Forking extension

In model theory, a forking extension of a type is an extension of that type that is not free^[clarify] whereas a non-forking extension is an extension that is as free as possible. This can be used to extend the notions of linear or algebraic independence to stable theories. These concepts were introduced by S. Shelah.

Definitions

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.

Axioms

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

If p≤ q then p⊂q. If f is an elementary map then p≤q if and only if fp≤fq
If p⊂q⊂r then p≤r if and only if p≤q and q≤ r
If p is a type of A and A⊂B then there is some type q of B with p≤q.
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.
For any p there is a cardinal λ such that there are at most λ non-contradictory types q with p≤q.

References

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, 44 (3), Association for Symbolic Logic: 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