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Generic derivations on o-minimal structures
Journal of Mathematical Logic ( IF 0.9 ) Pub Date : 2020-10-06 , DOI: 10.1142/s0219061321500070 Antongiulio Fornasiero 1 , Elliot Kaplan 2
Journal of Mathematical Logic ( IF 0.9 ) Pub Date : 2020-10-06 , DOI: 10.1142/s0219061321500070 Antongiulio Fornasiero 1 , Elliot Kaplan 2
Affiliation
Let T be a complete, model complete o-minimal theory extending the theory R C F of real closed ordered fields in some appropriate language L . We study derivations δ on models ℳ ⊧ T . We introduce the notion of a T -derivation: a derivation which is compatible with the L ( ∅ ) -definable 𝒞 1 -functions on ℳ . We show that the theory of T -models with a T -derivation has a model completion T G δ . The derivation in models ( ℳ , δ ) ⊧ T G δ behaves “generically”, it is wildly discontinuous and its kernel is a dense elementary L -substructure of ℳ . If T = R C F , then T G δ is the theory of closed ordered differential fields (CODFs) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that T G δ has T as its open core, that T G δ is distal, and that T G δ eliminates imaginaries. We also show that the theory of T -models with finitely many commuting T -derivations has a model completion.
中文翻译:
o-最小结构的一般推导
让吨 是一个完整的,模型完整的 o-minimal 理论扩展该理论R C F 一些适当语言的真正封闭有序字段大号 . 我们研究推导δ 在模型上ℳ ⊧ 吨 . 我们引入一个概念吨 -派生:与大号 ( ∅ ) - 可定义𝒞 1 - 功能开启ℳ . 我们证明了理论吨 - 模型与吨 -派生有一个模型完成吨 G δ . 模型中的推导( ℳ , δ ) ⊧ 吨 G δ 表现“一般”,它是非常不连续的,它的内核是一个密集的基本元素大号 - 的子结构ℳ . 如果吨 = R C F , 然后吨 G δ 是由 Michael Singer 介绍的封闭有序微分场 (CODF) 理论。我们能够在我们的环境中恢复许多关于 CODF 的已知事实。除其他外,我们表明吨 G δ 已吨 作为它的开放核心,吨 G δ 是远端,并且吨 G δ 消除想象。我们还证明了理论吨 -具有有限多通勤的模型吨 -derivations 有一个模型完成。
更新日期:2020-10-06
中文翻译:
o-最小结构的一般推导
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