更多关于树的属性
More on tree properties
摘要：
树的性质是由Shelah引入的，众所周知，一个理论有TP（树的性质）当且仅当它有TP $_1$或TP $_2$。在任何简单的理论中（即，一个没有TP的理论），分叉提供了一个很好的独立性概念，因为它满足对称性，传递性，扩展性，局部特征和类型合并。Shelah还引入了SOP $_n$（$n$ -强序性质）。最近证明了在任何NSOP $_1$理论（即不具有SOP $_1$的理论）中，Kim-forking也满足除基单调性（传递性的一个方向）以外的所有上述独立性.这些结果是本文的动机来源。我们主要为SOP $_2$（相当于TP $_1$）和SOP $_1$制定类型计数标准。此外，我们还研究了TP $_2$与Kim分叉之间的关系，得到了一个理论是超单的当且仅当不存在可数无限Kim分叉链.
英文摘要：
Tree properties are introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP$_1$ or TP$_2$. In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion as it satisfies symmetry, transitivity, extension, local character, and type-amalgamation. Shelah also introduced SOP$_n$ ($n$-strong order property). Recently it is proved that in any NSOP$_1$ theory (i.e. a theory not having SOP$_1$) holding nonforking existence, Kim-forking also satisfies all the mentioned independence properties except base monotonicity (one direction of transitivity). These results are the sources of motivation for this paper. Mainly, we produce type-counting criteria for SOP$_2$ (which is equivalent to TP$_1$) and SOP$_1$. In addition, we study relationships between TP$_2$ and Kim-forking, and obtain that a theory is supersimple iff there is no countably infinite Kim-forking chain.
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