Forking extension

Suppose that A and B are models of some complete ω-stable theory T. If p is a type of A and q is a type of B containing p, then q is called a forking extension of p if its Morley rank is smaller, and a nonforking extension if it has the same Morley rank.

Let T be a stable complete theory. The non-forking relation ≤ for types over T is the unique relation that satisfies the following axioms:

1. If p≤ q then p⊂q. If f is an elementary map then p≤q if and only if fp≤fq
2. If p⊂q⊂r then p≤r if and only if p≤q and q≤ r
3. If p is a type of A and A⊂B then there is some type q of B with p≤q.
4. There is a cardinal κ such that if p is a type of A then there is a subset A₀ of A of cardinality less than κ so that (p|A₀) ≤ p, where | stands for restriction.
5. For any p there is a cardinal λ such that there are at most λ non-contradictory types q with p≤q.

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