Where can you find intuitive explanation on oriented matroids?

Other perhaps relevant questions on this

  1. How do you get the chirotope of a oriented matroid from the signed circuits? (other than just work backwards)

  2. Also is there any nice characterization on the chirotope of a directed graph or the signed circuit structure of a linear oriented matroid?


My professor motivated me on matroids that they make easier to study things such as certain connectivities. My feeling is that there maybe surprising links between matrix theory, matroids and graph theory. Some elementary and contextual ongoing questions also contain

  1. Are Oriented Graphs Related to Oriented Matroids?

  2. Oriented graph VS directed graph?

  3. On terms "Orientation" & "Oriented" in different mathematical areas?

which are motivated by the sole importance of digraphs in different applications.

My Intuition

Oriented matroids make it easier to study more abstract things such as connectivity. In comparison to graph theory has focus on things such as vertices and edges. Then again Matroid theory by definition has focus on circuits and cocircuits making it easier to study things such as duality theorem between the cut space and cycle space (usually formulated in vector spaces and graph theory). The intuition is that because oriented matroids are by definition focused on paths rather than individial points, they are more liable on questions with connectivity and cuts.

Feel free to comment or challenge the intuition.


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