扫描、多胞体、定向拟阵与排列的容许图
Sweeps, polytopes, oriented matroids, and allowable graphs of permutations
AI总结：
本文引入扫描定向拟阵与排列的容许图两种推广，以建模高维点构型的扫描偏序集，揭示了其与模超平面、Dilworth截断及广义Baues问题的联系，并证明伪扫描偏序集形变收缩为球面。
摘要：
点构型的扫描是由线性泛函诱导的任何有序划分。平面点构型的扫描偏序集由Goodman和Pollack在排列的容许序列理论下形式化并抽象化。我们引入了两种推广，以对高维构型的扫描偏序集进行建模。模仿点构型的扫描多胞体（即相关zonotope的单调路径多胞体）是排列多面体的投影这一事实，我们将扫描定向拟阵定义为辫子定向拟阵的强映射。容许序列即为秩为2的扫描定向拟阵，其许多性质可推广至更高秩。我们展示了扫描定向拟阵与（无定向）拟阵理论中的模超平面及Dilworth截断之间的紧密联系。我们还探索了它们与元胞字符串的广义Baues问题的联系，其中扫描定向拟阵可扮演单调路径多胞体的角色，甚至对于不可实现的定向拟阵也是如此。特别地，我们证明对于容许扫描定向拟阵的定向拟阵，其伪扫描偏序集形变收缩为适当维度的球面。第二种推广是排列的容许图：由容许序列两两连接的排列对称集合。它们具有无环胚的结构，并包含扫描定向拟阵。
英文摘要：
A sweep of a point configuration is any ordered partition induced by a linear functional. Posets of sweeps of planar point configurations were formalized and abstracted by Goodman and Pollack under the theory of allowable sequences of permutations. We introduce two generalizations that model posets of sweeps of higher dimensional configurations. Mimicking the fact that sweep polytopes of point configurations (the monotone path polytopes of the associated zonotopes) are projections of permutahedra, we define sweep oriented matroids as strong maps of the braid oriented matroid. Allowable sequences are then the sweep oriented matroids of rank 2, and many of their properties extend to higher rank. We show strong ties between sweep oriented matroids and both modular hyperplanes and Dilworth truncations from (unoriented) matroid theory. We also explore their connection with the generalized Baues problem for cellular strings, where sweep oriented matroids can play the role of monotone path polytopes, even for non-realizable oriented matroids. In particular, we show that for oriented matroids that admit a sweep oriented matroid, their poset of pseudo-sweeps deformation retracts to a sphere of the appropriate dimension. A second generalization are allowable graphs of permutations: symmetric sets of permutations pairwise connected by allowable sequences. They have the structure of acycloids and include sweep oriented matroids.
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