论文标题
SOP $ _1 $，SOP $ _2 $和Antichain Tree Property
SOP$_1$, SOP$_2$, and antichain tree property
论文作者
论文摘要
在本文中，我们研究了一些树的特性及其相关的不明智。首先，我们证明可以通过公式可以看到SOP $ _2 $，其中有一棵树树，其中一棵树属于“任意均匀不一致”（例如，弱k-tp $ _1 $条件或其他可能的不一致配置）。我们介绍了树木印象的概念，该概念保留了SOP $ _1 $的证人，并通过使用此概念，我们研究了（in）平等的问题SOP $ _1 $和SOP $ _2 $。假设存在具有SOP $ _1 $的公式，以至于没有有限的连词具有SOP $ _2 $，我们观察到该公式必须见证某些类似树的类似树的现象，我们将其称为抗小树的属性（ATP，ATP，请参见定义4.1）。我们表明，ATP意味着SOP $ _1 $和TP $ _2 $，但是每个含义的相反都不成立。因此，NATP理论类（没有ATP的理论）包含NSOP $ _1 $理论的类别和NTP $ _2 $理论的类。在本文的最后，我们构建了一个结构，其理论具有具有ATP的公式，但是该公式的任何连词都没有SOP $ _2 $。因此，此示例表明，在公式的级别，即有一个具有SOP $ _1 $的公式，而其任何有限的连接都不会见证SOP $ _2 $（但这种变体的公式仍然具有SOP $ _2 $）。
In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP$_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$_2$).
