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.

翻译：1926年，Levi证明：对于任意伪线排列$\mathcal{A}$以及平面上的两个点，$\mathcal{A}$总可以被一条包含这两个指定点的伪线所扩张。随后，研究者们探索了高维空间中伪超平面排列的可扩张性。虽然$\mathbb{R}^d$中真超平面排列关于包含$d$个指定点的超平面的扩张性是平凡的，但Richter-Gebert发现$\mathbb{R}^3$中存在一种无法用包含两个特定指定点的伪平面进行扩张的伪平面排列。本文研究符号拓扑（一种编码伪超平面排列丰富子类的组合结构）的可扩张性。我们的主要结论是：奇数秩的符号拓扑在如下意义下可扩张——对于两个指定的交叉点，我们可以添加一个包含它们的元素。此外，我们猜想在所有偶数秩$r \geq 4$中，存在对两个指定点不可扩张的符号拓扑。这一猜想得到了基于SAT方法发现的秩4、6、8、10和12中实例的支持。