精炼抽象分离系统中的缠结树：非本质部分
Refining trees of tangles in abstract separation systems: inessential parts
AI总结：
本文将 Erde 统一图中缠结树定理与缠结-树对偶定理的结果推广到抽象分离系统，以刻画一般离散结构中 $k$-缠结的数量与位置。
摘要：
Robertson 和 Seymour 证明了关于图中缠结的两个基本定理：缠结树定理指出，每个图都存在一个树分解，使得可区分的缠结位于树的不同节点中；缠结-树对偶定理指出，不含 $k$-缠结的图具有一个树分解，可见证此类缠结不存在，因为 $k$-缠结本须栖居于某个节点，但没有任何节点大到足以容纳一个。Erde 将这两个基本定理合二为一：他构造了单个树分解，使每个节点要么容纳恰好一个 $k$-缠结，要么过小而无法容纳任何 $k$-缠结。因此，这种树分解一眼就能显示一个图有多少个 $k$-缠结以及它们位于何处。此后，这两个基本定理已被推广到抽象分离系统，该系统支持更一般离散结构中的缠结。本文将 Erde 的统一定理推广到此类一般系统。
英文摘要：
Robertson and Seymour proved two fundamental theorems about tangles in graphs: the tree-of-tangles theorem, which says that every graph has a tree-decomposition such that distinguishable tangles live in different nodes of the tree, and the tangle-tree duality theorem, which says that graphs without a $k$-tangle have a tree-decomposition that witnesses the non-existence of such tangles, in that $k$-tangles would have to live in a node but no node is large enough to accommodate one. Erde combined these two fundamental theorems into one, by constructing a single tree-decomposition such that every node either accommodates a single $k$-tangle or is too small to accommodate one. Such a tree-decomposition thus shows at a glance how many $k$-tangles a graph has and where they are. The two fundamental theorems have since been extended to abstract separation systems, which support tangles in more general discrete structures. In this paper we extend Erde's unified theorem to such general systems.
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。
