and realizations of {\displaystyle q}. For example, if {\displaystyle p} and {\displaystyle q} are the generic types of two strongly minimal sets {\displaystyle P} and {\displaystyle Q}, then {\displaystyle p} and {\displaystyle q} are orthogonal if and only if there are no finite-to-finite correspondences between {\displaystyle P} and {\displaystyle Q}, i.e., no definable sets {\displaystyle C\subset P\times Q} with {\displaystyle C} projecting onto {\displaystyle P} and onto {\displaystyle Q} 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 {\displaystyle p} and {\displaystyle q} are two non-orthogonal types of rank 1, then {\displaystyle p} and {\displaystyle q} 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). ::: ::: :::