Abstract Algebra/Matroids

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A matroid is an algebraic construct that is related to the notion of independence.

Matroids are an abstraction of several combinatorial objects, among them graphs and matrices. The word matroid was coined by Whitney in 1935 in his landmark paper "On the abstract properties of linear dependence". In defining a matroid Whitney tried to capture the fundamental properties of dependence that are common to graphs and matrices. Almost simultaneously, Birkhoff showed that a matroid can be interpreted as a geometric lattice. Maclane showed that matroids have a geometric representation in terms of points, lines, planes, dimension 3 spaces etc. Often the term combinatorial geometry is used instead of simple matroids. However, combinatorial geometry has another meaning in mathematical literature. Rank 3 combinatorial geometries are frequently called linear spaces. Matroids are a unifying concept in which some problems in graph theory, design theory, coding theory, and combinatorial optimization become simpler to understand.