AI 中文总结
该研究对基础拟阵为正则的有限中心排列的自由性给出完全独立于基域的组合分类，利用正则拟阵的Seymour分解定理证明自由性强制超可解性，还刻画了有限简单二元拟阵的良好划分条件。
AI 中文摘要
我们对基础拟阵为正则的有限中心排列的自由性进行分类。设$\boldsymbol{\text{𝓐}}$为任意域上的此类排列，记$M=M(\boldsymbol{\text{𝓐}})$，则$\boldsymbol{\text{𝓐}}$是自由的当且仅当$M$是超可解的，等价于$M$存在良好划分，等价于$M$是弦简单图的圈拟阵。因此，对于具有正则基础拟阵的排列，自由性具有完全独立于基域的组合分类。我们利用正则拟阵的Seymour分解定理证明自由性强制超可解性，还刻画了有限简单二元拟阵的良好划分：划分是良好的当且仅当它是独立的且没有直线包含在单个块中，由此得出有限无环二元拟阵存在良好划分当且仅当它是简单且超可解的。
英文摘要
We classify freeness for finite central arrangements whose underlying matroids are regular. Let $\mathcal A$ be such an arrangement over an arbitrary field, and put $M=M(\mathcal A)$. Then $\mathcal A$ is free if and only if $M$ is supersolvable; equivalently, $M$ admits a nice partition; equivalently, $M$ is the cycle matroid of a chordal simple graph. Thus, for arrangements with regular underlying matroids, freeness has a complete combinatorial classification independent of the base field. We use Seymour's decomposition theorem for regular matroids to prove that freeness forces supersolvability. We also characterize nice partitions of finite simple binary matroids: a partition is nice if and only if it is independent and no line is contained in a single block. Consequently, a finite loopless binary matroid admits a nice partition if and only if it is simple and supersolvable.
