Expansion of a model of a weakly o-minimal theory by a family of unary predicates
Expansion of a model of a weakly o-minimal theory by a family of unary predicates
复制标题
一元谓词族对弱 o-极小理论模型的扩展
DOI：
10.2307/2695114
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发表时间：
2001
影响因子：
0.6
通讯作者：
B. Baizhanov
中科院分区：
文献类型：
--
作者：
B. Baizhanov
全序结构M的子集A⊆M称为凸集，如果对任意a，b∈A：[a<；b→∀t(a<；Tb→t∈A)].一个一阶完备理论是弱o-极小的(M.Dickmann[D])，如果任一模型M完全按某种∅可定义公式排序，且M的任何子集可用来自M的参数定义是凸集的有限并。本文证明了对于弱o-极小理论T的任何模型M，M的任何一元谓词族展开M+都有弱o-极小理论当且仅当每个谓词的所有实现的集合是有限个凸集的并(定理63)。这解决了Cherlin-Macpherson-Marker-Steinhorn[MMS]对于弱o-极小理论的问题。
Abstract A subset A ⊆ M of a totally ordered structure M is said to be convex, if for any a, b ∈ A: [a < b → ∀t (a < tb → t ∈ A)]. A complete theory of first order is weakly o-minimal (M. Dickmann [D]) if any model M is totally ordered by some ∅-definable formula and any subset of M which is definable with parameters from M is a finite union of convex sets. We prove here that for any model M of a weakly o-minimal theory T. any expansion M+ of M by a family of unary predicates has a weakly o-minimal theory iff the set of all realizations of each predicate is a union of a finite number of convex sets (Theorem 63). that solves the Problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories.