FANDOM


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).