U-rank


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:
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 theory T,.
We then get a concise characterization of superstability; a stable theory T 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 than β, then it must be ∞.

    Examples

  • U > 0 precisely when p is nonalgebraic.
  • If T is the theory of algebraically 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.