Motivated by some computations of Feynman integrals and certain conjectures on mixed Tate motives, Bejleri and Marcolli posed questions about the ${\mathbb{F}}_1$-structure (in the sense of torification) on the complement of a hyperplane arrangement, especially for an arrangement defined in the space of cycles of a graph.

In this paper, we prove that an arrangement has an ${\mathbb{F}}_1$-structure if and only if it is Boolean. We also prove that the arrangement in the cycle space of a graph is Boolean if and only if the cycle space has a basis consisting of cycles such that any two of them do not share edges.

中文翻译：

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在 F1 上定义的费曼图和超平面排列

受费曼积分的一些计算和混合泰特动机的某些猜想的启发，Bejleri 和 Marcolli 提出了关于 ${\mathbb{F}}_1$-结构（在torification的意义上）在超平面排列的补充上，特别是对于在图的循环空间中定义的排列。

在本文中，我们证明了一个安排有 ${\mathbb{F}}_1$-结构当且仅当它是布尔值。我们还证明了图的循环空间中的排列是布尔的，当且仅当循环空间具有由循环组成的基，使得它们中的任何两个不共享边。