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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 , 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「U-rank」の詳細全文を読む スポンサード リンク
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