超平面的Chambers of Arrow's不可能定理
Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem
摘要：
设${\mathcal A}$是超平面的非空实中心排列，${\rm \bf Ch}$是${\mathcal A}$的腔室集合.每个超平面$H$定义一个半空间$H^{+} $和另一个半空间$H^{-}$。让PDMATHAMR。对于$H\in {\mathcal A}$，通过$ε_{H}^{+} (C)=+ \text{(if} C\subseteq H^{+}) \text{and} ε_{H}^{+} (C)= - \text{(if} C\subseteq H^{-}).$定义一个映射$ε_{H}^{+} : {\rm \bf Ch} \to B$定义$ε_{H}^{-}=-ε_{H}^{+}.$设${\rm \bf Ch}^{m} = {\rm \bf Ch}\times{\rm \bf Ch}\times...\times{\rm \bf Ch} (m\text{times}).$然后映射$ε_{H}^{\pm}$导出映射$ε_{H}^{\pm} : {\rm \bf Ch}^{m} \to B^{m} $。我们将研究与每个$ε_{H}^{\pm}$相容的容许映射$Φ: {\rm \bf Ch}^{m} \to {\rm \bf Ch}$。假设$|{\mathcal A}|\geq 3$和$m\geq 2$。然后证明了${\mathcal A}$是不可分解的当且仅当每个容许映射都是到一个分量的投影。当${\mathcal A}$是不可分解的辫状排列时，这个结果等价于经济学中的阿罗不可能定理。我们还确定了一组允许的地图明确每一个非空的真正的中心安排。
英文摘要：
Let ${\mathcal A}$ be a nonempty real central arrangement of hyperplanes and ${\rm \bf Ch}$ be the set of chambers of ${\mathcal A}$. Each hyperplane $H$ defines a half-space $H^{+} $ and the other half-space $H^{-}$. Let $B = \{+, -\}$. For $H\in {\mathcal A}$, define a map $ε_{H}^{+} : {\rm \bf Ch} \to B$ by $ε_{H}^{+} (C)=+ \text{(if} C\subseteq H^{+}) \text{and} ε_{H}^{+} (C)= - \text{(if} C\subseteq H^{-}).$ Define $ε_{H}^{-}=-ε_{H}^{+}.$ Let ${\rm \bf Ch}^{m} = {\rm \bf Ch}\times{\rm \bf Ch}\times...\times{\rm \bf Ch} (m\text{times}).$ Then the maps $ε_{H}^{\pm}$ induce the maps $ε_{H}^{\pm} : {\rm \bf Ch}^{m} \to B^{m} $. We will study the admissible maps $Φ: {\rm \bf Ch}^{m} \to {\rm \bf Ch}$ which are compatible with every $ε_{H}^{\pm}$. Suppose $|{\mathcal A}|\geq 3$ and $m\geq 2$. Then we will show that ${\mathcal A}$ is indecomposable if and only if every admissible map is a projection to a omponent. When ${\mathcal A}$ is a braid arrangement, which is indecomposable, this result is equivalent to Arrow's impossibility theorem in economics. We also determine the set of admissible maps explicitly for every nonempty real central arrangement.
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