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The weak-map order on the matroid base polytopes is the partial order definedby inclusion. Lucas proved that the base polytope of no binary matroid includesthe base polytope of a connected matroid. A matroid base polytope is said to bedecomposable when it has a polytopal decomposition which consists of at leasttwo matroid base polytopes.  We shed light on the relation between the decomposability and the weak-maporder of matroid base polytopes. We classify matroids into five types withrespect to the weak-map order and decomposability. We give an example of amatroid in each class. Moreover, we give a counterexample to a conjectureproposed by Lucas, which says that, when one matroid base polytope coversanother matroid base polytope with respect to inclusion, the latter matroidbase polytope should be a facet of the former matroid base polytope.

The Weak-Map Order and Polytopal Decompositions of Matroid Base Polytopes