Let T be a complete, model complete o-minimal theory extending the theory RCF 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 𝒞1-functions on ℳ. We show that the theory of T-models with a T-derivation has a model completion TGδ. The derivation in models (ℳ,δ)⊧TGδ behaves “generically”, it is wildly discontinuous and its kernel is a dense elementary L-substructure of ℳ. If T = RCF, then TGδ 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 TGδ has T as its open core, that TGδ is distal, and that TGδ 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：

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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RCF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𝒞1- 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Gδ. 模型中的推导(ℳ,δ)⊧吨Gδ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 = RCF， 然后吨Gδ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Gδ已吨作为它的开放核心，吨Gδ是远端，并且吨Gδ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。