Closed-set lattice of regular sets based on a serial and transitive
  relation through matroids
  Rough sets are efficient for data pre-processing in data mining. Matroids are
based on linear algebra and graph theory, and have a variety of applications in
many fields. Both rough sets and matroids are closely related to lattices. For
a serial and transitive relation on a universe, the collection of all the
regular sets of the generalized rough set is a lattice. In this paper, we use
the lattice to construct a matroid and then study relationships between the
lattice and the closed-set lattice of the matroid. First, the collection of all
the regular sets based on a serial and transitive relation is proved to be a
semimodular lattice. Then, a matroid is constructed through the height function
of the semimodular lattice. Finally, we propose an approach to obtain all the
closed sets of the matroid from the semimodular lattice. Borrowing from
matroids, results show that lattice theory provides an interesting view to
investigate rough sets.
