Tangle-tree duality in abstract separation systems

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