  The weak-map order on the matroid base polytopes is the partial order defined
by inclusion. Lucas proved that the base polytope of no binary matroid includes
the base polytope of a connected matroid. A matroid base polytope is said to be
decomposable when it has a polytopal decomposition which consists of at least
two matroid base polytopes.
  We shed light on the relation between the decomposability and the weak-map
order of matroid base polytopes. We classify matroids into five types with
respect to the weak-map order and decomposability. We give an example of a
matroid in each class. Moreover, we give a counterexample to a conjecture
proposed by Lucas, which says that, when one matroid base polytope covers
another matroid base polytope with respect to inclusion, the latter matroid
base polytope should be a facet of the former matroid base polytope.
