I such that for all y e Y, y e A f(y). , i e I. A chapter in the next volume will be devoted to these matroids. These matroids can also be thought of as arising from binary relations. 14. Some bases of a matroid defined by the incidence relation of a graph.
Whether or not such characterizations will be useful remains to be seen. 3. CLOSURE AND RANK It is now worthwhile to address our attention to two more concepts, namely, closure and rank. A (matroid) closure operator over the finite set E is an operator cl: 2E -+ 2E satisfying the following: (cll) For every X c E, X c cl(X) (increase). (c12) For every X, Y s E, if X c Y then cl(X) s cl(Y) (monotonicity). Axiom Systems 39 (c13) For every X c E, cl[cl(X)] = cl(X) (idempotence). (c14) For every X c E and for every y, z e E, if y e cl(X u z) - cl(X), (exchange).
A First Course in Graph Theory (Dover Books on Mathematics) by Gary Chartrand, Ping Zhang