Forking extension

In model theory, a forking extension is an extension that is not free 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.

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:

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