FANDOM


Let $ p(x) $ be a complete type over some set of parameters $ B $, and let $ A $ be a subset of $ B $. One says that $ p(x) $ splits over $ A $ if $ \phi(x;b_1) \in p(x), \quad \phi(x;b_2) \notin p(x) $ for some formula $ \phi(x;y) $, and $ b_1, b_2 \in B $ having the same type over $ A $. Splitting is a weaker condition than dividing, so not splitting is a stronger condition than not dividing. If $ M $ is a sufficiently saturated model containing $ A $, (for example, the monster model), then $ p \in S(M) $ doesn't split over $ A $ if and only if $ p $ is $ \operatorname{Aut}(M/A) $-invariant.