If p(x) and q(y) are two complete types over a set C, p and q are set to be almost orthogonal if there is a unique complete type in the variables (x, y) extending p(x) ∪ q(y). That is, if a, a′ ⊨ p and b, b′ ⊨ q, then ab≡Ca′b′.
If p(x) and q(y) are stationary types in a stable theory, then one can easily check that p(x) and q(y) are almost orthogonal if and only if a↓Cb for all a realizing p(x) and b realizing q(y).
In a stable theory T, two stationary types p and q are orthogonal if p|C and q|C are almost orthogonal for every set C containing the bases of p and of q. Here p|C and q|C denote the unique non-forking extensions of p and q to C. It turns out that if p|C and q|C fail to be almost orthogonal for some C, then p|C′ and q|C′ also fail to be almost orthogonal for all C′ ⊇ C. Therefore, it suffices to check the orthogonality at sufficiently large sets C, and orthogonality depends only on the parallelism class of p and q.
Roughly speaking, p and q are orthogonal if there are no interesting relations between realizations of p and realizations of q. For example, if p and q are the generic types of two strongly minimal sets P and Q, then p and q are orthogonal if and only if there are no finite-to-finite correspondences between P and Q, i.e., no definable sets C ⊂ P × Q with C projecting onto P and onto 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 p and q are two non-orthogonal types of rank 1, then p and 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). ::: ::: :::