Editing Discrete geometry (section)

===Oriented matroids===
{{main|Oriented matroid}}
An '''oriented matroid''' is a [[mathematical structure]] that abstracts the properties of [[directed graph]]s and of arrangements of vectors in a [[vector space]] over an [[ordered field]] (particularly for [[ordered vector space|partially ordered vector space]]s).<ref>[[Rockafellar]] 1969. Björner et alia, Chapters 1-3. Bokowski, Chapter 1. Ziegler, Chapter 7.</ref> In comparison, an ordinary (i.e., non-oriented) [[matroid]] abstracts the [[linear independence|dependence]] properties that are common both to [[Graph (discrete mathematics)|graphs]], which are not necessarily ''directed'', and to arrangements of vectors over [[field (mathematics)|field]]s, which are not necessarily ''ordered''.<ref>Björner et alia, Chapters 1-3. Bokowski, Chapters 1-4.</ref><ref>Because matroids and oriented matroids are abstractions of other mathematical abstractions, nearly all the relevant books are written for mathematical scientists rather than for the general public. For learning about oriented matroids, a good preparation is to study the textbook on [[linear optimization]] by Nering and Tucker, which is infused with oriented-matroid ideas, and then to proceed to Ziegler's lectures on polytopes.</ref>