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In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) which is totally ordered by < is called an o-minimal structure if and only if every definable subset ''X'' ⊂ ''M'' (with parameters taken from ''M'') is a finite union of intervals and points. O-minimality can be regarded as a weak form of quantifier elimination. A structure ''M'' is o-minimal if and only if every formula with one free variable and parameters in ''M'' is equivalent to a quantifier-free formula involving only the ordering, also with parameters in ''M''. This is analogous to the minimal structures, which are exactly the analogous property down to equality. A theory ''T'' is an o-minimal theory if every model of ''T'' is o-minimal. It is known that the complete theory ''T'' of an o-minimal structure is an o-minimal theory.〔Knight, Pillay and Steinhorn (1986), Pillay and Steinhorn (1988).〕 This result is remarkable because the complete theory of a minimal structure need not be a strongly minimal theory, that is, there may be an elementarily equivalent structure which is not minimal. ==Set-theoretic definition== O-minimal structures can be defined without recourse to model theory. Here we define a structure on a nonempty set ''M'' in a set-theoretic manner, as a sequence ''S'' = (''S''''n''), ''n'' = 0,1,2,... such that # ''S''''n'' is a boolean algebra of subsets of ''M''''n'' # if ''A'' ∈ ''S''''n'' then ''M'' × ''A'' and ''A'' ×''M'' are in ''S''''n''+1 # the set is in ''S''''n'' # if ''A'' ∈ ''S''''n''+1 and ''π'' : ''M''''n''+1 → ''M''''n'' is the projection map on the first ''n'' coordinates, then ''π''(''A'') ∈ ''S''''n''. If ''M'' has a dense linear order without endpoints on it, say <, then a structure ''S'' on ''M'' is called o-minimal if it satisfies the extra axioms
The "o" stands for "order", since any o-minimal structure requires an ordering on the underlying set. 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「O-minimal theory」の詳細全文を読む スポンサード リンク
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