We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs and matroids, but can be much more general or quite different. The theorem asserts a duality between the existence of high cohesiveness somewhere local and a global overall tree structure. We describe cohesive substructures in a unified way in the format of tangles: as orientations of low-order separations satisfying certain consistency axioms. These axioms can be expressed without reference to the underlying structure, such as a graph or matroid, but just in terms of the poset of the separations themselves. This makes it possible to identify tangles, and apply our tangle-tree duality theorem, in very diverse settings. Our result implies all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width or rank-width. It yields new, tangle-type, duality theorems for tree-width and path-width. It implies the existence of width parameters dual to cohesive substructures such as $k$-blocks, edge-tangles, or given subsets of tangles, for which no width duality theorems were previously known. Abstract separation systems can be found also in structures quite unlike graphs and matroids. For example, our theorem can be applied to image analysis by capturing the regions of an image as tangles of separations defined as natural partitions of its set of pixels. It can be applied in big data contexts by capturing clusters as tangles. It can be applied in the social sciences, e.g. by capturing as tangles the few typical mindsets of individuals found by a survey. It could also be applied in pure mathematics, e.g. to separations of compact manifolds.

中文翻译：

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抽象分离系统中的缠结树对偶性

我们证明了具有明确定义的内聚和分离概念的组合结构的一般宽度对偶定理。这些可能是图形和拟阵，但可以更通用或完全不同。该定理断言局部某处高内聚性的存在与全局整体树结构之间存在二元性。我们以缠结的格式以统一的方式描述内聚子结构：作为满足某些一致性公理的低阶分离的方向。这些公理可以在不参考底层结构（例如图或拟阵）的情况下表达，而只是根据分离本身的偏序集。这使得在非常多样化的环境中识别缠结并应用我们的缠结树对偶定理成为可能。我们的结果暗示了图次要理论中宽度参数的所有经典对偶定理，例如路径宽度、树宽度、分支宽度或秩宽度。它产生了新的、缠结型的、树宽和路径宽的对偶定理。这意味着存在与内聚子结构（例如 $k$-blocks、edge-tangles 或给定的 tangles 子集）对偶的宽度参数，对于这些子结构，以前不知道宽度对偶定理。抽象分离系统也可以在与图和拟阵完全不同的结构中找到。例如，我们的定理可以通过将图像的区域捕获为定义为像素集的自然分区的分离缠结来应用于图像分析。它可以通过将集群捕获为缠结来应用于大数据环境中。它可以应用于社会科学，例如 通过将调查中发现的个人的少数典型心态捕捉为缠结。它也可以应用于纯数学，例如用于紧凑流形的分离。