Monster模型子集的轨道与几何理论
Orbits of subsets of the monster model and geometric theories
摘要：
设$\mathbb{M}$是完备一阶理论$T$的怪物模型。如果$\mathbb{D}$是$\mathbb{M}$的子集，则在D. Zambella我们考虑$e(\mathbb{D})=\{\mathbb{D}^\prime\mid (\mathbb{M},\mathbb{D})\equiv (\mathbb{M},\mathbb{D}^\prime)\}$和$o(\mathbb{D})=\{\mathbb{D}^\prime\mid (\mathbb{M},\mathbb{D})\cong (\mathbb{M},\mathbb{D}^\prime)\}$。我们通常会问的问题是，$e(\mathbb{D})=o(\mathbb{D})$何时开始？$\mathbb{D}$对于某个小集合$A$是$A$ -不变的情况相当简单：它只是意味着$\mathbb{D}$是可定义的。我们研究了$\mathbb{D}$在任何小子集上都不不变的情况。如果T是几何的，并且$(\mathbb{M},\mathbb{D})$是$H$ -结构（在A的意义上）. Berenstein和E.瓦西里耶夫）或一个可爱的一对，我们得到一些答案。在$SU$ -秩1的情况下，$e(\mathbb{D})$总是不同于$o(\mathbb{D})$。在o-minimal情况下，一切都可能发生，这取决于可定义闭包的复杂性。
英文摘要：
Let $\mathbb{M}$ be the monster model of a complete first-order theory $T$. If $\mathbb{D}$ is a subset of $\mathbb{M}$, following D. Zambella we consider $e(\mathbb{D})=\{\mathbb{D}^\prime\mid (\mathbb{M},\mathbb{D})\equiv (\mathbb{M},\mathbb{D}^\prime)\}$ and $o(\mathbb{D})=\{\mathbb{D}^\prime\mid (\mathbb{M},\mathbb{D})\cong (\mathbb{M},\mathbb{D}^\prime)\}$. The general question we ask is when $e(\mathbb{D})=o(\mathbb{D})$ ? The case where $\mathbb{D}$ is $A$-invariant for some small set $A$ is rather straightforward: it just mean that $\mathbb{D}$ is definable. We investigate the case where $\mathbb{D}$ is not invariant over any small subset. If T is geometric and $(\mathbb{M},\mathbb{D})$ is an $H$-structure (in the sense of A. Berenstein and E. Vassiliev) or a lovely pair, we get some answers. In the case of $SU$-rank one, $e(\mathbb{D})$ is always different from $o(\mathbb{D})$. In the o-minimal case, everything can happen, depending on the complexity of the definable closure.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
