$\mathbb{Q}$ ACFA
$\mathbb{Q}$ACFA
摘要：
我们表明，许多很好的性质的理论$T$遵循相应的属性，其减少有限的子签名。如果$\{ T_i \}_{i \in I}$是一阶理论的保守展开的有向族，并且每个$T_i$都是稳定的（分别是简单的、玫瑰色的、依赖的、子模型完全的、模型完全的、可伴的），那么联合$T := \cup_i T_i$也是稳定的。在大多数情况下，$T$中的（thorn）-forking等价于某些$T_i$中代数闭包的（thorn）-forking。这适用于具有$(\mathbb{Q}, +)$的动作的域，其对有限子签名的约简与具有一个自同构的域的理论是可相互定义的。我们表明，该理论的模型伴侣$\mathbb{Q}$ ACFA是严格简单的，并具有相同的水平的量词消除和相同的代数特征的代数封闭和分叉的独立ACFA。命名的自同构的固定域的格打破了$\mathbb{Q}$ ACFA中的超简单性，但远离这些，我们发现许多（弱）极小公式。
英文摘要：
We show that many nice properties of a theory $T$ follow from the corresponding properties of its reducts to finite subsignatures. If $\{ T_i \}_{i \in I}$ is a directed family of conservative expansions of first-order theories and each $T_i$ is stable (respectively, simple, rosy, dependent, submodel complete, model complete, companionable), then so is the union $T := \cup_i T_i$. In most cases, (thorn)-forking in $T$ is equivalent to (thorn)-forking of algebraic closures in some $T_i$. This applies to fields with an action by $(\mathbb{Q}, +)$, whose reducts to finite subsignatures are interdefinable with the theory of fields with one automorphism. We show that the model companion $\mathbb{Q}$ACFA of this theory is strictly simple and has the same level of quantifier elimination and the same algebraic characterization of algebraic closure and forking independence as ACFA. The lattice of the fixed fields of the named automorphisms breaks supersimplicity in $\mathbb{Q}$ACFA, but away from these we find many (weakly) minimal formulas.
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