In model theory, a branch of mathematical logic, U-rank is one measure of the complexity of a type, in the context of stable theories. As usual, higher U-rank indicates less restriction, and the existence of a U-rank for all types over all sets is equivalent to an important model-theoretic condition: in this case, superstability.
Definition
U-rank is defined inductively, as follows, for any n-type p over any set A:
For any α = β + 1, U ≥ α precisely when there is a forking extensionq of p with U ≥ β
We say that U = α when the U ≥ α but not U ≥ α + 1. If U ≥ α for all ordinals α, we say the U-rank is unbounded, or U = ∞. Note: U-rank is formally denoted, where p is really p, and x is a tuple of variables of length n. This subscript is typically omitted when no confusion can result.
Ranking theories
U-rank is monotone in its domain. That is, suppose p is a complete type over A and B is a subset of A. Then for q the restriction of p to B, U ≥ U. If we take B to be empty, then we get the following: if there is an n-type p, over some set of parameters, with rank at least α, then there is a type over the empty set of rank at least α. Thus, we can define, for a complete theoryT,. We then get a concise characterization of superstability; a stable theoryT is superstable if and only if for every n.
Properties
As noted above, U-rank is monotone in its domain.
If p has U-rank α, then for any β < α, there is a forking extension q of p with U-rank β.
If p is the type of b over A, there is some set B extending A, with q the type of b over B.
If p is unranked, then there is a forking extension q of p which is also unranked.
Even in the absence of superstability, there is an ordinal β which is the maximum rank of all ranked types, and for any α < β, there is a type p of rank α, and if the rank of p is greater thanβ, then it must be ∞.
Examples
U > 0 precisely when p is nonalgebraic.
If T is the theory ofalgebraically closed fields then. Further, if A is any set of parameters and K is the field generated by A, then a 1-type p over A has rank 1 if p are transcendental over K, and 0 otherwise. More generally, an n-type p over A has U-rank k, the transcendence degree of any realization of it.
