Matroids are the underlying structures of many discrete objects (families of hyperplanes, for example). Oriented objects (families of oriented hyperplanes, for example) give rise to oriented matroids. We will define matroids and oriented matroids in detail.

To an (oriented) matroid we can associate several simplizial complexes (order complex, Bergman Komplex). We will look at a triangulation result by Ardila and Klivans (2003) and a generalization of it using nested sets (Feichtner und Sturmfels, 2004) . Using the orientation of a matroid we find positive analogues of these results.