Lesson plan
Oriented matroid
. A network with the value of flow equal to the capacity of an s-t cut 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. 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. 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). 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.
Article summary
. A network with the value of flow equal to the capacity of an s-t cut 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. 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. 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). 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.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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