Condition for neighborhoods in covering based rough sets to form a
  partition
  Neighborhood is an important concept in covering based rough sets. That under
what condition neighborhoods form a partition is a meaningful issue induced by
this concept. Many scholars have paid attention to this issue and presented
some necessary and sufficient conditions. However, there exists one common
trait among these conditions, that is they are established on the basis of all
neighborhoods have been obtained. In this paper, we provide a necessary and
sufficient condition directly based on the covering itself. First, we
investigate the influence of that there are reducible elements in the covering
on neighborhoods. Second, we propose the definition of uniform block and obtain
a sufficient condition from it. Third, we propose the definitions of repeat
degree and excluded number. By means of the two concepts, we obtain a necessary
and sufficient condition for neighborhoods to form a partition. In a word, we
have gained a deeper and more direct understanding of the essence over that
neighborhoods form a partition.
