Forking types and rank functions in stable theories
The authors use a concept of generalized rank functions to prove theorems on non-forking types. In this context a type is not necessarily complete. A mapping R from the set of types of a given big model of a stable theory T into a linearly ordered set W with a greatest element \(\infty\) is a rank function if R commutes with isomorphism of submodels and if \(p\subseteq q\) implies R(p)\(\geq R(q)\). R is totally normal if i) for every set A and every type p over A there is a complete type q over A with \(p\subseteq q\) and \(R(p)=R(q)\), ii) there is a \(\kappa\) s.t. for every set A and every type p over A \(| \{q| \quad q\) complete and \(R(p)=R(q)\}|<\kappa\), and iii) there exists a cardinal \(\mu\) s.t. for any set A and any p over A there is a set \(B\subseteq A\) s.t. \(| B| \subseteq \mu\) and \(R(p)=R(p\upharpoonright B)\). The main theorem states: If R is a totally normal rank function, \(A\subseteq B\) sets, p a complete type over B and \(R(p)<\infty\) then p does not fork over A iff \(R(p)=R(p\upharpoonright A)\). Hence for every totally normal rank function R with \(R(p)<\infty\), \(| \{q| \quad q\) complete extension over B, \(R(q)=R(p)\}|\) is independent from R. The main theorem has some nice consequences and it gives some short proofs of known theorems on non-forking types.
scientific article; zbMATH DE number 3865253