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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).〔Rockafellar 1969. Björner et alia, Chapters 1-3. Bokowski, Chapter 1. Ziegler, Chapter 7.〕 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''.〔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.〕 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).〔Björner et alia, Chapter 7.9.〕 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. ==Background== In order to abstract the concept of orientation on the edges of a graph to sets, one needs the ability to assign "direction" to the elements of a set. The way this achieved is with the following definition of ''signed sets''. * A ''signed set'', , combines a set of objects with a partition of that set into two subsets: and . : The members of are called the ''positive elements''; members of are the ''negative elements''. * The set is called the ''support'' of . * The ''empty signed set'', is defined in the obvious way. * The signed set is the ''opposite'' of , i.e., , if and only if and Given an element of the support , we will write for a positive element and for a negative element. In this way, a signed set is just adding negative signs to distinguished elements. This will make sense as a "direction" only when we consider orientations of larger structures. Then the sign of each element will encode its direction relative to this orientation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「oriented matroid」の詳細全文を読む スポンサード リンク
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