  We introduce and study filtrations of a matroid on a linearly ordered ground
set, which are particular sequences of nested sets. A given basis can be
decomposed into a uniquely defined sequence of bases of minors, such that these
bases have an internal/external activity equal to 1/0 or 0/1 (in the sense of
Tutte polynomial activities). This decomposition, which we call the active
filtration/partition of the basis, refines the known partition of the ground
set into internal and external elements with respect to a given basis. It can
be built by a certain closure operator, which we call the active closure. It
relies only on the fundamental bipartite graph of the basis and can be
expressed also as a decomposition of general bipartite graphs on a linearly
ordered set of vertices.
  From this, first, structurally, we obtain that the set of all bases can be
canonically partitioned and decomposed in terms of such bases of minors induced
by filtrations. Second, enumeratively, we derive an expression of the Tutte
polynomial of a matroid in terms of beta invariants of minors. This expression
refines at the same time the classical expressions in terms of basis activities
and orientation activities (if the matroid is oriented), and the well-known
convolution formula for the Tutte polynomial. Third, in a companion paper of
the same series (No. 2.b), we use this decomposition of matroid bases, along
with a similar decomposition of oriented matroids, and along with a bijection
in the 1/0 activity case from a previous paper (No. 1), to define the canonical
active bijection between orientations/signatures/reorientations and spanning
trees/simplices/bases of a graph/real hyperplane arrangement/oriented matroid,
as well as various related bijections.
