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.

• Harnik, Victor; Harrington, Leo (1984), "Fundamentals of forking", Ann. Pure Appl. Logic, 26 (3): 245–286, doi:10.1016/0168-0072(84)90005-8, MR 0747686
• Lascar, Daniel; Poizat, Bruno (1979), "An Introduction to Forking", The Journal of Symbolic Logic, Association for Symbolic Logic, 44 (3): 330–350, doi:10.2307/2273127, JSTOR 2273127
• Makkai, M. (1984), "A survey of basic stability theory, with particular emphasis on orthogonality and regular types", Israel Journal of Mathematics, 49 (1–3): 181–238, doi:10.1007/BF02760649, S2CID 121533246
• Marker, David (2002), Model Theory: An Introduction, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98760-6
• Ng, Siu-Ah (2001) [1994], "Forking", Encyclopedia of Mathematics, EMS Press
• Shelah, Saharon (1990) [1978], Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics (2nd ed.), Elsevier, ISBN 978-0-444-70260-9