SOP$_1$、SOP$_2$ 与反链树性质
SOP$_1$, SOP$_2$, and antichain tree property
AI总结：
本文通过树不可辨别性提出反链树性质 ATP，刻画其与 SOP$_1$、TP$_2$ 的蕴含关系，并构造公式级反例证明 SOP$_1$ 不等同于 SOP$_2$。
摘要：
本文研究若干树性质及其相关的不可辨别性。首先，我们证明 SOP$_2$ 可由一个带有元组树的公式见证，该树满足“任意齐次不相容性”（例如弱 k-TP$_1$ 条件或其他可能的不相容构型）。我们引入一种树不可辨别性概念，它保持 SOP$_1$ 的见证；利用这一点，我们考察 SOP$_1$ 与 SOP$_2$ 是否相等的问题。假设存在一个具有 SOP$_1$ 的公式，但其任何有限合取都不具有 SOP$_2$，我们观察到该公式必定见证某种类似树性质的现象，我们称之为反链树性质（ATP，见定义 4.1）。我们证明 ATP 蕴含 SOP$_1$ 和 TP$_2$，但这两个蕴含的逆均不成立。因此，NATP 理论类（不含 ATP 的理论）包含 NSOP$_1$ 理论类和 NTP$_2$ 理论类。在论文末尾，我们构造一个结构，其理论中有一个公式具有 ATP，但该公式的任何合取都不具有 SOP$_2$。因此该例子表明，在公式层面 SOP$_1$ 与 SOP$_2$ 并不相同：即存在一个公式具有 SOP$_1$，而其任何有限合取都不能见证 SOP$_2$（但该公式的一个变体仍具有 SOP$_2$）。
英文摘要：
In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP$_2$ can be witnessed by a formula with a tree of tuples holding 'arbitrary homogeneous inconsistency' (e.g., weak k-TP$_1$ conditions or other possible inconsistency configurations). And we introduce a notion of tree-indiscernibility, which preserves witnesses of SOP$_1$, and by using this, we investigate the problem of (in)equality of SOP$_1$ and SOP$_2$. Assuming the existence of a formula having SOP$_1$ such that no finite conjunction of it has SOP$_2$, we observe that the formula must witness some tree-property-like phenomenon, which we will call the antichain tree property (ATP, see Definition 4.1). We show that ATP implies SOP$_1$ and TP$_2$, but the converse of each implication does not hold. So the class of NATP theories (theories without ATP) contains the class of NSOP$_1$ theories and the class of NTP$_2$ theories. At the end of the paper, we construct a structure whose theory has a formula having ATP, but any conjunction of the formula does not have SOP$_2$. So this example shows that SOP$_1$ and SOP$_2$ are not the same at the level of formulas, i.e., there is a formula having SOP$_1$, while any finite conjunction of it does not witness SOP$_2$ (but a variation of the formula still has SOP$_2$).
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