On Variants of o-Minimality
On Variants of o-Minimality
复制标题
关于 o-极小性的变体
DOI：
10.1016/0168-0072(95)00037-2
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
C. Steinhorn
中科院分区：
文献类型：
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作者：
D. Macpherson;C. Steinhorn
零极小性和强极小性假设产生了丰富的模型理论结果。这两类模型的特点是，在任何基本等价结构中，一个变量的可定义集恰好是在给定结构的某些“基本”约化中可定义的无量词集。在o- minimal&y的情况下，约简是一个总顺序，而在强minimal&y中，它只是相等。这里我们的目标是开始研究，如果要考虑的所有结构都是其他一些基本关系结构的适当“最小”扩展，结果会是什么。指导我们工作的标准是基本结构在数学上有有趣的最小展开这类最小展开有一些合理的模型理论类似于在激励环境中可用的模型理论。由于Adeleke， Neumann和第一作者[1 - 3,20]，我们对基本结构的选择受到一些尚未发表的关于Jordan群的工作的影响。为了方便不熟悉的读者，我们现在对这项工作作一些详细的概述。让(G;X)是一个置换群，即G是集合X上的一个置换群。如果a cX，则GcA)，分别GIAl表示a在G上的点上的、集上的稳定子。如果不存在X的非平凡G不变分割，则称置换群为本原的。称置换群为k可传递的，其中k EN，如果G在X的不同元素的有序k元组的集合上可传递，如果子集A c X在A上可传递，则子集A c X是（G； X）的约当集，集A称为固有约当集
The hypotheses of o-minimality and strong minimality have yielded rich modeltheoretic consequences. The models in each of these classes are characterized by the property that the definable sets in one variable in any elementarily equivalent structure are exactly those which are quantifier-free definable in some “basic” reduct of the given structure. In the case of o-minimal@ the reduct is a total order, and in strong minimal&y it is just equality. Our goal here is to begin to investigate what results hold if all the structures to be considered are appropriately “minimal” expansions of some other basic relational structure. We are guided in our work by the criteria that the basic structures have mathematically interesting minimal expansions and that the class of minimal expansions has some reasonable model theory analogous to that available in the motivating contexts.Our choice of basic structures is influenced by some as yet unpublished work on Jordan groups, due to Adeleke, Neumann, and the first author [l-3, 20]. We now sketch this work in some detail for readers unfamiliar with it. Let (G; X) be a permutation group, that is, G is a permutation group on a set X. If A cX, then GcA), respectively GIAl, denotes the pointwise, respectively setwise, stabilizer of A in G. The permutation group is called primitive if there does not exist a non-trivial G-invariant partition of X. It is said to be k-transitive, where k EN, if G is transitive on the set of ordered k-tuples of distinct elements of X, and is called highly transitive if it is k-transitive for all k E N. A subset A c X is a Jordan set for (G; X) if IAl> 1 and GCXiA) is transitive on A. The set A is called a proper Jordan set