In model theory

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

, a forking extension of a type is an extension that is in some sense not free, and 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

Stable theory

In model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases...

. 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

Morley rank

In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry.-Definition:Fix a theory T with a model M...

is smaller, and a nonforking extension if it has the same Morley rank.

Axioms

Let T be a stable complete theory. The nonforking 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.