Structures having o-minimal open core
Structures having o-minimal open core
复制标题
具有最小开放核心的结构
DOI：
10.1090/s0002-9947-09-04908-3
复制
发表时间：
2009
影响因子：
1.3
通讯作者：
C. Steinhorn
中科院分区：
文献类型：
--
作者：
Alfred Dolich;Chris Miller;C. Steinhorn
稠密线性序的展开的开放核心是它的约简，在可定义性的意义上，由它的所有开放可定义集的集合生成。本文研究了具有o-极小开核的稠密线性序的展开式，重点研究了稠密序群的展开式。第一个主要结果建立的条件下，一个密集有序群的扩展有一个O-最小的开放核心。具体地说，证明了：设R是稠密序群（R，<，*）的扩张，它是可定义完备的，并且满足一致有限性。则R的开核是O-极小的。两个例子类的结构，不是O-最小的，但有O-最小的开放的核心进行了讨论：稠密对O-最小的扩张有序群，O-最小结构的一般谓词的扩展。特别地，这样的结构具有与原始的o-最小结构可相互定义的开放核心。这些例子的区别在于存在可定义的一元函数，其图形在平面上是稠密的，这种现象可能发生在稠密对中，但不会发生在一般谓词的扩展中。研究了无稠密图的性质与一致有限性、可定义完备性和具有o-极小开核的关系。
The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions under which an expansion of a densely ordered group has an o-minimal open core. Specifically, the following is proved: Let R be an expansion of a densely ordered group (R, <, *) that is definably complete and satisfies the uniform finiteness property. Then the open core of R is o-minimal. Two examples of classes of structures that are not o-minimal yet have o-minimal open core are discussed: dense pairs of o-minimal expansions of ordered groups, and expansions of o-minimal structures by generic predicates. In particular, such structures have open core interdefinable with the original o-minimal structure. These examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane, a phenomenon that can occur in dense pairs but not in expansions by generic predicates. The property of having no dense graphs is examined and related to uniform finiteness, definable completeness, and having o-minimal open core.