In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory).

在本文中，我们介绍了一个 AEC 框架，用于研究具有通勤自同构的域。具有通勤自同构的域与差域密切相关。一些作者将差分环（或域）定义为一个环（或域）加上几个通勤自同态，而另一些作者只研究一个自同态。Z. Chatzidakis 和E. Hrushovski 深入研究了ACFA 的模型论，ACFA 是一个自同构的差场模型伴。我们具有通勤自同构的领域概括了这种设置。我们有几个自同构，它们需要通勤。Hrushovski 已经证明，在具有两个或多个通勤自同构的域的情况下，存在封闭模型不一定形成一阶模型类。在本文中，我们介绍了 FCA 类，一个 AEC 框架，用于研究具有通勤自同构的场论的存在封闭模型。我们证明一个 FCA 类有 AP 和 JEP，因此是一个怪物模型，Galois 类型与存在封闭模型中的存在类型一致，该类是齐次的，并且有一个类型合并定理的版本允许组合三个特定条件下的类型。最后，我们使用这些结果表明我们的怪物模型是 S. Buechler 和 O. Lessman 意义上的简单同质结构（这是简单一阶理论的分类理论概念的非基本类比）。Galois 类型与存在封闭模型中的存在类型一致，该类是同质的，并且有一个版本的类型合并定理允许在特定条件下组合三种类型。最后，我们使用这些结果表明我们的怪物模型是 S. Buechler 和 O. Lessman 意义上的简单同质结构（这是简单一阶理论的分类理论概念的非基本类比）。Galois 类型与存在封闭模型中的存在类型一致，该类是同质的，并且有一个版本的类型合并定理允许在特定条件下组合三种类型。最后，我们使用这些结果表明我们的怪物模型是 S. Buechler 和 O. Lessman 意义上的简单同质结构（这是简单一阶理论的分类理论概念的非基本类比）。