符号位的一个扩张定理
An extension theorem for signotopes
摘要：
1926年，Levi证明，对于每个伪线排列$\mathcal{A}$和平面上的两个点，$\mathcal{A}$可以通过包含两个指定点的伪线来扩展。后来的可拓性研究安排的pseudohyperplanes在更高的维度。虽然在$\mathbb{R}^d$中的适当超平面的安排的可扩展性与包含$d$规定点的超平面是微不足道的，Richter-Gebert发现在$\mathbb{R}^3$中的伪平面的安排不能与包含两个特定规定点的伪平面一起扩展。在这篇文章中，我们调查的可扩展性signotopes，这是一个组合结构编码的一个丰富的子类的伪超平面安排。我们的主要结果是，奇数秩的signotopes是可扩展的意义上说，对于两个规定的交叉点，我们可以添加一个元素包含它们。此外，我们还猜想在所有偶数秩$r \geq 4$中，存在对两个指定点不可扩的符号表位.我们的猜想得到了秩4、6、8、10和12的例子的支持，这些例子是用基于SAT的方法发现的。
英文摘要：
In 1926, Levi showed that, for every pseudoline arrangement $\mathcal{A}$ and two points in the plane, $\mathcal{A}$ can be extended by a pseudoline which contains the two prescribed points. Later extendability was studied for arrangements of pseudohyperplanes in higher dimensions. While the extendability of an arrangement of proper hyperplanes in $\mathbb{R}^d$ with a hyperplane containing $d$ prescribed points is trivial, Richter-Gebert found an arrangement of pseudoplanes in $\mathbb{R}^3$ which cannot be extended with a pseudoplane containing two particular prescribed points. In this article, we investigate the extendability of signotopes, which are a combinatorial structure encoding a rich subclass of pseudohyperplane arrangements. Our main result is that signotopes of odd rank are extendable in the sense that for two prescribed crossing points we can add an element containing them. Moreover, we conjecture that in all even ranks $r \geq 4$ there exist signotopes which are not extendable for two prescribed points. Our conjecture is supported by examples in ranks 4, 6, 8, 10, and 12 that were found with a SAT based approach.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
