Lattice structures of fixed points of the lower approximations of two
  types of covering-based rough sets
  Covering is a common type of data structure and covering-based rough set
theory is an efficient tool to process this data. Lattice is an important
algebraic structure and used extensively in investigating some types of
generalized rough sets. In this paper, we propose two family of sets and study
the conditions that these two sets become some lattice structures. These two
sets are consisted by the fixed point of the lower approximations of the first
type and the sixth type of covering-based rough sets, respectively. These two
sets are called the fixed point set of neighborhoods and the fixed point set of
covering, respectively. First, for any covering, the fixed point set of
neighborhoods is a complete and distributive lattice, at the same time, it is
also a double p-algebra. Especially, when the neighborhood forms a partition of
the universe, the fixed point set of neighborhoods is both a boolean lattice
and a double Stone algebra. Second, for any covering, the fixed point set of
covering is a complete lattice.When the covering is unary, the fixed point set
of covering becomes a distributive lattice and a double p-algebra. a
distributive lattice and a double p-algebra when the covering is unary.
Especially, when the reduction of the covering forms a partition of the
universe, the fixed point set of covering is both a boolean lattice and a
double Stone algebra.
