Splitting

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.