An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields.[1] 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.[2] [3]
All oriented matroids have an underlying matroid. Thus, results on ordinary matroids can be applied to oriented matroids. However, the converse is false; some matroids cannot become an oriented matroid by orienting an underlying structure (e.g., circuits or independent sets).[4] The distinction between matroids and oriented matroids is discussed further below.
Matroids are often useful in areas such as dimension theory and algorithms. Because of an oriented matroid's inclusion of additional details about the oriented nature of a structure, its usefulness extends further into several areas including geometry and optimization.
- ^ R. Tyrrell Rockafellar 1969. Anders Björner et alia, Chapters 1-3. Jürgen Bokowski, Chapter 1. Günter M. Ziegler, Chapter 7.
- ^ Björner et alia, Chapters 1-3. Bokowski, Chapters 1-4.
- ^ 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.
- ^ Björner et alia, Chapter 7.9.