Orthogonal Surfaces and Their CP-Orders
Orthogonal Surfaces and Their CP-Orders
复制标题
正交曲面及其 CP 阶
作者：
S. Felsner;Sarah Kappes
正交曲面是很好的数学对象，与各种领域有着有趣的联系，例如，整数规划，单项理想和序维数。虽然正交表面在一个或两个维度是相当平凡的三维情况下已经有一个丰富的结构与连接到施奈德森林，平面图和三个多面体。我们的目标是检测更多的正交表面的结构在四维和更高的维度。特别是，我们被驱动的问题，非通用的正交表面有一个多面体结构。我们回顾了三维情况的知识状态。在此基础上，我们引入了高维正交曲面的术语，并继续研究了正交曲面的特征点和cp阶，即，特征点上的优势度。在一般情况下，这些订单（几乎）面格的多面体。例子表明，在一般情况下，cp序可能缺乏面格的关键属性。我们调查额外的要求，这可能有助于有CP-订单，这是面对格。最后，我们把焦点转向正交曲面上多面体的可实现性。有一些准则阻止大类单纯多面体的可实现性。另一方面，我们确定了一些家庭的多面体，可以实现正交曲面。
Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with connections to Schnyder woods, planar graphs and three-polytopes. Our objective is to detect more of the structure of orthogonal surfaces in four and higher dimensions. In particular we are driven by the question which non-generic orthogonal surfaces have a polytopal structure. We review the state of knowledge of the three-dimensional situation. On that basis we introduce terminology for higher dimensional orthogonal surfaces and continue with the study of characteristic points and the cp-orders of orthogonal surfaces, i.e., the dominance orders on the characteristic points. In the generic case these orders are (almost) face lattices of polytopes. Examples show that in general cp-orders can lack key properties of face lattices. We investigate extra requirements which may help to have cp-orders which are face lattices. Finally, we turn the focus and ask for the realizability of polytopes on orthogonal surfaces. There are criteria which prevent large classes of simplicial polytopes from being realizable. On the other hand we identify some families of polytopes which can be realized on orthogonal surfaces.