Regular matroid

In mathematics, a regular matroid is a matroid that can be represented over all fields. A matroid is defined to be a family of subsets of a finite set, satisfying certain axioms. The sets in the family are called "independent sets". One of the ways of constructing a matroid is to select a finite set of vectors in a vector space, and to define a subset of the vectors to be independent in the matroid when it is linearly independent in the vector space. Every family of sets constructed in this way is a matroid, but not every matroid can be constructed in this way, and the vector spaces over different fields lead to different sets of matroids that can be constructed from them. A matroid is regular when, for every field , can be represented by a system of vectors over . If a matroid is regular, so is its dual matroid, and so is every one of its minors. Every direct sum of regular matroids remains regular. Every graphic matroid (and every co-graphic matroid) is regular. Conversely, every regular matroid may be constructed by combining graphic matroids, co-graphic matroids, and a certain ten-element matroid that is neither graphic nor co-graphic, using an operation for combining matroids that generalizes the clique-sum operation on graphs. The number of bases in a regular matroid may be computed as the determinant of an associated matrix, generalizing Kirchhoff's matrix-tree theorem for graphic matroids. The uniform matroid (the four-point line) is not regular: it cannot be realized over the two-element finite field GF(2), so it is not a binary matroid, although it can be realized over all other fields. The matroid of the Fano plane (a rank-three matroid in which seven of the triples of points are dependent) and its dual are also not regular: they can be realized over GF(2), and over all fields of characteristic two, but not over any other fields than those. As showed, these three examples are fundamental to the theory of regular matroids: every non-regular matroid has at least one of these three as a minor.