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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.