In model theory, a branch of mathematical logic, U-rank is one measure of the complexity of a (complete) 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.
, where p is really p(x), and x is a tuple of variables of length n. This subscript is typically omitted when no confusion can result.
That is, suppose p is a complete type over A and B is a subset of A.
If we take B (above) 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 (stable) theory T,