On the surface geometry of ordered sets.

• Main Author: Hashemi, S. Medhi T.
• Other Authors: Rival, Ivan
• Format: Others
• Published: University of Ottawa (Canada) 2009
• Subjects: Mathematics.
• Online Access: http://hdl.handle.net/10393/9638; http://dx.doi.org/10.20381/ruor-7894

This dissertation has two aims. The first is to study upward drawings of ordered sets on two-dimensional surfaces and secondly to study the geometry of the surfaces on which ordered sets can be drawn without crossing edges. Critical points, in particular, saddle points of ordered sets will play a decisive role. The Discrete Index Theorem, too, is fundamental. We present a characterization, in terms of critical points, of spherical ordered sets--ordered sets which have upward drawings without crossing edges on the sphere. A class of ordered sets, each member of which is called a spiral, plays an important role in the complexity of upward drawings without crossing edges on a specific surface. For a subclass of spirals we characterize those surfaces on which members of this subclass have upward drawings, and we apply spirals, as "gadgets", in the reduction to prove that this decision-making problem, whether an ordered set is spherical, is NP-complete. In the process, we derive a new and simpler proof for NP-completeness of upward planarity testing. From a different point of view, we explore the interrelationship between the upward drawing of ordered sets and the geometry of surfaces. For each smooth, two-dimensional surface S embedded in R$\sp3$, we shall construct an ordered set P to fit S, that is, (i) P has an upward drawing, without crossing edges on S, (ii) P contains the ordered set critical(S) of critical points of S and (iii) if $S\sp\prime$ is any two-dimensional surface of genus g on which P has an upward drawing, without crossing edges, then critical(S) $\subseteq$ critical($S\sp\prime$). A leading unsolved problem is whether an ordered set has a realizable upward drawing on a surface of genus of its covering graph. We consider what seems to be a likely candidate for a counterexample--the upward drawing of the lattice of subspaces of the projective plane of order two.