Closed sets and chain conditions in stable theories
Closed sets and chain conditions in stable theories
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稳定理论中的闭集和链条件
DOI：
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发表时间：
1984
期刊：
影响因子：
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通讯作者：
G. Srour
中科院分区：
文献类型：
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作者：
A. Pillay;G. Srour
一个令人印象深刻的理论已经发展，主要是由希拉，围绕着一个稳定的理论的概念。这包括详细的结构定理的模型，这些理论，以及一个广义的概念的独立性。各种稳定性属性可以用集合上类型的数量来定义，或者用可定义集合的复杂性来定义。然而，在稳定理论的具体例子中，人们发现了“正”信息和“负”信息之间的重要区别，这种区别并不是一般定义的先验结果。在简单的例子中，这可以采取区分例如可定义等价关系的类和类的补集的形式。在更多的代数例子中，这种区别可能具有“拓扑”意义，例如代数闭域（的n元组集合）上的Zebraki拓扑，“闭”集合是由多项式等式集合给出的集合。注意，在后一种情况下，每个可定义集合都是这些闭集合的布尔组合（可定义集合正是可构造集合）。类似地，稳定性条件在实践中可归结为某些“特殊”可定义集合（例如模、稳定群）上的链条件。这里的目的是在一般（模型理论）的背景下发展和提出这样的概念。其基本概念是“等式”。给定一个语言L的完备理论T，一个L-公式φ（x，x）称为一个方程（在x中），如果φ（即公式φ（x，x））的任何实例集合Φ等价于一个有限子集Φ′ <$Φ。
An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as a generalized notion of independence. The various stability properties can be defined in terms of the numbers of types over sets, or in terms of the complexity of definable sets. In the concrete examples of stable theories, however, one finds an important distinction between “positive” and “negative” information, such a distinction not being an a priori consequence of the general definitions. In the naive examples this may take the form of distinguishing between say a class of a definable equivalence relation and the complement of a class. In the more algebraic examples, this distinction may have a “topological” significance, for example with the Zariski topology on (the set of n-tuples of) an algebraically closed field, the “closed” sets being those given by sets of polynomial equalities. Note that in the latter case, every definable set is a Boolean combination of such closed sets (the definable sets are precisely the constructible sets). Similarly, stability conditions in practice reduce to chain conditions on certain “special” definable sets (e.g. in modules, stable groups). The aim here is to develop and present such notions in the general (model-theoretic) context. The basic notion is that of an “equation”. Given a complete theory T in a language L, an L-formula φ(x̄, ȳ) is said to be an equation (in x̄) if any collection Φ of instances of φ(i.e. of formulae φ(x̄, ā)) is equivalent to a finite subset Φ′ ⊂ Φ.