Orthogonal types (stability theory)

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 $$p(x) \cup q(y)$$. That is, if $$a, a' \models p$$ and $$b, b' \models q$$, then $$ab \equiv_C a'b'$$.

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 $$a \downarrow_C b$$ 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 $$C' \supseteq C$$. 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 $$C \subset P \times Q$$ 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).