  We introduce the matroid-minor coalgebra C, which has labeled matroids as
distinguished basis, and coproduct given by splitting a matroid into a
submatroid and complementary contraction all possible ways. We introduce two
new bases for C; the first of these is is related to the distinguished basis by
Mobius inversion over the rank-preserving weak order on matroids, the second by
Mobius inversion over the suborder excluding matroids that are irreducible with
respect to the free product operation. We show that the subset of each of these
bases corresponding to the set of irreducible matroids is a basis for the
subspace of primitive elements of C. Projecting C onto the matroid-minor Hopf
algebra H, we obtain bases for the subspace of primitive elements of H.
