U-rank

U-rank is defined inductively, as follows, for any (complete) n-type p over any set A:

• U(p) ≥ 0
• If δ is a limit ordinal, then U(p) ≥ δ precisely when U(p) ≥ α for all α less than δ
• For any α = β + 1, U(p) ≥ α precisely when there is a forking extension q of p with U(q) ≥ β

We say that U(p) = α when the U(p) ≥ α but not U(p) ≥ α + 1.

If U(p) ≥ α for all ordinals α, we say the U-rank is unbounded, or U(p) = ∞.

Note: U-rank is formally denoted ${\displaystyle U_{n}(p)}$, 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.

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(q) ≥ U(p).

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, ${\displaystyle U_{n}(T)=\sup\{U_{n}(p):p\in S(T)\}}$.

We then get a concise characterization of superstability; a stable theory T is superstable if and only if ${\displaystyle U_{n}(T)<\infty }$ for every n.