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

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.