  A matroid has been one of the most important combinatorial structures since
it was introduced by Whitney as an abstraction of linear independence. As an
important property of a matroid, it can be characterized by several different
(but equivalent) axioms, such as the augmentation, the base exchange, or the
rank axiom.
  A supermatroid is a generalization of a matroid defined on lattices. Here,
the central question is whether a supermatroid can be characterized by several
equivalent axioms similar to a matroid. Barnabei, Nicoletti, and Pezzoli
characterized supermatroids on distributive lattices, and Fujishige, Koshevoy,
and Sano generalized the results for cg-matroids (supermatroids on lower
locally distributive lattices).
  In this study, we focus on modular lattices, which are an important
superclass of distributive lattices, and provide equivalent characterizations
of supermatroids on modular lattices. We characterize supermatroids on modular
lattices using the rank axiom in which the rank function is a directional
DR-submodular function, which is a generalization of a submodular function
introduced by the authors. Using a characterization based on rank functions, we
further prove the strong exchange property of a supermatroid, which has
application in optimization.
  We also reveal the relation between the axioms of a supermatroid on lower
semimodular lattices, which is a common superclass of a lower locally
distributive lattice and a modular lattice.
