U-rank

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.

Definition

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

• $U(p)\geq 0$
• If $\delta$ is a limit ordinal, then $U(p)\geq \delta$ iff $U(p)\geq \alpha$ for all $\alpha <\delta$
• If $\alpha =\beta +1$, then we say $U(p)\geq \alpha$ iff there is some $B\supset A$ and some q, an n-type over B, a forking extension of p with $U(q)\geq \beta$

We say that the U-rank of p is precisely $\alpha$ when the U-rank is at least, but not more than, $\alpha$.

If there is no $\alpha$ above the U-rank of p, we say the U-rank is unbounded, or $U(p)=\infty$.

Note: U-rank is formally denoted $U_{n}(p)$, where n is the arity of p. This subscript is typically omitted when no confusion can result.

Ranking Theories

U-rank is monotone in its domain. That is, if $p\in S_{n}(A)$ and $B\subset A$, then for $q=p\upharpoonright B$, $U(q)\geq U(p)$.

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

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

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 (that is, p has U-rank ∞), 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(p)>0 precisely when p is nonalgebraic.
• If T is the theory of algebraically closed fields (of any fixed characteristic) then U_1(T)=1. 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 (all realizations of) p are transcendental over K, and 0 otherwise. More generally, an n-type p over A has U-rank k, the transcendence degree (over K) of any realization of it.