A trichotomy theorem for o‐minimal structures
A trichotomy theorem for o‐minimal structures
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o 最小结构的三分定理
DOI：
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发表时间：
1998
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影响因子：
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通讯作者：
S. Starchenko
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作者：
Y. Peterzil;S. Starchenko
设M = M，<，. M是线性序结构.我们定义M是o极小的，如果M的每个可定义子集是区间的有限并。经典的例子是有序可分阿贝尔群和真实的闭域。我们证明了任意极小M在M中任意a的邻域上诱导的结构的一个可分性定理。粗略地说，以下之一成立：（i）a是平凡的（技术术语），或（ii）a有一个凸邻域，在该邻域上M诱导有序向量空间的结构，或（iii）a包含在一个开区间中，在该开区间上M诱导真实的闭域的展开的结构。证明使用'几何演算'，它允许一个恢复一个微分结构，由纯粹的几何方法。1991年数学学科分类：小学03 C45;中学03 C52、12 J15、14 P10。
Let M = 〈M, <, …〉 be alinearly ordered structure. We define M to be o‐minimal if every definable subset of M is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo‐minimal M can induce on a neighbourhood of any a in M. Roughly said, one of the following holds: (i) a is trivial (technical term), or (ii) a has a convex neighbourhood on which M induces the structure of an ordered vector space, or (iii) a is contained in an open interval on which M induces the structure of an expansion of a real closed field. The proof uses ‘geometric calculus’ which allows one to recover a differentiable structure by purely geometric methods. 1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.