Lascar Rank
The Lascar, or U, rank of a complete type p ∈ S(A)
over a subset A ⊆ M of the universe of some model
M is defined by
- U(p) ≥ 0 always
- U(p) ≥ α + 1 ⇔ p has a
forking extension of U-rank at least
α. That is, there is an elementary extension
N of M,
a set A ⊆ B ⊆ N, and a type q ∈ S(B)
a forking extension of p with
U(q) ≥ α.
- U(p) ≥ λ for λ a limit just in
case (∀ &alpha < &lambda) U(p) ≥ &alpha.
- U(p) := α if U(p) ≥ &alpha but U(p)
is not greater or equal to α + 1. If no such
ordinal exists, then U(p) := ∞.