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'') ≥ 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 $U_n(p)$, 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.

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''(''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'', $U_n(T)=\sup \$.

We then get a concise characterization of superstability; a stable theory ''T'' is superstable if and only if $U_n(T)<\infty$ 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 (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.

References

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Model theory