View source for U-rank

In [[model theory]], a branch of mathematical logic, '''U-rank''' is one measure of the complexity of a (complete) [[type (model theory)|type]], in the context of [[stable theory|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, [[Stable theory#Superstable theories|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'')&nbsp;≥&nbsp;''δ'' precisely when ''U''(''p'')&nbsp;≥&nbsp;''α'' for all ''α'' less than ''δ''
* For any ''α''&nbsp;=&nbsp;''β''&nbsp;+&nbsp;1, ''U''(''p'')&nbsp;≥&nbsp;''α'' precisely when there is a forking extension ''q'' of ''p'' with ''U''(''q'')&nbsp;≥&nbsp;''β''

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

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

Note: U-rank is formally denoted <math>U_n(p)</math>, 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 '''[[Monotonic function#Monotonicity in order theory|monotone]]''' in its domain.  That is, suppose ''p'' is a complete type over ''A'' and ''B'' is a subset of&nbsp;''A''.  Then for ''q'' the restriction of ''p'' to ''B'', ''U''(''q'')&nbsp;≥&nbsp;''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&nbsp;''α''.  Thus, we can define, for a complete (stable) theory ''T'', <math>U_n(T)=\sup \{ U_n(p) : p\in S(T) \}</math>.

We then get a concise characterization of superstability; a stable theory ''T'' is superstable if and only if <math>U_n(T)<\infty</math> for every&nbsp;''n''.

== Properties ==

* As noted above, U-rank is monotone in its domain.
* If ''p'' has U-rank ''α'', then for any ''β''&nbsp;<&nbsp;''α'', there is a forking extension ''q'' of ''p'' with U-rank&nbsp;''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''&nbsp;<&nbsp;''β'', there is a type ''p'' of rank ''α'', and if the rank of ''p'' is greater than ''β'', then it must be&nbsp;∞.

== Examples ==

* ''U''(''p'')&nbsp;>&nbsp;0 precisely when ''p'' is nonalgebraic.
* If ''T'' is the theory of [[algebraically closed field]]s (of any fixed characteristic) then <math>U_1(T)=1</math>.  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 ==

{{cite book |last1=Pillay |first1=Anand |title=An Introduction to Stability Theory |year=2008 |origyear=1983 |publisher=Dover |isbn=978-0-486-46896-9 |page=57}}

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[[Category:Model theory]]