Oriented matroid
![](/Images/godic/202501/31/Max-flow_min-cut_example.svg2220.png")
![This is an example of a pseudoline arrangement that is distinct from any line arrangement.](/Images/godic/202501/31/Pappos_pseudo.svg2220.png")
![A zonotope, which is a Minkowski sum of line segments, is a fundamental model for oriented matroids. The sixteen dark red points (on the right) form the Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus signs (+): The right plus sign is the sum of the left plus signs.](/Images/godic/202501/31/Shapley–Folkman_lemma.svg2220.png")
![In convex geometry, the simplex algorithm for linear programming is interpreted as tracing a path along the vertices of a convex polyhedron. Oriented matroid theory studies the combinatorial invariants that are revealed in the sign patterns of the matrices that appear as pivoting algorithms exchange bases.](/Images/godic/202501/31/Simplex_description2220.png")
An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space over an ordered field (particularly for partially ordered vector spaces). In comparison, an ordinary (i.e., non-oriented) matroid abstracts the dependence properties that are common both to graphs, which are not necessarily directed, and to arrangements of vectors over fields, which are not necessarily ordered.